/ CJ "'/? ...z..z~/l) AJo, ~ IVll ENGINEERING STUDIES

STRUCTURAL RESEARCH SERIES NO. 210

SE Ke+z Reference Room

"Ci -IiI Engineerir.g Department BIOS C. E. Building University of Illinois Urbana,: Illinois 61801

By R. T. EPPINK

and

A. S. VELETSOS

Report to

RESEARCH DIRECTORATE

AIR FORCE SPECIAL WEAPONS CENTER

Air Research and Development Command

Kirtland Air Force Base, New Mexico

CONTRACT AF 29(601)-2591

PROJECT 1080

UNIVERSITY OF ILLINOIS

URBANA, ILLINOIS

DECEMBER 1960

'::,'C

AFSWC-TR-60 ... .s3

RESPONSE OF ARCHES. UNDER DYNAMIC 'LOADS

by

R. T. Eppink

A. S. Veletsos

University of Illinois p~partment of Civil Engineering

August 1960

Research Directorate AIR FORCE SPECIAL WEAPONS C.ENTER Air Res earch and Development. Command Kirtland Air Force Base, New Mexico

Approved:

a/~ P~ L. HUIE ~:tr2~l USAF '

Project No. 1080 Chief, Structures Division

Contract No. AF 29(601)-2591

L--J.aa~. t7f:EONARD A. ED Y Colonel USA Di r e c to r , Res e a rc h Di r e c tor at e

TR-6Q-- 53

ABSTRACT

This report is concerned with a 'study of the response of arches sub-jected to the influence of transient forces. 0

A numerical procedure is presented for the computation of the dynamic respons e of-arches for both the elastic and the inelastic ranges of d'eforma­

°tion. The procedure is applicable to arches having any shape and any distri­bution of mass and stiffness. The distribution of the pressure along the arch and its timewise variation may be arbitrary.

The analysis is simplified by replacing the actual arch which has an infinite numbe~ of degrees of freedom. by a discrete framework consisting of a series of rigid bars, flexible joints, and concentrated point masses. For the computation of the response iri the inelasti~ range, the cross­sectional area of the arch is considered to consist 'of two concentrated flange areas connected, by a thi.n rigid web. The equations of :motion of the replace­ment system are solved by use of a. step-by- step method of numerical inte-gration. .

Computer programs are described for the analysis of two general classes of problems: (a) circular elastic arches subjected to a uniform nor­mal pressure of arbitrary timewise variation; and (b) arches of arbitrary . shape subjected to a triangular moving pressure pulse. For the latter case it ,is possible to evaluate the response in' the inelastic range of behavior.

Numerical solutions are presented for a wide range of the parameters involv.ed, and the effects of the various parameters are discussed.

PUBLICA TION REVIEW

This report has been reviewed and is approved.

EY .. IBRYAN, JR. Colonel USAF Deputy Commander

iii

TABLE OF CONTENTS

I. INTRODUCTION. . • • •

1.1 Object and Scope

1.2 Notation

1.3 Acknowledgment

II . METHOD OF ANALYSIS . .

2.1 Characteristics of Replacement Str~cture

2.2 Designation of Joints and Bars

2.3 Equations of Motion in Terms of Forces 2 . 3.1 General . . . . . . . . . • . . 2.3.2 Components of Resisting Forces. 2.3.3 Components of External Forces

2.4 Displacement-Deformation Relationships

2.5 Deformation-Force Relationships •. 2.5.1 Elastic Range of Behavior 2.5.2 Inelastic Range of Behavior

2.6 Method of Numerical Integration .

1110 COMPUTER PROGRAMS

3.1 General • . . . . .

3.2

3.3

Arch Subjected to a Uniform All-Around Pressure . . 3.2.1 Description of Programs . . •. 3.2.2 Outline of Programs •.•..•...•.

Arch Subjected to a Triangular Moving Pressure . 0 • • • •

3.3.1 Description of Programs 3~3.2 Outline of Programs ..•. 0 0 ••• 0 •

TI. NUMERICAL SOLUTIONS FOR ARCH SUBJECTED TO A UNIFORM PRESSURE PULSE . . • 0 • •

4.1 Problem Parameters

4.2 Presentation pf Results . 4.2 ... 1 General . • . . • 0 • • • • • • • • •

4.2.2 Typical Response Characteristics • 4.2.3 Effect of Length of Time Interval of Integration

and Number of Bars • • . . • . . • 0 • • • • • •

1

1

3

5

7

7

7

8 8

10 10

11

12 12 13

16

19

19

19 19 21

24 24 26

29

29

30 30 30

32

iv

TABLE OF CONTENTS (Continued)

40204 Effect of Duration of Loading " 0 0 0 0

40205 Effect of Parameters 'Influencing Tendency "for Buckling " " " 0 0 " 0 " " "

40206 Effect of Arch Dimens'ions " 0 " 0 0 0 0 " 0 0 0

v " NUMERICAL SOLUTIONS FOR ARCH SUBJECTED TO A TRIANGULAR MOVING

35

37 38

PRESSURE 0 " " 0 0 0 0 " " 0 0 " " 0 " 41

501 Problem Parameters 0

502 Presentation of Results 50201 Typical Response Characteristics . 0 0 c

50202 Effects of Characteristics of Pressure Pulse 5 0 2,,3 Effect of Arch Dimensions 0

50204 Effect of Inelastic Action

SUMMARY

601 Summary 60101 General 0 0 0 " 0

60102 Solutions for Arches Subjected to Around Pressure 0 0 0 0 0 • 0 " 0

60103 Solutions for Arches Subjected to Moving Pressure 0

REFERENCES

a Uniform All-

a Triangular

APPENDIX A "EXACT tI DEFORMATION-FORCE RELATIONSHIPS FOR INELASTIC

41

41 41 43 46 47

50

50 50

50

51

53

BEHAVIOR 0 0 0 0 0 0 0 0 0 0 0 0 '0 0 0 0 0 0 0 0 0 0 54

APPENDIX B EQUATIONS FOR CIRCULAR .ARCHES IN POLAR COORDINATES 57 Bol Equations of Motion 0 0 " " 0 0 " 0 " " " 0 e 0 e 0 57 Bo2 Displacement-Deformation Relationships 58

APPENDIX C MODAL METHOD OF .ANALYSIS " 0 0 " 0 0 0 • 0 " 59

v

LIST OF TABLES

Table No.

4.1 Natural Periods of a Two-Hinged Uniform Circtilar Arch . 61

4.2 Modal Solution for a Two Hinged Circular Arch . . . . 62

4.3 Maximum Values of Response and Times of Occurrence for a Two-Hinged Uniform Circular Arch . . . . . . . . .. 63

4.4 Static Effects in a Two-Hinged Uniform Circular Arch for a Uniform All-Around Pressure . . . . . . . . . . . . . .• 63

4.5 Effect of Time Int~rval of Integration on Maximum Values of Response and Times of Occurrence ~ . . . . -. . . . . .. 64

4.6 Comparison of Solutions by Numerical Integration Procedure and Modal Method . . . . . . • . . . . 66

4.7 Eff~ct of Load Duration on Maximum Values of Response and Times of Occurrence . . . 68

4.8 Comparison of Approximate and Exact Maximum Stresses 70

4.9 Effects of Peak. Pressure and Ittitial Out-of-Roundness on MaximUm Values of Response 71

4.10 Effect of Arch Dimensions on Maximum Values of Response and Times of Occurrence . 72

C.l Symmetrical Modes of Vibration for a Two-Hinged Uniform Circular Arch and Participation Factors for Modal Solution. 74

vi

LIST OF FIGURES'

Figure No ..

2.1 Model Considered in Analysis . " . e

2.2 Free Body Diagrams for a Typical Bar and Joint 0 •

2.3 Diagram Used in Determining Displacement-Deformation Rela tionships . . " . . . . " ~ • 0 0 • • '

2" 4 Bilinear Str~ss.-Deformation Relationship "

2.5 Deformed Joint " . . . . . . . 0 " • • • •

3.1 Coordinate: System for Circular Arch Model

Page

75

76

76

77

77

78

3.2 Triangular":Shaped Moving Pressure Pulse 78

3.3 Flow Diagram of Computer Program for Circular Arch Subjected to a Uniform All-Around PressUre ..•. 0 • 0 • G, 0 • " •• 0 79

3.4 Flow Diagram of Computer Program for Arch Subjected to a Triangular Moving Pressure . • " . . • . . . . 80

4.1 Triangular Pressure Pulse with Initial Peak 81

4.2 Natural Modes of Vibration for a Two-ainged Uniform Circular Arch . . . . . . . . . ." . 0 • 0 0 0 • • • • • • • 0 •• 82

4.3 Typical Response Curves for an Arch Under a Uniform All-Around Pressure Pulse . 0 0 • • • • • •

4.4 Response Curves of Bending Moment in an Arch for a Uniform

84

All-Around Pressure Pulse ." 0 •• ••••• 86

4.5 Effect of NUmber of Bars on Response 88

4.6 Response Curves for an Arch Under'a Uniform All-Around Pressur~ Pulse--td/To = 0025 ..•...... 0 0 , •••• ". 90

4.7 Response Curves for an Arch Under a Uniform All-Around Pressure pulse--td/To, = ?50 0 " • • • • • • •••• 0 • " •• 92

4.8 Response Curves for an Arch Under a Uniform All~Around Pres~u:re Pul~e~-td/To = 0.75 . . . . . . 0 • • • 0 • • 94

4.9' Response Curves' for an Arch Under'a'Uniform All:Arolind Pressure Pulse--td/To = 1.0 ' .......... 0 ••••• 0 96

vii

LIST OF FIGURES (Continued)

4.10 Response Curves for an Arch Under a Uniform All-Around Pressure Pulse--td/To = 1.5 ...... C • 0 •••••

Response Curves for an Arch Under a Uniform All-Ar,ound Pressure Pulse--td/To = 4.0 ...... 0 •••• ' ••• 100

4.li

4.12 Response Spectra for an Arch Under a Uniform All-Around \

Pre s'sure Pulse . . . . . . . . . • 102

4.13 Effect of Peak Pressure on Response for a Vniform All-Around Pressure Pulse 104

4.14 Effect of Initial out-of-Roundness on Response for a Uniform All-Around Pressure Pulse . 0 • • 0 • • 0 • • • • • • • • • ~. 106

4.15 Respons'e Curves for an Arch Under a Uniform All-Around Pressure Pulse--L /t = 50 . . 0 . . · . · 0 . · . . . . .. · 108

0

4.16 ~esponse Curves for anArch U~der a Uniform All'-Around Pressure Pulse--L /r = 200 · . · ,. · 0 . . . · .. . liO

0'

4.17 Respons'e Curves for an Arch Under a Uniform All-Around' Pressure Pulse - -f /L = 0.1 · · . · 0 .. . . · . 112

0

4.18 Response Curves for an Arch Under a Uniform All-Around Pressure Pulse--f/L = 0.5 ...... 0 •••••••

0, 114

5.1 Typical Response Curves for an Arch Su1:;>jec,ted to a Triangular Moving Pressure . . . 0 ••••••• 0 • • • .' • • " 0 o. ll6

5.2 Response Curves of Bending Moment ,in an Arch for a Triangular Moving Pressure .............•. 0 • • • • •• 118

5.3 Response Curves for an Arch Subjected to a Triangular Moving pressure--td/tt = 0.5. . . 0 • • • 0 0 0 • 0

Response Curves for an Arch Subjected to a Triangular Moving pressure-.:..td/tt = -2.0 ... 0 0 0 •••••••

Response Spectra for an Arch Subjected to a Triangular Moving Pressure $.... ~ . ~ . . . . . . . . .

Response Curves for an Arch Subjected to a Triangular .Moving pressure--po/pcr = 0.001 . ~, 0 ••• 0 0 •••

Response Curves for an Arch Subjected to a Tri~gular Moving Pressure--p /p = 2.0 . 0 •• 0 • • • ~ 0 •• o cr

121

l23

l26

127

l30

Viii

LIST OF FIGURES (Continued)

Response Curves for an Arch Subjected to ~ Triangular Moving pressure--tt/To = 2o~ 0 0 • • • • <I • • • • 0 •

5.9 Response Curves for an Arch Subjected to a Triangular Moving Pressure-tt/To = 005' '. ~ 0 • 0 0 0 ••• 0 ••

5.10 Response Curves for an Arch Subjected to a Triangular Moving Pressure--L Ir = 50 0 • • .0 • .. • • • 0 • • •

o

5.12 Response Curves 'for an Arch Subjected to a Triangular Moving Pressure--L /r = 200 <I • 0 0 0 • 0 • • • • • •

o

5.12 Response Curves for an Arch Subjected to a Triangular Moving Pressure~-f/L = 001 . . 0 • 0 0 • • 0 0 • • • •

. 0

5.13 Response Curves for 'an Arch Subjected to a Triangular Moving Presshre--f/L = 005 • 0 ••••••••••••

, 0

5.14 Curves for Inelastic Response of an Arch Subjected to a Triangular Moving Pressure--€ = 100 P R/AE .•.. 0 • y 0

5015 Curves for Inelastic Response of anArch Subjected to a Triangular Moving Pre s sure - -€ ~ o. 5 p RI AE 0 • 0 • • • y 0

132

135

137

139

· · 0 · 141

· · · · 143

· 0 · 0 145

· · · · 147

RESPONSE OF ARCHES UNDER DYNAMIC LOADS

I" INTRODUCTION

1.1 Object and Scope

The objectives of this study ~e: (a) to develop a method for the

cOIDF~tation of the response of arches deflecting in their own plane under the

influence of transient forces, and (b) 'With the aid of this method, to obtain

numerical solutions for a range of the parameters involved in order to gain

insight into the dynamic behavior of this structural type.. The ultimate purpose

of this investigation is the development of a rational method of design for

arches subjected to dynamic loads of the type ariSing from bomb blasts.

Although of an exploratory nature, the numerical data presented in

this report provide information on the effects of the major variables that

enter into the problem. Furthermore, they may be used to evaluate the adequacy

of existing methods of' design for arches under dynamic loads~ [41[61*

The method is a:pJ?licable to arches having any shape and any distri­

bution of' ma,ss and, stiffness.. The distribution of the forces along the arch

and their timewise' variation may be arbitrary.. The response of the structure

for both the elastic and the inelastic ranges of deformation may be investigated 0

The problem is analyzed approximately by replacing the continuous

arch~ whi ch has an infinite number of degrees of freedom, by a discrete frame­

'WOrk consisting of' a series of rigid bars and flexible joints.. The actual

distributed mass of the structure is lumped into a series of point masses at

the joints. For the computation of the response in the inelastic range, the

cross-sectional. area of the arch is approximated by two flanges connected by

a thin rigid web 0 The resistance of each flange is represented by a bilinear

stress~strain ni agram.. The equations of motion for the analogous framework

are solved by use of a step-by-step method of numerical integration.

Mathematically, the replacement of the original continuous system by

a discrete system is analogous to the solution of' the governing differential

equation by means of difference equations.. The major advantage of a physical

* Unless otherwise identifiedJ numbers in brackets refer to the corresponding items in the list of References given at the end of the text.

analogue is that it attaches physical significance to the various assumptions

and approximations introduced in the analysis c· This is particularly true for

the inelastic range of deformation for which the problem becomes considerably

more involved. In several respects, the present analysis is similar to one

reported previously by Jo Ao Brooks [1]0

The method has been programmed for the ILLIAC J the high-speed elec­

tronic digital computer of the University of Illinoiso Programs have been

prepared for two general classes of problems~ (a) circular elastic arches

subjected to a uniform all-around pressure having any timewise variationJ and

(b) arches of arbitrary shape subjected to a triangular moving pressure pulseo

The latter program can be used to evaluate the response in both the elastic

and post-elastic ranges of behavioro ~nere are no restrictions as to the

distribution of mass or stiffness along the arch~ and the arch supports may

be either hinged or fixedo For a moving pressure~ it is also possible to

consider the effects of partial fixity at the supports 0 In addition, a special­

ized program has been prepared which accounts for the effect of an initial out­

of-roundnesso This program is restricted to uniform circular arches subjected

to a uniform pressureD

The computer programs have been used to obtain numerical solutions

for the response of two-hinged circular archeso Results were obtained (a) for

a uniforffi all-around pressure with a timewise variation represented by a triangle

with an initial peak, and (b) for a triangular moving pressure pulse 0 While

most of the results are for the elastic range of behavior, a few solutions have

also been obtained to investigate the response into the post-elastic range of

deformation. 'Ib.th~se soltLtions r ·~the·:~ cr.osSf!·sectd.ona.l : a.r~a. : andth.~ maSi s ,

per unit of length of the arch were considered to be constant 0 Although limited

in number~ the numerical solutions cover a wide range of load and arch para­

meters .. The load parameters include the duration of the pressure 'pulse, the

magnitude of the peak pressure as compared to the critical buckling pressure,

andJ for the moving pressure solutions, the velocity of propagation of the

pressure front 0 The arch parameters include the geometric and physical pro­

perties of the archa The effect of ~~ initial out-of-roundness was investigated

by considering a pressure pulse uniformly distributed around the arch 0

3

The characteristics of the analogous framework used in the analysis

are described in Chapter IIo Also included in this chapter are the derivations

of the e~uations necessary for dynamic response calculations and an outline of

the procedure for their solution for both the elastic and inelastic ranges of

behavior. The capabilities and the general organization of the computer pro­

grams are described in Chapter III. In Chapters IV and V, the numerical results

are presented and the effects of the various variables are discussed briefly.

Chapter IV is devoted to the uniform all-around pressure solutions, and Chapter

V to the triangular moving pressure solutions. A summary of the more important

results is ,given. in Chapter VI.

1.2 Notation

The letter symbols used in this report are defined where they are

first introduced in the text. The most important of these are summarized below

in alphabetical order.

= area of the cross-section at joint j A. J

t Ab Aj' J

= areas of the top and bottom flanges at joint j

b = instaneous set

t 1) c .j c.

J J = distance from the centroidal axis of the cross-section at

joint j to the centers of the top and bottom flanges; respectively.

d. J

= distance between the centers of flanges at joint j

E. J

= modulus of elasticity of material at joint j

f = rise of arch

F. J

= concentrated external force acting at joint j

I. = moment of inertia of the cross-section of the arch at joint j , <.J

L 0

= span length of arch

L. J

= length of bar j

m. J

= mass at joint j

M. J

= moment at center of joint j

1 M~ Mj' J = moments immediately to the left and right of joint j

p

4

= axial force in bar j

axial forces in the top and bottom flanges to the left and

right of joint j

= intensity of external pressure

= peak value of external pressure

= [4~2 _ IJE~ = critical buckling pressure for a hinged arch subjected \Po ,R

to a uniform static press'Ureo T;he corresponding deflec-

tion configuration is antisymmetrical.

= components of F. immediately to the left and right of joint j and J

normal to the respective bars

Qj

= transverse shearing force in bar j

r = radius of gyration of cross-section

R = radius of circular arch

td = duration of pressure pulse

tt = time of transit of pressure front across the arch

T = fundamental period of vibration of complete ring, the breathing period o

v. = tangential component of displacement of joint j J

V. = x-component of internal forces at joint j J

w. = radial component of displacement at joint j J

w = amplitude of initial out-of-roundness m

w . = y-component of internal forces at joint j J

x. = x-coordinate of joint j in undeformed position J

X. = x-component of external forces at joint j J

Yj

= y-coordinate of joint in undeformed position

Y. = y-component of external forces at joint j J

z

a. J

t3 j

5 . J

£ or 5 j' J

Ol.t, o~t J J

f/b , O~b J J

11 • J

e . J

cr. J

£t rt cr. , cr.

J J

£b rb

5

= number of bars in arch model

= angle between the y-axis and the line of action of F. when first J

applied, positive when clockwise

= direction angle of bar j with positive x-axis

= total change in distance between joints j-l and j

= portions of the total deformation of joint j to the left and right

of the joint

= deformations of the top and bottom flanges to the left and right

of joint j, respectively

= component of displacement of joint j in y-direction

= angle between the line of action of F. when first applied to the J

arch and its line of action after deformation

= mass per unit of length of the arch

= component of displacement of joint j in x-direction

= stress at joint j

= stresses in the top and bottom flanges to the left and right of

OJ ,OJ joint j, respectively

cp 0 = total angle of opening of circular arch

X. = angle change at joint j J

xi.' J Y! = angle changes to the left and right of joint j J J

1jr • = rotation of bar j J

1.3 Acknowledgment

This study was performed as part of an investigation conducted in

the Civil Engineering Department of the University of Illinois under contracts

6

AF 29(601)-464 and AF 29(601)-2591 with the Air Force Special Weapons Center,

Albuquerque, New Mexicoo This report is based on Dro EppinkTs doctoral dis­

sertationo It was prepared under the general direction of Dro No Mo, Newmark,

Head of the Department of Civil Engineering, and the immediate supervision of

Dro Ao So Veletsos, Professor of Civil Engineeringo Appreciation is expressed

to the Air Force for the support that has made this study possible.

7

II. METHOD OF ANALYSIS

201 Characteristics of Replacement Structure

A schematic drawing o~ the substitute framework is shown in Fig. 2.1.

It consists of a number o~ flexible joints connected by rigid bars, which, for

convenience, are assumed to be straight. Inasmuch as the ~lexibility of the

structure is assumed to be concentrated at the jOints, all axial and angular

deformations take place at the joints. The individual bars are considered to

be massless, and the actual distributed mass of the structure is lumped into

a series of point masses at the joints. The characteristics at a joint of the

substitute framework are determined from the characteristics of the section of

the true arch included between the midpoints of the adjoining segments. For

a non-unifor.m arch the flexibilities of the joints and the magnitudes of the

concentrated masses will vary from one joint to the next.

This system is in some respects similar to a physical system used

by Clough [2] in an experimental investigation of buckling of arches.

In considering the behavior of the structure in the inelastic range, I

it is further assumed that the cross-sectional area of the arch is made up of

two flanges connected by a thin web which is rigid in shear. It is assumed

that the web resists no direct forces. A cross~section of this type is an

idealization of a steel I-beam or wide-flange beam. It also approximates a

section of a cylindrical shell consisting of an I-beam with a concrete cover.

The resistance of each of the flanges is given by a bilinear stress-deformation

diagram. The areas of the top and bottom flanges may be different at anyone

jOint, and the total areas of the sections may vary from one joint to the next.

The physical properties of the top and bottom flanges may differ, but a varia­

tion in these properties from joint to joint is not considered in the equations

presented here.

202 Designation of Joints and Bars

The joints of the analogous framework are numbered consecutively

starting with zero at the left end and terminating with z at the right end.

The bars are numbered in the same order from 1 to z. Bar j is the bar connect­

ing joints j-l and j.

8

As shown in Figo 201, the configuration of the undeformed arch is

specified by the rectangular coordinates, x. and Y.} of the joints. The con-J J

figuration of the deformed structure is defined by the displacements of the

jOints, measured from the undeformed position of the structure. The components

of displacement of joint j in the positive x and y directions are designated

by ~j and ~j~ respectively.

2.3 Equations of Mation in Terms of Forces

20301 General. The forces acting at the ends of a representative

bar and joint are shown in ,Fig. 2.2. They include the axial thrust, N, the

transverse shearing force, Q, and the bending moment, Mo An axial thrust is

positive when it produces tension. A shearing force is positive when it tends

to rotate an element of the arch in a clockwise direction 0 A bending moment

is positive when it produces compression in the exterior fiber of the arch.

Thecconcentrated external force at joint j is denoted by F., and the angle J

between its line of action and the y axis, measured in a clockwise direction,

is denoted by a.o The x and y components of the external force are considered J

to be positive when they act in the positive x and y directions. In Fig. 202

all forces are shown in their positive directions.

Referring to Fig. 2.2a and considering the equilibrium of bar j,

one finds that both N. and Q are constant aJ..ong the length of the bar and J j

that Q. is related to the moments acting at the ends by the equation, J

£ r M. "" M. 1 J J~

L. J

where L. is the length of bar j. The superscripts 1 and r designate, respec­J

ti vely, values of the moment to the left and to the right of ~be~~oint.. ~.The

total change in the distance between joints j-l and j is denoted by 0 .. Also, i. r J

5. and 5.1

designate the components of the total deformation occurring at J J-

joints j and j-l, respectively. Thus:

r ~£. 5.=5· 1 +u J J- J

(2.2)

The quantities 5., 5~, and 5~ are measured aJ..ong a reference axis which passes J J J

through the centroid of the cross-sectional area of the analogous framework.

9

If the cross-section consists of more than one material, the centroid is

determined from the transformed cross-section.

Consideration is next given to the e~uilibrium of moments at the

joint about an axis normal to the 'plane of the framework. Referring to Fig.

202b and defining M. to be the moment at the center of joint. j, one finds J

that M~ and M~ are given by the e~uations, J J

The last term in each of these e~uations represents the moment produced by

the shears as a result of the deformations at the joint. These terms are

likely to be important only for large axial deformations. Substituting these

expressions into E~. 201 and making use of E~o 202, one obtains,

M. - M. 1 Q = J .J-

j L. + 5. J J

From the preceding discussion it follows that the axial thrusts and shearing

forces are essentially defined at the centers of the bars, whereas the bending

moments are defined at the centers of the joints.

The e~uations of motion for mass m. in the x and y directions are J

m.~.=V.+X. J J J J

m.ij.=W.+Y. J J J J

where Vo represents the component of all internal forces acting at joint j in J

the positive x direction, and X. represents the corresponding component of J

the external forces. Similarly, W. designates the component of all internal J

forces acting at joint j in the positive y direction, and Y. designates the J

corresponding component of the external forces. A dot superscript denotes

differentiation with respect to time. Thus, ~j and ~j denote the accelerations

in the x and y directions of the jth mass.

10

20302 Components of Resisting Forceso In the following derivations,

the effects of the deformation of the structure are taken into accounto Refer­

ring to Figo 2020 and summing the components of all internal forces in the x

direction, one obtains the following expression for V.: J

Vj = -N.cos(~. - *.) +N. lCos(~. 1 - ~J+' 1) J J J J+ J+

-Q.sin(~. - *.) + Q. Isin(~. - - *J'+l) J J J J+ J+.L (2.6)

where ~, with the appropriate subscript, denotes the rotation of a bar and

is positive in the clockwise direction 0 The direction angle of bar j with the

positive x-axis for the undeformed arch is denoted by ~.o J

In a similar manner, summation of the components of all internal

forces in the y-direction leads to the following e~uation for We: J

*j+l)

It may be recalled that the shearing forces Q. and Q. 1 in the above e~uations J J+

are related to the bending moments M. l' M.J and M. 1 by EClo 2030 J- J J+

2.303 Components of External Forceso In order to express X. and J

Y. in J

termS of the total force F it is necessary to kllOW or assume the line j'

of action of F j

as the structure deformso For example, if in the process of

deformation, F. J

remains parallel to its original direction, then,

X. = F .sin o. J J J

(208)

Y. = F ,cos o. J J J

To obtain general expressions, let e. represent the angle between the line J

of action of F. when first applied to the structure and its line of action J

after deformation, as shown in Figo 2 .. 2bo Then,

x = F. sine a. + e.) j J J J

(2010)

Y. = F.cos(o. + e.) J . J J J

11

An important practical case is one in which the structure is sub­

jected to a pressure which remains normal to the surface. In this case, it

is convenient to express the concentrated force F. in terms of a component

p~ which is normal to the bar j immediately to th~ left of joint j, and a J r

component P. which is normal to the bar j+l immediately to the right of the J

joint. These components, considered to be positive in the outward direction,

can readily be evaluated in terms of the normal pressure 0 Then,

X.= -P~ sin(t3. - 'If.) - p~ sin(t3. 1 - Ijrj+l) J J J J J J+

(2.12)

£ - Ijr.) r cos (t3. 1 - Ijrj+l) Y. = P. cos(t3. + P.

J J J J J J+ .... (2.13)

204 Displacement-Deformation Relationships

In Fig. 2.3, ab and aib' represent, respectively, the positions of

bar j and joints j-l and j before and afte·r deformation of the arch. Let the

projection atc l of the deformed system on the undeformed system be designated

by L. + 0.; then· 5. can be expressed in terms of the displacements of joint J J J

j-l and j as follows:

6. = (~. - ;. l)cOS t3. + (~. - ~. l)sin t3. J J J- J J J- J

(2.l4a)

Similarly, let the projection cib 1 of the deformed system in a direction normal

to the undeformed system be denoted by L .l.; then, J J

~. = Ll . [(~. - ;. 1) sin t3. + (~. - ~. l)cOS t3.J J j J J- J J J- J

(2ol5a)

In terms of O. the length aib! can be expressed as, J

whence

-2 ( - / )2 L.+o.=L. 'If. + l+o.L. J J J J J J

2 ( - / ) 2 J ".. - 1 ..... ·2 'If. + 1 + O. L. = o. + -2 L. V~. 'I>

J J J J J J (2 .14b)

The rotation of the bar, Ijr., can likewise be expressed in terms of J

O. and W. as follows: J J

-1 *. Ijr. = tan _~.J,---_ J 1 + 5./L.

J J

12

It is noted that the quantities 5. and w. represent the first order approxima-J J

tions to the change in distance 5. and to the rotation W.o J J

and that

sion~

Refe~ing_to·F1,g~ -2.),.,one also. finds that

cos t3j

= x. - x

.1 ,i-l L.

J

(2 .. 16)

sin YJ• - Y i-I

t3.= h s/

J L. J

cos (t3. - Ijr.) J J

The angle change at joint jJ denoted by X., is given by the expres­J

x = ,Ir ,I, j 'I' j+l - 'I' j

205 Deformation-Force Relationships

The equilibrium equations and the geometrical relationships developed

in the preceding sections are valid irrespective of the properties of the

material composing the structureD However, in the formulation of the deforma­

tion-force relationships, a distinction must be made between the expressions

applicable to the elastic and inelastic ranges of behavior.

2.501 Elastic Range of Behavior. The deformations to the left and

to the right of joint j are given by the following equations:

L N 5~ = .1+1 .i+l

J 2 E.A. J J

13

The symbols E. and A. represent the modulus of elasticity of the material and J J

the cross-sectional area at joint jo If the cross-section consists of more

than one material, the values of these quantities are those of the transformed

cross-sectiono Combining Eqs~ 2019 by use of Eqo 202 and solving for N., one J

obtains the expression,

2 EQA • . J J E.A.

1 + J J E j_1Aj _l

For a uniform arch the coefficient 2 cancels with the terms in the denominator.

The bending moment, M.~ may be evaluated from the equation, J

M.= J L. + L. 1 Xj

J J+

:2 E .1. sl J

where I. represents the moment of inertia of the cross-section at joint j. J

The bending moment produced by a uniform extension of the arch axis is not

taken into account in Eq$ 2021.

20502 Inelastic Range of Behavior? As already discussed in Section

201, the analysis for the inelastic range of behavior is restricted to cross­

sections which can be approximated by a thin" rigi.d.-web.~~ ::t'Wd~:,fia.nge~s.

The relationship between the stress and the deformation for each flange is

considered to be represented by the bilinear diagram shown in Fig. 2.4. The

curve is shown in dimensionless form such that the slope of the elastic

resistance line is unity and yielding occurs for % equal to one. The slope y -

of the second line which represents the inelastic resistance is E/E, the ratio

of the stiffness of the material in the post-elastic range to that in the

elastic range. For an elasto-plastic stress-deformation relationship, E is

equal to zero 0 Unloading is considered to take place along a line parallel

to the initial. elastic line, such as line AB in. Figo 2040 The intercept of

this line on the O/Oy-axiS is the instantaneous setJ b/o~ expressed in dimen­

sionless form. Unloading will continue along this line until a new ffelasticft

limi t is reached at BoThe total change in deformation between points A and

Further unloading will occur along a line having a slope equal to B is 20 0

Y E"/Eo In the figure, a possible path for a loading and unloading cycle is

represented by OABCD o·

14

Whereas for elastic response the axial force, N., and the moment, J

M., can be determined directly from O. and 'X 0' the same procedure cannot be J J J

used if inelastic behavior is to be considered. In this case it is first

necessary to compute the deformations of the flanges. The forces in the flanges

at each of the joints can then be determined from the stress-deformation re-

lationships. Once these have been evaluated, N. and M. can readily be determined 0

J J

For the anaJ.pgous system conSidered, an rtexactU determination of the

flange deformations can be made ; however, the procedure is extremely time­

consuming even in terms of the time required on a high-speed computer 0 (The

details of this analysis are included in Appendix A.) It is therefore desirable

to introduce a further approximation 0 Figo 205 shows schematically a deformed 1 r joint. As before, 5. and 5. denote the deformations to the left and right of J J

,joint jo The superscripts t and b have been added to these symbols to identify

the deformations in the top and bottom flanges, respectively. For example,

5~ denotes the deformation in the top flange on the right side of jOint jo J

If it is assumed that the angle change at a joint is distributed to

each side of the joint acc?rding to the lengths of the bars adjacent to the

joint (ioe., according to the relative stiffnesses of the adjacent bars) and

the;.;totalcha.bge~iit:L·l.eng.th betwe.en add&::ent 1joints.!:i.1e():proP01":ti-one~:to.~

the two joints inversely according to the areas of their cross-sections, then

the deformations are given by the following equations~

Ib A. 1 L. b 5. = A

J- 5 J c . ~X. J j + A. 1 j L

j + L. 1 J J J- J+

5It =

A. 1 L t J- 0.+ rj C 0 x,. j A. + A. 1 J L.+ L' l J J

J J- J J+

Aj±l L. (2.22)

5~b 5. 1 J b

:X. = - c. J A. + A. 1 J+ L. + L. 1 J J

J J+ J. J+

5~t = A. 1 L. t J+ 5. 1 + J c . :~.

J A. + A. 1 J+ L. + L. 1 J J J J+ J J+

t and b denote the distances from the centroidal axis to the top The symbols c. c .

J J

15

and bottom flanges, respectively. These e~uations yield exact results when

all flange deformations remain within the elastic range.

At the supports, these e~uations must De specialized to account for

the appropriate boundary conditions. For example, for joint ° the expressions

are:

For a hinged boundary,

orb ort = A

1 0 = 0 0 Ao + Al 1

For a fixed boundary,

orb A b 1

= Ao + Al 01 -c IVI 0 0

(2.24a)

If the ends of the arch are elastically restrained by moment-resisting

springs, the expressions for the flange deformations at the supports can be

generalized further. For a spring of stiffness k ~I, the resisting moment is

given by the expression, o

where

M = k g:~e L. -

0 .. :

e = the rotation at the support

(2.25)

E = the modulus of elasticity of the arch material for a reference

section

I = the moment of inertia of the cross-section at the reference

section

k = stiffness coefficient for the spring.

For an elastically restrained boundary at joint 0, the expressions for the flange

deformations are:

(2 .. 26a)

16

(2.26b)

where I is the moment of inertia of the cross-section at joint O. For hinged o

ends, k is equal to zero, and for fixed ends, k is equal to infinity. Substi-

tution of these values into Eqso 2026 leads to Eqso 2023 and 2024, respectively.

When the deformations of the individual flanges are determined from

Eqs. 2.22 and the appropriate expressions for the boundary conditions, the

moments on each side of the joints and t~e axial forces at each end of the

bars will not be equal as required for equilibrium 0 This difficulty is over­

come by evaluating these quantities by the following averaging procedure,

1 ( rb rt -~b Ni.t) N. = -2 N. 1 + N. 1 + N. +

J J- J- J J (2.27)

1 ,_J.b b i. t t Nr.hcb. _J"t t) Mj = 2 (N j c j - Nj c j + J J - ~j c j (2.28)

206 Method of Numerical Integration

The problem of determining the dynamic response of the structure con­

sidered herein is essentially one of satisfying the two equations of motion,

Eqso 2.4 and 2.5, for joints 1 through z-lo With the aid of the relationships

established in the preceding sections, it is possible to express these equations

in terms of displacements and to solve the resulting equations numerically using

a step-by-step method of integration 0 In passing, it may be of interest to

note that the resulting equat.ions constitute a system of 2(z~1) simultaneous.~

nonlinear differential equations of the second order.

In this method of solutionj the time during which the response of

the system is to be determined is divided into a number of s~all time intervals,

6t, and for each interval the equations are solved by means of an iterative

scheme 0 The procedure is described by assuming that at some time t the values n

of the displacements, velocities, and accelerations of all joints are known,

and that it is desired to determine the corresponding quantities for a time

tn+l' which differs from tn by the interval ~to

17

1. The solution is started by assuming the values of the accelera­

tions in the x- and y-directio~s for each joint at the end of the time interval

considered. These values can be taken equal to those for the beginning of the

interval, or they can be estimated on some other basis.

2. An assumption is then made regarding the manner in which the

accelerations vary during the small time interval, and the values of the veloc­

ities and displacements at the ends of the interval are evaluated i~ terms of

the assumed accelerations and the known accelerations~ velocities, and.dis­

placements at the beginning of the interval. The following equations due to

N. Mo Newmark [5] were used:

x = x + n+l n (8t) .0

x + n o. x n+l

. (2 .. 29a)

(2 .29b)

In these expressions x may be interpreted as the component of displacement of

joint j either in the x-direction, g., or in the y-direction, ~ .• As before, J J

a dot superscript denotes differentiation with respect to time. The subscripts

n and n+l refer to times t and t l' respectively. The dimensionless para-n n+

meter ~ depends on the assumption made regarding the variation of the accelera-

tion within the interval 6t. For a linear variation, ~ = 1/6.

3. From the values of the displacements at the end of the interval,

the changes of length and rotations of the bars and the angle changes of the

joints are evaluated using Eqso 2014, 2.15, and 2.l8.

4,~-!.- For each joint the forces X. and Yare then evaluated by use J j

of Eqso 2.12 and 2.13 (It is assumed that the load on the arch arises from a

normal pressure) 0 Next the axial forces.~ bending moments, shears, and the

forces V. and W. are determined. J J

For an elastic arch this is accomplished by

use of Egs. 2.20, 2.21, 2.3, 206 and 207, respectively. When inelastic behavior

is to be considered, it is first necessary to determine the deformations of

the flange elements at each joint. This is accomplished approximately by use

of Eqs. 2022 and 20260 Having evaluated the deformations at time tn+l~ ~ne

computes the forces in the flanges by use of the appropriate stress-deformation

18

curves. FinallYJ N. and M. are calculated from E~s. 2.27 and 2.280 The shears J J

and the forces V. and W. are found from E~s. 2.3, 206, and 2.7 as before. J J

5. The resulting values of Wj, V

j' Xjj and Y

j are then substituted

into E~so 2.4 and 2.5, and a new set of accelerations is determined. The

accelerations in the x-direction are obtained from E~o 2.4, and those in the

y-direction are obtained from E~o 205.

60 The derived accelerations are finally compared with the assumed

values. If the agreement is not satisfactory, the process is repeated with

the derived accelerations taken as the new assumed accelerations. When a

satisfactory degree of agreement is reached between assumed and derived values,

one proceeds to the next time interval and repeats the procedure.

19

III. COMPUTER PROGRAMS

·3.1 General

Computer programs have been prepared for two classes of problems:

(a) circular elastic arches subjected to a uniform all-around pressure, and

(b) arches of arbitrary shape subjected to a moving pressure pulse~

For the circular arches considered under (a)J the equations used

in the analysis were formulated in terms of polar coordinates. A schematic

drawing of a circular arch model is shown in Fig. 3.1. In this case the

arch is divided into bars of equal length; each bar subtends a central angle

~o The displacement of joint j is expressed in terms of its radial component,

wj ' and its tangential component, vjo The component Wj acts along a radius

through joint j and is positive when outward. The component v. acts along J

an axis perpendicular to the radius through joint j and is positive in a

clockwise direction. The equations pertaining to this analysis have been

presented in Reference [3]0 For convenience, a summary of these e~uations for a normal pressure is given in Appendix Bo These equations parallel those

presented in Chapter II.

For this class of problems it is also possible to consider the effect

of an initial out-of-roundnesso The out-of-roundness is defined in terms of

the deviations of the unstressed arch from a perfectly circular shape.

For the second class of problems, the loading is an initially peaked

triangular pressure pulse moving across the arch at a constant velocity as

illustrated in Fig. 3.20 The pressure is considered to act normal to the arch

at all times. There is no restriction as to the shape of the arch that may

be considered, and the response of the system may be investigated for both

the elastic and inelastic ranges of deformationo

3.2 Arch Subjected to a Uniform All-Around Pressure

3.2.1 Description of Programs. Rather than developing a single

general program for the analysis of arches under a uniform pressure~ it was

found desirable to prepare a separate program for the solution of problems

in which an initial out-of-roundness is to be specifiedo The pressure-time

20

relationship may be arbitrary, but i t must~ be approximated by a series of'

straight-line segments.. Both programs have been coded to solve either hinged

or fixed-ended arches. The program 'Which was prepared specifically to handle

an :i.nitial out-of ... roundness is restricted to 'U!liform arches3 however, the other

program is completely general in this respect, and it is possible to obtain

solutions for arChes with varying cross~sectionsG

The quantities evaJ.uated include N 0' M., 'W oj and v... In addi tionJ J J J J

for a uniform arch the extreme fiber stresses in the cross-section may be cal-

culated.. These results a.:t."e printed in dimensionless form at specified inte:rvals

of time together with the maximum values of these quantities and the times at

which they occur e For cross-sections Which are symmetrical about the arch axis"

the maximum stress in the section at each joint may be evaluated for two values

of c/r~ where c is the distance from the center of gravity of the cross-section

to the extreme fiber and r is the radius of gyration of the cross-section 0 The

quantity c/r depepds on the shape of the cross=section; for a rectangular section

it is equal to ~J and for a section consisting of two concentrated flanges~ it

is equal to 2 .. - For sections 'Which are not symmetricaJ. about the arch axis, the

program evaluates the stresses in both the top and bottom fibers of the cross-

section ..

In their present forms, the programs utilize the entire Williams

(fast) memory of the ILLIAC but only a part of the magnetic drum (slow) memory 0

The computer time required to obtain a solution depends obviously on the number

of segments into 'Which the arch is divided" the length of the time interval

used in the integra.tion procedure, the number of time intervals for which the

problem is to runJ and to a certain extent on the arch dimensionso For a

perfectly circular arch the average computer time required for each of the

t'Wel ve-bar solutions presented herein 'WaS about 'twenty minutes 0 The time r<e=

quired when an initial out .... of .,.,roundness is specified is approxi.mately twice

that amount.

The problem parameters which must be specified in using these programs

include the geometric and physical characteristics of the archo These are

specified in terms of the angle of opening, cp J and the quantity L /r:; which o 0 represents the ratio of the span length of the arch to the radius of gyration

of a reference cross-section. Throughout ~~e remaining parts of this report,

21

the Cluantity L Ir will be referred to as the "slenderness ratio"o ParaJ:lleters o

which describe the variation in cross-section and mass from joint to joint

must also be given. An initial out-of-roundness is defined in terms of the

radial and tangential components of the deviation of each joint from a per­

fectly circular configuration. The timewise variation of pressure is specified

in terms of the coordinates of the end points of each straight-line segment.

The magnitude of the pressure at any time is defined in terms of the dimension­

less parameter, PR3/EI, where p is the intensity of pressure and R is the

radius of the arch. Other parameters which must be designated are the number

of bars, z, the time interval of integration, .6t, the total number of time

intervals the problem is to run, and the number of time intervals between

print-out of results. If stresses are to be calculated, the values of c/r must

be specified.

In the use of these programs, limitations must be placed on the values

of these parameters. The maximum number of segments into which the arch can be

divided is twelve when it is desired to solve a problem in which an initial out­

of-roundness is to be introduced. For a perfectly circular arch, it is possible

to consider up to twenty bars in the analogous framework. The largest value

that can be input for the pressure parameter, PR3/EI, is 100. The maximum value

of L Ir which can be considered is 1000, that ~f ~ is 1800• Limitations also

o 0

exist with regard to the minimum values of these Cluantities; however, the lower

limits are not so well defined because they are a result of the scaling of the

intermediate quantities. Enough solutions have not been obtained to def'ine:~ae~­

rately these limits. It is believed that the range of these parameters is

sufficiently large to include all practical cases.

3.2.2 Outline of Programs. In this section a general description

is given of the computer programs for the uniform all-around pressure. A more

detailed description of these programs and of those described in Section 3.3, including copies of the codes, have been placed in the Library of the Struc­

tural Research Group of the Civil Engineering Department at the University

of Illinois ..

The complete program is divided into several subroutines each of

which performs a specific function. A flow chart of the program indicating

22

the se~uence of orders executed is presented in Fig. 3030 The roles of the

individual subroutines are also indicated in +~e flow charta

The program is started by transferring control to the first instruc­

tion of the main input routine (Ml) which reads in the problem parameters from

the data tape, sets the quantities necessary to begin a solution) and prints

a heading which identifies the specific problem to be solved.

The integration procedure is started after control is transferred

to routine (21). Except for the initial cycle corresponding to t = 0 when a

special techni~ue is re~uired, trial values are first assumed for the radial

and tangential components of the accelerations of the joints for the end of

the time interval considered. These values are taken e~ual to those at the

beginning of the interval. Next the velocities and displacements are evaluated

by use of Eqs. 2.29 with f3 taken e'lual to 1/6.

The quantities required to determine improved values for the accel­

erations are calculated by routines (l2), (10)) and (11). Since the velocities

and displacements are known initially, the accelerations for t = 0 are cal­

culated by entering these routines directly after leaving (Ml). Routine (12)

evaluates the uniform pressure coefficientJ PR3/EI. This is determined for

the end of the time interval under consideration. Routine (10) calculates

the quantities o./L, V., sin(~/2 + V.), cos(~/2 + V.), and 1/(1 + o./L). In J J - J . - J J

the calculation of o./L and V., the second order approximations given in J J __ Eqs. 2~14b and 2.15b are used. The quantities O. and V. are determined by

J J use of E~s. B.2.14 and B.2.15 in Appendix B. The sine and cosine terms are

evaluated by use of the trigonometric identities for the sum and difference

of two angles, with the terms sin V. and cos V. determined from the second J J

order approximations

sin

cos 1 -2 ,1,. =- 1 ,1, 'I' J. - 2" 'I' j

23

Similarly, the following second order expression is used in the calculation

of the remaining quantity:

__ l~ ~ 1 - ~L· + (t.)2 1 + '6 aiL

J

Routine (11) calculates the angle changes X., by use of Eqo 2018, and the axial J

forces, N., and moments, M., by use of Eqso 2020 and 2.21, respectively. Routine J J

(01) serves as the control for routines (10), (ll), and (12)0

The improved accelerations are computed by routine (20) which evaluates

Eqso Bo204 and Bo2.5. The terms on the right-hand side of these equations

are evaluated from Eqso B.206; Bo2.7; B.2.12, and B.2.l3, wherein the shearing

forces have been expressed in terms of the bending moments by use of Eqo 203.

These accelerations are compared with the assumed accelerations, and the pro­

cedure is repeated if the difference between the derived and assumed accelera­

tions exceeds a specified tolerance. It has been found convenient to make this

comparison after calculating the velocities and displacements from the deri~~d

accelerations.

When convergence is accomplished, control is transferred to the out­

put routine (M2) which first calculates the deformations corresponding to the

final displacements. Following this, a test is made to determine whether or

not the extreme fiber stresses are to be ~alculatedo If the stresses are desired,

they are determined from the following equation which is applicable to uniform

* arches :

cr =! [N j + N j+l ± ~ ( c ') J' j A 2 ... r 'r

The maximum values of the various quantities are then evaluated by comparing

the results at the end of each time interval against the previously stored

* For non-uniform arches the corresponding expression is

This expressions has not been included. in the programso

24

maxima. FinallYJ the routine prints the values of the various quantities in

tabular form at specified intervals of time.

Subroutine (MP) is used to print the absolute maximum ~~ues of the

response. It is entered at the end o~ each problem and also at the instant

the pressure pulse terminates if the termination of the pulse occurs in the

interval of time for which the response is evaluated 0 A test is provided in

routine (12) which detects the end of the pulse and transfers control to routine

(MP)o After these maxima have been printedj the memory locations in which they

are stored are set equal to zero. Consequently~ in this case, the maximum

values printed at the end of the problem are for the era of free vibration.

The basic features of the program which allows consideration of an

initial out-of-roundness are the same as for the program applicable to a per­

fectly circular arch. However, in the latter case account is taken of symmetry,

and only one-half of the arch is considered, while the former requires con­

sideration of the entire arch.. Ano·ther difference concerns the manner in which

the deformations are computed. For a perfectly circular arch, the deformations

are calculated directly from the displacements measured with reference to the

undeformed configuration of the structure 0 However, for an arch with an initial

out-of-roundness, the deformations are calcul.ated in two steps: First, a

fictitious set of deformations is determined in terms of displacements measured

not from the undeformed configuration of the arch but from the perfectly cir­

cular configuration 0 Second, the actual deformations are determined by

subtracting from these fictitious deformations the i¥deformations" corresponding

to the initial out-of-roundness. The first step is accomplished by use of

routine (lO)* Subroutine (13)J not shown in the flow chart, has been added

immediately after routine (10) to perform the subtractions of the second step 0

The ttdeformations tt corresponding to the initial. out~of'-round.ness are calculated

and permanently stored with the aid of subroutine (10), which is entered from

routine (Ml) during the initial stages of the solution.

303 Arch Subjected to a Triangular Moving Pressure

30301 Description of Programs. Two separate programs have been pre­

pared for the solution of an arch subjected to a triangular-shaped mOVing

25

pressure. Because the computation of the response of arches in the post-elastic

range of behavior is very time-consuming, it was found desirable to prepare a

specialized program for the solution of elastic arches.

The ~uantities evaluated by both programs include the forces N. and J

Mj

and the displacements ~j and Tljo In addition, the ftinelastic program U

determines the average strains at each joint in the top and bottom flanges.

All of these results are printed in dimensionless form at specified intervals

of time.

Non-uniform arches having either hinged or fixed boundaries can be

considered by both programs. In addition, the t1inelastic;' program"" can,:be

used to study the effects of partial fixity at the supports.

The programs make use of the complete Williams memory and part of

the magnetic drum memor.f of the ILLIAC. The time required for a solution

depends on the same factors which have been enumerated in Section 3.2.1. For

each of the elastic solutions presented herein, the computer time ~veraged

about forty-five minutes. For the inelastic solutions, the average time was

roughly twice that for a corresponding elastic solution.

The problem parameters which must be specified in using the programs

can be classified into four groups. The first group determines the shape of

the arch and includes the rectangular coordinates of the joints of the arch.

For circular ~~d parabolic arches, subroutines have been prepared for calculat­

ing these coordinates within the machine so that it is not necessary to input

the coordinates of the individual joints directly. The second group of parameters

describes the physical properties of the arch and includes the slenderness ratiO,

L Ir, the variations in cross-section and mass from joint to joint, the charac-o

teristics of the bilinear stress-deformation relationships for the top &~d

bottom flanges, and the boundary conditions of the arch. The third group of

parameters defines the characteristics of the moving pressure pulse and includes

the peak value of the pressure and parameters related to the duration and speed

of the pressure wave. Finally, the time interval, ~t) the number of bars, z,

the total number of time intervals the problem is to run, and the number of

time intervals between print-out of results must also be specified.

26

The maximum number of bars that can be considered in the analogous

arch is eighteen for an elastic solution, and fourteen when use is made of the

ttinelastic :progr.amt~ .. The largest value of the peak pressure which can be

considered corresponds to a value of p L3 lEI equal to 1000 c Other limitations o 0

exist with regard to some of the other parameters; however,. there is sufficient

latitude to permit consideration of most problems of practical interest.

303c2 Outline of Programsc A general description of the computer

programs is included in this sectiono A flow chart of the program indicating

the functions of the various subroutines, is shown in Figure 3.40 The principal

features of these programs are the same as those of the programs described in

Section 30202. The significant differences lie in the determination of the

external forces at the joints and in the additional subroutines necessary to

determine the axial forces and moments for the ninelastic' programi'since these

cannot be calculated directly from the total deformations and rotations of the

bars"

The program begins when control is transferred to the input routine

(Ml) which, in conjunction with subroutines (Mlo2) and Ml.3), reads in the

problem parameters from the data tape, sets the quantities neces~ary to begin

a solution, and prints a heading used to identify each problem. The specific

function of subroutine (Mlo2) is to set the rectangular coordinates of the

joints into their proper locations in the memor~o Several versions of routine

(Mlo2) have been developed. One of these is applicable to arches of arbitrary

shape; in this case, the coordinates x. and y. are specified on the input tape. J J

The subroutine, then, merely reads these values and places them in their

correct locations in the machineo Another version applies to circular arches

for the case in which the geometry is specified by the rise-span ratiO, tiL 0

o Using this parameter, the subroutine evaluates the coordinates of the joints

and then stores them 0 A siroil.ar rout,ine has also been developed for parabolic

arches; other shapes can be evaluated in a similar manner by simply replacing

this subroutine with a proper substitute.

The integration procedure is accompl.ished by routine (-21) 0 As in

the case of the programs for a uniform all-around pressure, the initial trial

values of the accelerations at the end of each time interval are taken equal

27

to those at the beginning of the interval except for the special case when t = o. The velocities and displacements then are determined by use of E~s. 2.29 with

f3 = 1/60

Improved values of accelerations are determined using ~uantities

evaluated in routines (ll), (12)" (13) and (-44), and (14). Using 'the dis­

placements derived from the assumed accelerations, routine (11) calculates the

~uantities 5.iL., *., cos(f3. - W.), sin(f3. - ~.) and 1 ! IL. The quantities J J J J J J J + ..

o./L. and v. are determined by use of the second order apptoxfmations given in J J J

E~s" 2014b and 2.150, where 5 j and 'ifj

are given by E~s. 2.14a an9- 2.15a. The

sine and cosine terms are evaluated by use of Eqs. 2.17, where l/(L. + 5.) is J J

determined by use of the second order approximation,

1 = 1 (1 _ ~ + ( ~ )2 ] L. L. L.

J J J L. + 5.

J J

In the "inelastic programTt, the deformations of the flanges, oit etc., are j ,

determined by routine (12) using E~s. 2.22 and 2.26. In routines (13) and

(-44) the corresponding stresses are found by use of the bilinear stress­

deformation relationship described in Section 2.5.2. The axial forces, N., J

and moments, M., are then calculated in routine (14) by use of E~s. 2.27 and J

2.28. For an elastic arch, routines (12) and (13) are omitted, and a different

routine (14) is used which calculates N. and M. directly by use of Eqso 2020 J J

and 2.21. Routine (01) serves as the control for routines (11), (12), (13),

and (14).

The external forces at each joint; p~ and p~, are determined in J J

subroutine (21) which is controlled from routine (22). These quantities are

calculated by determining the instantaneous static reactions at the ends of

the bars; the bars are assumed to be simply supported. It may be noted that

these reactions are different for each time interval, since the pressure

distribution changes as the load moves across the arch. Routine (22) calcu­

lates the improved accelerations by evaluating the equations of motion, Eqso

2.4 and 2.5. The terms on the right-hand side of these e~uations are calcu­

lated by use of Eqs. 2.6, 2.7, 2.12, and 2.13. In E~s. 2.6 and 2.7 the shearing

forces have been expressed in terms of the moments by use of Eq. 2.3. These

28

accelerations are compared with the assumed accelerations, and, if satisfactory

agreement is not attained, the integration procedure is repeated 0

At the end of each time interval, control is transferred to routine

(M2) which evaluates the final deformations and stresses by use of subroutines

(11), (12), and (13). At the times when the results are to be printed, the

final values of the moments and axial forces are calculated by use of subroutine

(14)0 Then the results are printed in tabular form.

29

IV 0 NUMERICAL SOLUTIONS FOB .ARCH SUBJECTED TO A UNIFORM PRESSURE PULSE

401 Problem Parameters

In this chapter numerical solutions are presented for the elastic

response of two-hinged circular arches subjected to a uniform' all-around pressure

pulse. The cross-section and mass per unit of length of the arch are considered

to be constant 0 The pressure-time relationship is represented by a triangle

with an initial peak as shown in Figo 4.10

For the benefit of the reader who'may have bypassed Chapter III, the

parameters used to define the problem are summarized belowo The dimensions ,of

the structure are expressed in terms of the rise-span ratio, f/L , and the s.l.en­o

derness ratio, L /r 0 (Note that tan(cp /4) = 2f/L , where cp is the central angle o 000

of opening of the arch.) The characteristics of the pressure pulse are expressed

in terms of the ratios p /p and td/TO. The quantity p is the peak value of o cr 0

the pressure; Pcr represents the critical buckling pressure corresponding t.o an antisymmetrical mode of deformation 'and is given by the equation,

Pcr = [

41(2 _ 1 J EI-2 3 ' cp R o

where R is the radius of the arch. The symbols E and I represent the modulus

of elasticity of the material and the moment of inertia of the cross-section

of the arch, respectively. The quantity td denotes the duration of loadingJ and

T , the fundamental period of vibration of a complete ring. The latter quantity o

is given by the equation,

TO = 2.r V:.2

= 2.r ~ where C is the speed of sound in the material of the arch, A is the cross­

sectional area, and ~ is the mass per unit of length of the, arch.

The initial out-of-roundness considered in some of the solutions to

be presented is taken in the form of a complete sine wave with nodes at the

supports and at the crown. The magnitude of the out-of-roundness is expressed

in terms of w /L a where w is the amplitude of the deviation. , m 0'" m

4.2 Presentation of ResUlts

4c2.l General. The majority of the solutions presented are for a

perfectly circular two-hinged arch of uniform cross-section having the following

dimenSions,

f/L= 0.20 o

L Ir = 100 o

( c'orre sponds to cp = 87.21 0 ) o

(corresponds to R/r = 72.5)

Except where otherwise indicated, these solutions were obtained by considering

twelve bars of eClual length in -the analogous framework. The natural freCluencies

and modes of vibration for this structure have been reported in Reference [3]

For convenience, the natural periods are summarized in Table 4.1, and the first

three antisymmetrical and symmetrical modes of vibration are given in Fig. 4.2.

4.2.2 Typical Response Characteristics. In Figs. 4.3a through 4.3d

are plots of the response of the arch analogue referred to above when subjected

to a loading defined by the parameters tdlT = 2 and P /p = 10 Included are o 0 cr the deflection configurations at a few selected times and time histories of the

radial displacement, axial force, and bending moment. The radial displacements

and bending moments are plotted for the crown and Cluarter point, and the axial

force, for the crown. No tangential displacements are presented because they

are generally small compared to the radial displacements. The displacement

configurations are shown to an exaggerated scale.

The axial forces are eXpressed in terms of p R, and the displacements, o

in terms of p R2/AEo These Cluantities represent, respectively, the axial force o

and the radial displacement caused by a uniform pressure p acting statically o

on a complete ring. The bending moments are expressed in terms of p Rr. For- a o

section conSisting of two concentrated flange areas, clr is eClual to one and,

conseCluently, the stress at the extreme fiber of a section is proportional to

the sum of the ordinates of the axial force curve and the corresponding moment

curve.

The response curves presented in Figs. 4.3 are generally smooth and

regular. The period of the oscillations in the curve for the axial force is

31

approximately e~ual to that of the second s~~etrical mode of vibration of the

arch 0 The period of this mode, which is predominantly extensional, is close

to T J the lIbreathingn period of a complete ring 0 The periods of the oscilla-o

tions in the curves for bending moment are not directly identifiable with any

of the natural periods of the arch, and, in fact, those for moment at the

crown and ~uarter point are between the periods of the second and third modeso

The principal oscillations in the response curves for radial displacement at

the quarter point have a period e~ual to that of the second symmetrical mode;

however, oscillations of a period e~ual to that of the first and third natural

modes are also noticeable in the response curves for displacement at the crown.

The cont,ributions of these three modes can also be seen by comparing the dis­

placement configurations in Figo 403a wlth the natural modes given in Figo 402b.

The participation of the various modes can be determined more readily

by a modal method of analysis 0 Such a solution was obtained primarily for the

purpose of checking the results of the computer calculations. The basic e~ua­

tions used for this analysis are given in Appendix C" The results of this

modal solution are presented in Table 4 0 2 where t.he contributions of each mode

to the values of displacementJ axial force, and moment are listed. The function

f (t) represents the dynamic amplification factor for the rthmode and depends r ' on the shape of the pressure pulse., The structure considered in this solution

is the same twelve-bar frameworkJ and the load is a step-pulse of infinite

duration 0 The solution applies only to values of Po which are small in com­

parison to the critical buckling load 0 From an examination of the resul,ts

presented in this table, it can be seen that the major part of the response

arises from the participation of the first three symmetrical modes of vibrationo

This is consistent with the obse~~ations made above on the basis of the response

curves 0

The results show that at any time the distribution of the axial force,

N.9 is essentially uniform t.hroughout the ,arch. ,Conse~uently" the curve in

Figo 403c may'be interpreted as representing the time variation of N for any

point on the arch. This result also holds for the solutions presented in the

following sections of this chapter 0 The distribution of bending moment across

the arch can best be seen from Figso 404 which show the variation of the moment

at each joint 0 It can be seen that the bending moment reaches a maximum first

32

near-the supports and that the absolute maXimum moment occurs at the crown 0

In Figo 403 it is of interest to note that the maximum displacement at the

quarter point is larger than that at the croWllo It may also be noted that

the absolute maximum value corresponds to the first maximum for both the dis­

placements and axial forces.

In Table 403 "are listed the maximum values o~ the radial displacement,

axial force, and bending moment and the times at which these maxima occur.

Displacements and moments are given for each joint of the analogous framework

and axial forces are given for each bar. Also included in this table are values

of the maximum stresses at each joint for cross-sections with clr = 1 and

clr = 2. The corresponding static values are given in Table 4040 For conven­

ienceJ the maximum dynamic effects for the quarter point and crown are compared

with the corresponding static values in the following table.

w N M R2/AE P R p Rr Po 0 0

114 point * * 1/4 point crown l/4 point crown crown

Static -loilO -10578 -00994 -0.994 0.063 0.086

Max. Dynamic -20385 -10875 -1.684 -10679 0.934 -10137

The displacement at the crown increases only 19 percent over the static value,

whereas that at the quarter point increases 125 percento The increase in the

axial force due to the dynamic load is 69 percento Because of the small static

bending moments in the arch, the dynamic amplification of moment is quite

large. It should be noticed that the maximum bending stresses are of the same

order of magnitude as those due to the axial forces.

40203 Effect of Length of Time Interval of Integration and Number

of Barso For the solution presented in the preceding section the arch was

replaced by a framework of twelve bars and. the time interval of integration

was taken as ~t = 0001 T 0 The purpose of this section is to discuss how the o

solution is influenced by the choice of the time interval and the number of

segments used in the replacement system.

* The axial forces are defined for the bars to the immediate left" of the points indicated.

33

The interval of integration should be small enough such that suc­

cessive cycles of iteration convergeand.tb.~ res,ulting','soJtu:tions',are',

stable and accurate. In general, the smaller the time interval of integration,

the more accurate will be the solution; however, the time required to obtain

the solution will be correspondingly longer. In the integration procedure used

in this analysis, the acceleration was assumed to vary linearly during each

time interval. For a linear problem, it has been shown [5] that the procedure

is convergent and leads to a stable solution when

where T is the smallest natural period of the structure. While not strictly

applicable to the present non-linear problem, this relation may be used as a

guide in the selection of the time interval of integration, with the quantity

T inte~reted as the smallest natural period of the unloaded arch. For the

particular twelve-bar system considered, this corresponds to

and for a sixteen-bar system, to

b.t = 0.026 T o

At = 0.019 T • 0'

Comparisons were made of the results obtained for (a) a twelve-bar

system using time intervals of 00005 T , 0.01 T , and 0002 T and (b) a o 0 0

sixteen-bar system using time intervals of 0.005 T and 0.01 T. The maximum o 0

values for these solutions are presented in Tables ~.3 and 405. It can be

seen that, for all'practical purposes, the results for the same number of bars

are identical 0 In other words, the accuracy of the solution does not appear

to be sensitive to the choice of time interval, provided the convergence and

stability requirements are satisfied.

As an indication of how the time required for solution is influenced

by the choice of time interval, the number of iterations required in the solu­

tion of the problems mentioned in the preceding paragraph are listed in the

following table. In this table are given the number of iterations required

in the computation of the response for an interval of time equal to T and o

the average number of iterations for each step of integration, ~t.

Number of Bars

12

16

34

.6t T

0

00005 0001 0002

00005 0001

Iterations per Iterations per T .6.t

0

583 2091 402 4~02 755 15011

600 3.00 571 5.71

It is noted that the larger the time interval, the larger is the number of itera­

tions re~uired for convergence per time interval. However, the number of iterations

per T andJ eonse~uently, the time re~uired to obtain a solution decreases with o

increasing .6t/T until values close to the convergence limit are reached. The o

results of these solutions suggest that the optimum time interval, which leads

to the minimum time for a solution~ is of the order of one-third to one-half

the convergence limit. It is found that for this time interval convergence is

accomplished in approximately four or five iterations. These conclusions have

also been verified by a few additional solutions which are not included here.

The effect of increasing the number of bars in the analogous frame­

work was investigated to determine the adequacy of the twelve-bar solution in

depicting the response of the continuous structure. Results were obtained for

arches with 10, 12, 16 and 20 bars. Time histories of the radial displacements

at the 1/4 point and crown are compared in Figs. 4.5a and 405b, and the axial

forces and bending ~oments are shown in Figs. 4.5c and 4c5d. It is assumed that

the significant characteristics of the response of the continuous arch are

present in the 20-bar solution.

The results for the axial force and the radial displacement at the

~uarter point are in ~ood agreement, except for a slight phase shift. The agree­

ment is not as good for bending moment or displacement at the crown. For values

of tiT larger than 2.0, which correspond to the free vibration era, there are o

differences in the detailed characteristics of the curves shown in Fig. 4.5d in

addit.ion to the phase shift. The phase differences are attributed to the fact

that the natural frequencies of the modes which contribute to the response are

not the same for the different systems compared. The differences in the

35

response characteristics of the curves in Fig. 4.5d for the larger times can

be explained on the basis that the structures with the higher number of bars

can account better for the participation of the higher modes. However, it

is important to note that there are no major differences in the values of the

maximum effects or in the over-all characteristics of the various solutions.

The times required for solution of the 12-bar and20-bar problems

were approximately 20 and 40 minutes, respectively 0 For the purpose of the

present exploratory study, it was felt that the twelve-bar system would pro­

vide results of sufficient accuracy, and it was used to obtain the remainder

of the solutions.

In order to determine the reliability of the solutions obtained from

the digital computer, an independent check of the arch problem was provided by

obtaining comparative solutions by a modal method of analysis for an arch sub­

jected to a s~ep-pulse loading of infinite duration. This solution was obtained

on a desk calculator. The system considered was the twelve-bar analogous

framework described in Section 4.2" Since the modal method is applicable only

to structures which deflect in a linearly elastic manner, it yields results

only for 11 small" values of p /p . The computer solution used in the compari-. 0 cr .

son was obtained for a value of p /p = 0001. o cr

The results of these solutions are compared in Table 4.6. Radial

displacements, axial forces, and bending moments are tabulated at correspond­

ing times for both methods of solution. The agreement between the two sets of

solutions is quite satisfactorJo

40204 Effect of Duration of Loadingo In Figso 406a through 4.11d

are presented response curVes for an arch subjected to a triangular pressure

pulse for durations represented by values of td/To up to 40 The arch is the

same as that considered in Section 4.201, and p /p = 1 for all solutionso a cr The absolute maximum values for the various effects are summarized in Table

The general features of these curves are similar to those presented

in Fig. 4030 As before, the period of the predominant oscillations in the

curves for displacement and axial force is the period of the second symmetrical

mode of vibration (close to the period T of the Ubreathingft mode for a complete o

ring) 0 The period of the oscillations in the curves for bending moment are not

identifiable with any of the natural periods of the arch 0 The axial force is

found to be relatively uniform throughout the archJ whereas the distribution of

bending moment is non-uniformJ with the maximum effect occurring at the crown 0

The value of the maximum displacement at the quarter point is found to be con­

sistently larger than that at the crown.

In Figs. 4.12a through 4.12d are presented spectrum curves of radial

displacements J stresses, axial forces, and bending momentso These are plots

of the absolute values of the 'maximum effects versus the duration of loading,

td/Too The stress at a joint of the substitute framewor4 is determined from

the moment at that joint and the average value of the axial forces in the

adjoining barse The spectrum curves for stress have been calculated for clr = 1.

Within the range of load durations conSidered, the maximum displacement is 2.58

times the corresponding static value for a complete ring under uniform pressure.

The maximum stress reaches a value 2.43 times the static value for a ring. The

maximum axial force is 1.83 p RJ and the maximum moment is 1.47 p Rr 0

o . 0

To indicate the relative importance of the bending effects, in the

following table are listed the ratios of the true maximum stresses to the max­

imum stresses determined,by disregarding the component due to bendingo Results

are tabulated for the quarter-point and the crown for different values of tdlTo'

and for clr = 1 (two concentrated ~lange areas) and {3 (~rectangular cro~s­

section) 0

Ratio of true cr to cr disregarding bending max max

clr = 1 clr = 3 tdlTo

1/4 point 1/4 point crown crown

00 ;"::; 1.79 1.86 2.55 2.68 0050 1078 1.94 2053 2.80 0075 1,,58 1082 2022 2 .. 63 1000 1032 1045 1.75 2.09 1050 1.32 1015 1.57 ~.67 2~00 1.32 1023 1056 1072 4.00 1·32 1.35 1065 1080

It may be noted that this ratio varies from 1015 to 1.94 for clr = 1 and from

1056 to 2080 for clr = ~ and that there is a tendency for the ratios to

37

increase with decreasing values of td/Too Obviously, the bending effects are

more important for the cross-section having the larger clr ratio.

The absolute maximum displacement occurs on the first maximum for

the longer durations of load, and on the third or a later maximum for the

shorter durations. It follows that the effect of damping in reducing the

m~gnitude of the maximum values will be more pronounced for the shorter dura­

tions of loading.. The absolute maximum value of the axial force occurs on

the first maximum, whereas that for bending moment occurs on the third maximum.

Since the maximum axial force and maximum moment do not occur at

the same time, and since the stress calculation is dependent on the ratio clr,

it is desirable to have an approximate method for calculating the maximum

stresses. In Table 408 the exact maximum stresses are compared with those

computed by use of the following e~uations:

(J max

_ 1

A

M 2 (N )2 + ( max)

max r

2 ( c )

r

(a)

(b)

Comparisons are made for two values of c/re It can be seen that E~. (a) over­

estimates the maximum stress by a maximum of 49 percent and a minimum of 9

percent" The results obtained from Eq" (b) are, for the most part,. too low.

The maximum error is -20 percento

40205 Effect of Parameters Influencing Tendency for Bucklingo The

factors which govern the tendency of a circular arch to buckle under dynamic

loading include the magnitude of the peak pressure, the duration of the loading,

and the amount of initial out-of-roundness" Solutions are presented in this

section for a perfectly circular arch and an arch with an initial out-of­

roundness for different values of p Ip 0 The characteristics of the arch are o cr

considered to be the same as before, and the duration of loading, td

, is taken

equal to 2T 0

o

The effect of an increase in the peak pressure is indicated in Figs.

4.13 where the results for p /p e~ual to 005 and 200 are compared. It can ,ocr be seen that the peak values of the response are affected only slightly by

changes in the value ',of p /p '" The features of the response are also very " 0 cr

similar except for a small decrease in frequency with an increase in pressure.

This change is attributed to the reduction in stiffness of the arch due to

the increased axial thrust. The curves presented in Figs. 403 for p /p = 100 o cr

fall between those for p /p = 005 and p, /p = 200 0 o cr 0 cr

In Figs" 4014 are shown response curves for an arch having an initial

out-of-roundness in the form of a complete sine wave and an amplitude

w = 00001 L m 0

For the value of p /p = 1 consideredJ this corresponds to o cr

Comparing these results with those presented in Figso 4.3 for a perfectly

, circular arch j one may observe that the response

3/4 point) for the circular arch lie between the corresponding curves for the

arch with the initial out-of-roundnessJ and that the differences in the values

of the maximum effects are small 0 The maximum values for these solutions are

compared in Table 4.9. Also included in this table are the results for the

different values of p /p discussed above and one additional solution for an ,ocr arch with an initial out-of-roundness. These results indicate that for load

durations less than td/To = 2j the effect of the magnitude of the peak pressure

or of a small initial out-of-roundness is small 0 However j it is expected that

for longer durations of loading the buckling tendency may become more impor­

tanto Of e<lualimportance, howeverJ may be the effect of damping which was

neglected in these solutions 0

4.206 Effect of Arch Dimensions. The influence of the dimensions

of the arch on the response was investigated by obtaining a few solution$ for

different values of the slenderness ratio, L /r j and the rise-span ratio, o

f/L" The values of L Ir considered were 50, 100 and 200 for a value of f/L 000

equal to 0020 In additionJ rise-span ratios of 001 (cp = 450240

), '", o

O.,2(cp = 970'21 °L and 005 (cp = l800) were considered for L /r e<lual to 1000 , 0 0 0

In each of these solutions 0 the load parameters were taken as p /p = 100 and ,. 0 cr

39

The results for L /r equal to 50 and 200 are given in Figs. 4015 and o

4.16. The maximum values are sUIImlarizedin Table 4.10. From an examination

of these results and those presented in Figs. 4,,3 for L /r = 100, it can be o

seen that the detailed characteristics of these curves are, in general, differ-

ent. However, insofar as the ~-ial forces are concerned, the period of the

principal oscillations in the response curves are approximately equal to T J o

and the magnitude of the maximum forces are essentially the same. The maximum

values range from 1071 p R to 1.74 p Ro o 0

As might be expected, the effect of changing L /r is most pronounced o

on the response curves for moment and displacement. It can be seen that the

moment builds up more rapidly for the stiffer arches. There does not appear

to be an appreciable difference in the maximum values of the dimensionless

bending moment, M/p Rr, for L Ir = 100 and 200; however, for L /r = 50, the 000

value is appreciably larger. This is also true for the radial displacements.

For convenience, these results, in addition to those for the absolute maximum

stresses, are s~arized in the following tableo The values presented repre­

sent the absolute maximum effects for the entire archG

L /r o

50 100 200

Absolute Maximum Values

M max

p Rr o

1.65 1014 l,,19

CJ max

p RIA 0

(c/r=l)

?o30 2.22 2013

CJ max

P RIA 0

(c/r= 3)

3016 2.88 2.69

It should be noted that the maximum stresses are not as sensitive to changes

in L /r as are the moments. o

Response curves corre~ponding to different values of the rise-span

ratio are presented in Figs. 4.17 and 4018, and the maximum values of the

response are given in Table 4.10. These results together with those presented

in Figs 0 4.3 and Table 4.3 indicate that an increase in flL ha's the same o

general effect as an increase in L Iro o

A comparison of Figs. 4.l7 and 4015

shows a striking similarity between the corresponding time history curves.

Comparing Figs. 4.18 and 4.16 one may notice the same tendencies in the response

40

curves even though the correspondence is not as pronounced in this case as in

the previous comparison 0 It appears that an empirical relationship might be

developed which combines the rise-span ratio and the slenderness ratio into

a single parameter. On the basis of the meager data presented here, a possible

choice for this parameter is

f Lo f -- = L r r o

Additional solutions are necessary to determine both the adequacy and range

of applicability of this parameter 0

41

v 0 NUMERICAL SOLUTIONS FOR ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE

501 Problem Parameters

Numerical solutions are presented in this chapter for two-hinged

circular arches of uniform cross-section subjected to a triangular-shaped

moving pressure as shown in Figo 302¢ The characteristics of this type of

loading are defined by the parameters p Ip J t't/T J and tdft. The parameter o cr 0 t P Ip describes the peak value of the pressure wave, p :; in terms of the

o cr 0

buckling pressure, p c The quantity tt represents the transit time, which cr is the time required for the front of the wave to move ,from one support of

the arch to the other; td represents the duration of the pulse, ioeo, the

time required for the entire pressure pulse to pass over a fixed point on the

arch. As before, T is the "breathingt! period of vibration of a complete ring., o '

Constant values of the parameter td/tt represent pressure waveschaving the

same physical length. The total length of time the pressure wave acts on the

arch is given by the sum of tt "and tdo As in the preceding chapter, the arch

dimensions are expressed in terms of tiL and L Ir. o 0

The yield strain for the arch 9 € 0 is expressed in the dimensionless , yJ

form

€

P R7AE o

A static uniform pressure of intensity p will initiate yielding in a complete o

ring when the value of the above parameter is equal to unity. The yield def-

ormation for a flange is equal to the product of € and half the length of the y

bar adjacent to the flange. For example,

=

(ort) = j y

L. € -.J. Y 2

L· 1 E ....J±:!:. Y 2

Similar expressions for the bottom flanges are obtained by replacing the super­

script t with bo

5. 2 Presentation of Results

50201 Typical Response Characteristicso Most of the solutions pre­

sented in this chapter are for the arch described in Section 4.201. Except

42

for the results discussed in Section 502.4, the response is restricted to the

elastic range of behavior. The response curves for a loading characterized

by the parameters po/pcr = 1, tt/To = 1 and td/tt = 1 are presented in Figs.

5.10 In these figures are shown representative displacement configurations

and time histories of the radial and tangential displacements, axial forces,

and bending moments for the 1/4 poin~, crown, and 3/4 point. The curves for

axial force are actually for the bars of the analogous framework just to the

left of the designated points. All of these quantities are expressed in the

same terms described in Section 4.2.2.

As might be expected, a major contribution to the response arises

from the participation of the first antisymmetrical mode of vibration. This

can be seen clearly from the displacement configurations shown in Fig. 5.1a

which have the same general shape as the first antisymmetrical mode of vibra­

tion illustrated in Fig. 4.2a. The same observations can be made from the

displacement curves for the l/4 point and 3/4 point for which the period of

the major oscillations seem to be equal to that of the first antisymmetrical

mode 0

Although the most critical section of the arch is away from the

crown, the effects at the crown are also appreciable. The maximum radial

displacement at the crown is about 59 percent of that at the 1/4 point, and

the maximum moment at the crown is about 63 percent of that ,at the 3/4 point.

It is of interest to note that the period of the predominant oscillations in

the curves for radial displacement and moment at the crown appears to corre~

spond to that of the first symmetrical mode of vibration. Response in the

symmetrical modes is apparent also in the curves for axial force for which

the principal period of oscillation is approximately equal to the second

symmetrical, or extensional, mode of vibration.

It is of interest to note that for all of the curves for moment and

axial force, higher frequency oscillations appear praminantly. This may be

contrasted to the very smooth and regular response curves for the uniform all­

around pressure discussed in the preceding chapter.

The time histories for the axial force reveal that N does not'vary

appreciably throughout the arch. This result is in agreement with the results

obtained for a uniform all-around pressure. The distribution of bending

43

moment across the arch can be seen from Fig. 502, in which are plotted the

time histories of the moment for each joint. As might be expected for a moving

pressure, the moment builds up progressively from the near support to the far

support as the pressure wave moves across the arch.. The maximum moment occurs

at the 3/4 point at a time when approximately one-half of the load is off the

arch.

For the solution discussed in this section the pressure acts on the

structure for a total time of 2 T . o

Although it is conceivable that the maximum

values of response beyond the time considered in the solution may be larger

than those obtained, this does not appear to be the case. In this connection,

it should be emphasized that the effect of damping will be to reduce the later

maxima.

The maximum values for a uniform all-around pressure with a duration

equal to 2 T are given in Table 4.3. Comparing the maximum values for the o

two types of loading, one may observe that the larger displacements result from

the moving pressure. For the particular case considered; the largest radial

displacement for the moving pressure is approximately 5 p R2jAE, while for the

uniform all-around pressure it is 2,,4 P R2

/AE. The maxim~ bending mom~nt is o 1.63 p Rr for the moving pressure and 1014 P Rr for the uniform pressure. The

o 0 maximum axial force, on the other hand, is larger for the uniform-pressure

solution, being equal to 1.71 p R as compared to a value of 0.71 p R for the o 0

moving-pressure solution.

5.2.2 Effects of Characteristics of Pressure Pulse. The influence

of the duration, velocity, and magnitude of the pressure wave on the response

of arches is discussed in this section. In Figs. 5.3 and 5.4 are presented

response curves f6r two load durations re~resented by values· of td/tt equal

to 0.5 and 2.0 respectively. For both solutions, the arch is the same as

that considered before, and the characteristics of the pressure pulse are de­

fined by the parameters tt/T = 1 and p /p = 1. o 0 cr

In Fig. 5.5, the values of the first major maxima for moment, axial

force, and radial displacement at selected points on the arch are plotted

versus the duration of loading. In addition, the absolute maximum tangential

displacement, which occurs at the crown, is shown. It can be seen that in­

general the maximum effects increase with increasing td/ttO For td/tt = 2,

44

t,he largest duration considered, the maximum radial displacement at the 1/4

point is e<1.ual to 8077 P R2; AE" The absolute maximum moment at the 3/4 point o is 2 0 1 7 p Rr, and the maximum axial force at the crown is 0" 80 p R. Except

o 0

for the axial force, these effects are appreciably larger than the values

computed for a uniform all-around pressureQ Conse<1.uentlYJ the bending effects,

as indicated by a comparison of the values of the maximum moments and axial

forces, are more important for the triangular moving pressures.

In order to study the effect of the magnitude of the pressure, a

few solutions were obtained for different values of the parameter, p /p . o cr

In Fig. 5.6 are shown the response curves obtained for p /p = 0.001 with o cr tt/To = 1 and td/tt = 20 For a small pressure of this magnitude, the large

deflection effects are negligible and. the response is essentially linear.

An indication of the ·buckling tendency which occurs for the larger magnitudes

of pressure may be obtained by comparing these results with those given in

Fig05e4 which is for p /p = 1. It can be seen that the larger pressure o cr

causes an over-all magnification in the amplitude of the response curves as

well as a decrease in their fre<1.uencieso Maximum values for these solutions

are compared in the following tableo Also included in this table is a com­

'paris~n of the maximum values for two pressures of larger mangitude acting

on t.he arch for a shorter duration of time. For these solutions peak pressures

of po/pcr = 1 and 2 were considered with td/tt and tt/To both e<1.ual to unity 0

(The time histories for these solutions are presented in Figso 5.1 and 5.7J

respectivelyo) For this comparison it was necessary to consider only the

first maxima since the second and later m~xima were not attained in most of

the solutions 0

td/tt = 100 td/tt = 2 .. 0

Quantity Location p Ip = 100 P /p = 200 P /p = 0.001 P Ip = o cr o cr o cr o cr

1/4 point -4.90 -6001 -5075 -8077 Vi -2090 =3046 -3.03 -4013 R2/~

crown 3/4 point 4087 5022 5.89 7025 Po

M 1/4 point 1 002 1019 1008 1.52

P Rr crown lo03 lo02 0093 0.98

0 3/4 point ~lo63 =1077 -1075 -2018

N 1/4 point -0061 -0060 -0.76 -0.76

p R crown -0064 -,0.62 -0081 -0.80 0 3/4 point -0066 -0093 -0·91 -0086

100

For·the longer duration it can be seen that a change in p /p from o cr

0.001 to 1.0 causes an increase in the absolute maximum displacement from

5.89 p R2/AE to 8.77 p R2/AE or about 49 percent. The max~ moment at the o 0

3/4 point increases from 1075 p Rr to 2.18 p Rr,or about 24 percent. These o 0 "

changes are appreciably larger than corresponding increases for a uniform all-

around pressure as reported in Chapter IV. The increases will obviously be

even larger for longer durations of load or higher pressures. For the shorter

duration, a change in pressure from p /p = 1 to 2 produces an increase in o cr

maximum displacement of about 23 percent and an increase in maximum moment of

only 8 percent. It may be inferred that the effect of an increase in pressure

is not as significant for short durations of load even for fairly large values

of P /p . o cr

The influence of the velocity of the pressure pulse was studied by

comparing solutions for pressure waves having the same physical length (same

value of td/tt)' but different velocities (different values of tt/To). In

Fig. 508 are presented solutions for td/tt = 0.5, which corresponds to a

pressure wave having a length equal to one-half the span of the arch, and

tt/T equal to 2. As before, p /p = 1" These curves can be compared with o 0 cr those given in Fig. 5.3 for tt/To = 1. In Fig. 5.9 are presented solutions

for td/tt = 2 (length of pressure wave equals 210

) and tt/To = 0.5 with

p /p = 1. The se re sul ts are comparable .tt> those shown in Fig. 5.4 for o cr tt/To = 1. The most prominent difference between the corresponding curves is

the reduction in magnitude of the response with an increase in velocity, i.eo,

with a decrease in tt/To. The absolute maximum values for these solutions are

listed in the following table.

Quantity Location td/tt = 0.5 td/tt = 2.0

(1) (2) (3) (4) tt7To = 2.0 tt/!r = 1.0 ttfTo = 1.0 tt7To = 0.5 . .0

Vi 1/4 point -7.33 -2.70 -8 .. 77 -3.82

R2

/AE crown -7.13 -1·70 -4.13 1.62 3/4 point 6.41 2.74 7.25 3.79 Po

M 1/4 point .. 1.21 0 .. 69 1.52 0.87

p Rr crown -2.39 0.70 -1.01 0083 0 3/4 point -2.72 -1.08 -2.18 -1.21

N 1/4 point -1.03 -0039 -0.76 -1.18

p R crown -0.96 -0042 -0.80 -1.16 0 3/4 point -0.97 -0.41 -0.86 -1.16

46

The influence of the velocity may also be studied by comparing

solutions for which the velocities of the pulse differ but t·he total time that

the pressure acts on the arch is constant~ In the above table this amounts to

comparing column (1) with column (3) and colunL~ (2) with column (4). In the

first comparison the time that the pressure acts is 3.0 T J and in the second, o

it is 1.5 T Q

o

On the basis of only four solutionsJ it is not clear which of these

methods of comparison is mere desirable or whether a comparison on some other

basis might not prove to be superior.

5.203 Effect of Arch Dimensions. The effects of a variation in the

dimensions of the arch for a triangular moving pressure were investigated in

the same manner as for a uniform all-around pressureD Solutions were obtained

for slenderness ratios, L Ir, equal to 50, 100 and 200 for a value of f/L o 0

equal to 0.2, and rise-span ratios of 0.1, 0.2, and 0.5 were considered for

L Ir = 1000 The load parameters considered for each of these solutions were: o

polp cr = 1, tt/To = 1, td/tt = 10

The results for L Ir equal to 50 ~~d 200 are presented in Figs. 5.10 o

and 5.11; for L Ir = 100, the response curves are given in Figo 501. A com-o

parison of these curves reveals no basic similarities in the response character-

istics. In the following table are listed the absolute maximum radial

displacements, axial forces, and moments for these solutions 0

L o

r

50 100

, \200

Absolute

w max

R2

/AE Po

4.51 4~90 6.07

Maximum Values

N M max max

:PIr 0 pEr

0

1035 2.42 0071 1063 0069 1.60

It can be seen that the maximum displacements increase with increasing values

of L Iro A substantial decrease in the maximum axial force and maximum moment o

occurs for an increase in L Ir from 50 to 100, but very little change is , 0

noticeable for slenderness ratios of 100 and 2000 This large variation in

47

moment for the smaller values of L Ir was also noticeable for the uniform all-o

around pressure results . However, It will be recalled that for the uniform

pressure solutions there was no change in the maximum axial force with changes

in L Ir .. o

The response curves for values of flL equal to 0.1 and 0.5 are o

presented in Figs. 5.12 and 5.13, and those for flL = 0.2 are given in Fig. o

5.1. The effect of an increase in the value of the rise-span ratio appears

to be the same as for an increase in the value of slenderness ratio. This is

the same result which was found to be valid for the uniform all-around pressure

solutions. A similarity can be noticed in the response features of the two

solutions shown in Figs. 5.10 and 5.12 for which the parameter fir is equal

to 10. This similarity is not as pronounced as for the uniform pressure;

nevertheless, it is quite apparent. Also, there is some resemblance in the

trends observable in the solutions shown in Figs. 5.11 and 5.13 for which fir is equal to 40 and 50, respectively. These results appear to substantiate the

observation made in Section 4.2.6 to the effect that the two parameters flL o

and L Ir may be replaced by the single parameter, fir. o

5.204 Effect of Inelastic Action. To illustrate the effect of

inelastic action, two solutions were obtained using an elasto-plastic stress­

deformation relationship for the flange areas. Two different yield levels

were considered. The arch dimensions and loading are the same as for the elastic

solution presented in Section 5.2.1. For this ~lastic solution, the maximum

stress developed in the arch is equal to 1097 p RIA and the corresponding strain o

is equal to 1.97 p R/ AE. The maximum radial displacement is 4.90 P R2 / AE • o 0

In Fig. 5.14 are presented response curves for a yield level prescribed

by the parameter

This corresponds to a yield stress in the fla.ri.ges, (Jr' equal to 1.0 poRIA or

approx~ately one-half the maximum stress obtained for the elastic case o

Included in this figure are the radial displacements, axial forces, moments,

and average strains for the 1/4 pOint, crown, and 3/4 point. The strain is

shown only for the flange for which the maximum value is largest. At the 1/4

point the largest strains occur in the top flange, whereas, at the crown and

48

3/4 pointj they occur in the bottom flange 0 In Figs 0 50 14a . and b are shown

also the curves for the corresponding elastic solutionQ These results cover

an interval of time of only 1" 4 T because of the large amount of computer o

time reCluired to obtain the solutiono This particular problem was run on the

ILLIAC for a total of one hour and forty minutes before the solution was dis­

continued.

The second t1inelasticTl solution was obtained for a yield :J.,evel given

€

R7~ = 005 Po

by

This corresponds to a yield stress equal to approximately one-fourth the maximum

stress obtained for the elastic system 0 The results are summarized graphically

in Figo 5,,150

Of major interest is a comparison of the maximum values of the two

"inelastic" solutions with the corresponding elastic solution. The absolute

maximum values for all three solutions are listed in the following table:

Absolute Maximum Values

€ € 0' W v N M

P R7AE max max max max max max

p R/AE P RIA R2

/AE R2

/AE PH p Rr 0 0 0 Po Po 0 0

Elastic 1097 1097 4~90 2032 0.71 1063 1.0 2091 1.00 6023 2044 0064 0.78 005 8.56 0050 14.82 . 3.51 0.50 00·47

For E = 100 p R/PJ5 it can be seen that the maximum strain is 48 percent larger y 0

than the corresponding strain in the elastic sys·tem" The maximum inelastic

strain is 1,,91 times the limiting elastic strain. Of course, the maximum stress

is equal to the yield stress, cr = 100 P RIAo It is of interest to note that y 0

the maximum radial displacement is only 27 percent larger than that for the

elastic solution 0 For € = 005 p R/AEJ the maximum inelastic strain is 1601 y 0

times the limiting elastic strain, whereas the maximum radial displacement is

3.0 times the corresponding displacement for the el.astic system.

The values of the maximum strain at each of the joints of the

analogous arch are tabulated below for both inelastic solutions. These results

are expressed in the form € Ie: Q The values in parentheses do no·t repres.ent . max' y true maxima; they are the largest computed values for the interval investigated.

Maximum Strains J €mru/€y

€ €

. R7AE = 100 P R7AE = 0,,5 Joint Po 0

top flange bottom flange top flange bottom flange

0 -0 .. 60 -0 .. 60 -4,,71 -4071 1 -0·99 -0 .. 77 -6.66 -5041 2 -2.20 -0050 ( -14 .. 33) -1·90 3 -2·91 -0 .. 36 -17,,13 -1.36

4 -2 .. 44 -0 .. 63 -6e77 ... 4000 5 ( -1 .. 81) -0088 -1 .. 23 -3·39 6 (-0.76) -1014 -1,,32 -~'o16 7 -0055 -1026 1022 -3.42 8 -0 .. 54 ( .... 1068) -1079 -1·96

9 ( 0.66) -2084 -3,,59 -3 .. 71 10 -0 .. 38 -2017 -2072 (-9.47) II -0.56 -1.28 -5020 -9.83 12 -0 .. 70 -0 .. 70 -8 .. 70 -8.70

It can be seen that for € = 0.5 p R/.AE yielding extends over the complete arch; y 0

even for e = 1.0 P R/AE yielding covers all of the arch except for sections y 0

close to the supports.. For both solutions the maximum strains occur at the

quarter point" It may be noted that the strains in the top and bottom flanges

are unequal over a major portion of the arch indicating the inrportance of the

bending effects"

50

Vlo SUMMARY

601 Summary

601.1 General 0 This investigation was concerned with the computation

of the response of arches subjected to time-dependent forces 0 A method of

analysis has been developed which is applicable to both the elastic and post­

elastic ranges of deformation 0 In this analysis the'continuous arch is approxi­

mated by a discrete framework consisting of a series of rigid bars and flexible

joints. For the computation of the response in the inelastic range, the cross­

sectional area of the arch is considered to consist of two concentrated flange

areas connected by a thin rigid web. The stress~deformation relationship for

each flange is represented by a bilinear diagram 0 The equations of motion for

the analogous framework are solved by use of a step-by-step method of numerical

integration.

Computer programs have been prepared for obtaining numerical solutions

on the ILLIAC, the high-speed digital computer of the University of Illinoiso

These programs are applicable to two general classes of problems~ (a) circular

elastic arches subjected to a uniform all-around pressure having any timewise

variation, and (b) arches of arbitrary shape subjected to a triangular moving

pressure pulse. For the latter case response into the post-elastic range of

behavior may be evaluated'~ The effect of an initial out-of-roundness may also

be considered in the case of the uniform all=around pressure.

Numerical solutions were obtained for two=hinged circular arches of

uniform cross-section subjected to a uniform all-around pressure having a

pressure-t~e relationship represented by a triangle wlth an initial peak and

also to a triangular-shaped moving pressureo Observations based on these results

are summarized belowo Unless otherwise indicated, these comments refer to

response in the elastic range of behavioro

60102 Solutions for Arches Subjected to a Uniform All-Around Pressure 0

The major aspects of the response for arches subjected to a uniform all-around

pressure can be explained in terms of the participation of the first three

symmetrical modes of vibrationo The mode which participates most significantly

is the extensional mode of vibration for which the period is close to the

"breathing" period of a complete ringo' The predominance of this mode can be

51

seen most clearly in the history curves of the axial force. This force is

essentially uni~or.m throughout the arch.

In spite of the predominance of the extensional mode, the bending

effects are significant, particularly for the smaller durations of loading.

For durations, td

, less than about 0.75 times the "breathing" period, T , of a . 0

complete ring, bending contributes roughly 40 percent of the total maximum

stress for a cross-section composed of two concentrated flange areas, i.e.

for clr = 1, and about 60 percent of the total stress for a rectangular cross­

section for which clr, = f3.. For load durat·ions of the order of td/To = 2 or

4, the contribution of bending is about 25 percent for clr = 1 and about 40

percent for clr = ~

As may be expected, the ab~olute maximum axial force increases with

an increase in the duration of the pressure pulse. The same is generally: true

of the maximum values of stress and displacement, altho~ rhe increase is not

continuous as in the case of the axial force. For values of td/To in the

range between 2 to 4, the maximum value of the radial displacement is of the

order of 2.5 to 3.0 times the static deflection produced in a complete ring by

a uniform pressure having a magnitude equal to the peak dynamic pressure" The

maximum axial force is approximately 1.8 times the static,value in a complete

ring, and the absolute maximum stress is of the order of 2 or 2.5 times the

static value for clr = l,and 3 times the static value for clr = ~.

For the range of parameters considered, the maximum effects were not

significantly affected by variations in the magnitude of the peak pressure and

by initial imperfections in the arch of the order of w /L = 0.001. This m 0

observation may not be valid for longer durations of loading.

The effects of variations in the slenderness ratiO, L Ir, or the o

rise-span ratio, flL , on the maximum effects were not o ' large for the limited

IDlmber of c&ses studiede An increase in the value of L Ir has essentially the o

same effect as an increase in f/L. It appears that the parameters f/L and o 0

L Ir can be incorporated into the single parameter (f/L ) (L Ir) = ~/r. 000

6.1.3 ~olutions for Arches' Subjected to a Triangular Moving Pressure.

For the case of an arch subjected to a triangular moving pressure, the major

contribution to the response appears to be due to the participation of the

first antisymmetrical mode of vibration.

52

Although the c;l.bsolute maximum effects in the arch occur at sections

away from the crown; the effects at the crown are also appreciable. For the

solutions presented, it was found that the maxL~um radial displacements and

bending moments at the crown are from 30 percent to 90 percent of the corre­

sponding absolute maximum values of these ~uantitieso The axial force remains

relatively uniform for all sections along the arch 0

A compar:Lson of the, maximum: effects, for a moving pressure and a

uniform al.l=around pressure both of which have the same magnitude and duration

shows that the largest displacements and moments occUr for the moving pressure,

whereas the maximum axial force occurs for the uniform pressure. For pressures

which act on t,he arch for a total. time of from 2 to 3 T j the maximum displace= o

ments for a moving pressure are of the order of 2 to 3 times those for a uniform

press~TeJ and the maximum moments are of the order of 105 to 2 times the corre­

sponding values for a uniform pressure ~ On the other hand, the maximum axial

forces for the moving pressure are about one=half the corresponding maxima for

a unifor!1! pressureo It follows from these results that the bending effects

due t,o .a moving pressure are more significant than for a uniform all-around

pressure pulse.

For durations of lo.ad t d of' the order of 2 T 0.1' it was found that the

buckling tendencies are appreciable for high magnitudes of pressure. For a

value of p /p = 12 the maximum displacements were found to be about 50 o ·cr

percent larger, and the maximum moments about 25 percent larger than would have

been the case had the buckling te~dency been neglectedo For shorter durations~

however.? the maximum effects are not influenced significantly by changes in

the magnitude of the pressure even for large 'Values of p Ip . 0

o cr

The effects of changes in the values of the slenderness ratio.? L Ir, o

were found to be the same as corresponding changes in values of the rise-span

ra.tios f/L" Since this relationship between the sl.enderness ratio and the o

rise-span ratio is the same as that observed from the results for a uniform

all-around pressure~ it offers additional assurance that the significant

structural parameter may be fir 0

Because of the' limited number of solutions obtained, it was not

possible to make any ,definite observations concerning the effects of the

velocity of the pressure pulse or the effects of inelastic action 0 However,

a few example solutions have been presented which illustrate, to a certain

ex-tent.? the influence of these variables 0

1. Brooks, Digital 19550

2. Clough, Science

53

REFERENCES

J. A., "Solution of StructuraJ. Mechanics Problems on High Speed Computing Machines", Ph.D. Dissertation, University of Illinois,

R. W., "Elastic Stability of Arch Ribs tt, Thesis for Doctor of

Degree, Massachusetts Institute of Technology, 1949.

3. Eppink, R. T., and Veletsos, A. S., ttAnalys:Ls and Design of Domes, Arches, and Shells, Volume II: Analysis of Circular ArChes", AFSWC-TR-59-9, Air Force Special Weapons Center, Kirtland Air Force Base, New Mexico, July, 1959.

4. Newmark, N. M., ttVulnerabili ty of Arches - Preliminary Notes tl, Department

of Civil Engineering, University of Illinois, 21 May 1956 (Unpublished manuscript) •

5. Newmark, N. M., t1A Method of Computation for Structural Dynamics ft, JournaJ.

of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, Vol. 85, no. EM 3, July 1959, ppm 67-94.

6. Manuals -Corps of Engineers, U. S. Army, "Engineering and Design; Design of Structures to Resist the Effects of Atomic Weapons - Arches and Domes", EM lllO-345-420, January l5, 1960.

54

APPENDIX Ao UEXACT~t DEFORMATION~FORCE RELATIONSHIPS FOR INELASTIC BEHAVIOR

The purpose of this section is to formulate ,the equations for deter­

mining the actual deformations of t:Q.e individual' flanges in terms of O. and x'.o J J

From the geometry of the deformed joint j shown in Fig. 205, it can

be seen that

t b

c/ ~ tb c. It = dO. + ~o

J • "J d. j J J

t b

o~ ~ rb c . + 2 r,.,

=: d 0; + dO. J i v j J

v

where d. denotes the distance between flanges. Application of Eq. 2~2 yields , J the following relationship:

t b c j-l o~b d. 1 J-l J-

"c j-l rt + -- 0 +

d. 1 j-l J~

b c. it to. =

j J O.

J

An analogous procedure is followed to relate the flange deformations

to X. 0 Wi th Xl. and x.~ denoting the angle changes to the left and right of J J J

joint j.9 respectively, the following expressions are obtained~

Xi. 1 (=c/b + ol.t) =: d. J

J J J

r ..l. (~o~b + rt Xj = OJ ) d. J

J

Since, Xi.. r

X. = + X. J J J

it follows that

= X. J

(A.2)

Two additional relationships are obtained by conSidering the equilib­

rium of the jth bar and the jth joint. The forces in the flanges are indicated

55

in a manner similar to that used to identify deformations. Thus, ~t denotes J

the force in the top right flange at joint j. Considering the equilibrium of

bar j, Qne obtains the equation,

If.bl

+ ~tl = J- J-

If the small moments Q5 produced by the shearing forces on either

side of a joint are assumed to be negligible, the following equilibrium

equation is obtained for the jth joint:

b b t t w.c. -rfc. J J J J

(A.4)

It is ~ortant to notice that the forces appearing in Eqso (Ao3) and (A.4)

are actually functions of the deformations 5~~ 5~, etc., which are to be

evaluated. These forces are related to the deformations through the stress-

deformation relationships and the following equations:

N1.b

= (/bA~ J J J

N£.t = (ltA~ J J ~.

(A .. 5) N~b = o~bA~

J J J

However, in order to express them in terms of the corresponding deformations,

it is necessary to know the region of the stress-deformation curve in which

the deformation is located. In general this region is determined from a know­

ledge of the past history of the deformations •

. For a structure with z bars, there is a total of 4z unknown flange

defor.mations, four at each intermediate joint and two at each of the end joints.

By application of Eqs. (A.2) and(A .. 4) to each joint and Eqs. (A.I) and (A.3)

to .;each bar, one obtains 4z -2 equations.. The boundary condition equations are

specialized forms of Eqs.. (A .. 2) and (A .. 4) .. For example, with a hinge at joint

0, M is equal to zero, and Eqo (A.!t) becomes, o

If joint 0 is fixed, E~. (A.2) becomes

1 (_orb d 0 o

+ =

Thus one obtains 4z .e~uations which relate the 4z unknown deformations to O. J

and X.o The solution of these equations results in a complete determination J

of the changes in length of the flanges.

Once the flange deformations have been calculated, the corresponding

forces can be computed by use of the stress-deformation diagrams. The total

axial force, N., is found from the expression on either the left or right side J

of Eqo' (A.3), and the moment M. is determined by use of the left or right J

side of Eq. (Ao4).

As stated above, the computation of Nj

and Mj from the quantities

o. and X. involves the solution of a set of simultaneous equations. The J J

computer time required for just one such solution may amount to several minutes.

For a dynamic response analysis, these steps are repeated several hundreds of

times. Consequently, the time re~uired to effect a complete solution of the

problem makes this Uexacttf method entirely impractical on present-day digital

computers such as the ILLIAC. .

57

APPENDIX B 0 EQUATIONS FOR CIRCULAR ARCHES IN POLAR COORDINATES

The e~uations presented in this section are for the coordinate system

shown in Fig. 3.1. They are derived in a manner analogous to that used in the

main body of the report, and they are numbered so as to correspond to the

equations given in Chapter II.

B.l E~uations of Motion

The e~uations .of motion for mass m. in the radial and tengential J

directions are

m . .y . = Z". + W. J J J J

(B.2.4)

m.v. = Y. + V. J J J J

(B.2.5)

The quantities Zj and Yj

represent the components of all internal forces at

joint j acting in the positive w. and v. directions, respectively; W. and V. J J J J

represent the components of the external forces at joint j acting in the

positive radial and tangential directions, respectively. The quantities W. J

and V. denote the radial and tangential accelerations of the jth mass. J

By considering the equilibrium of joint j and by summing the com­

ponents of all internal forces first in the radial direction and then in the

tangential direction, one obtains the follOwing two equations:

+ Q. cos( ~ - 'lr j ) - Q. 1 cos( 92.+ "'j+l) J J+ 2

Y. = - N. cos( ~-1jr.)+N·l cos( ~ + 1\rj+l) J J J J+

\

~ - 1\rj ) - Q. 1 sin ( ~ + 1jr. 1) - Q'~ sin( J J+ J+

(B.2.7)

For the case in which the analogous arch is subjected to a pressure

which remains normal to the surface, the components of the external forces

acting in the radial and tangential directions are given by the equations,

58

(B.2.l3)

For a uniform all-around pressure p,

p~ = p~ = -00 5 pL. J oJ

Bo2 Displacement-Deformation Relationships

In terms of the radial and tangential displacements, the quantities

5. and f. are expressed as follows: J J

The expre s s ions for the quanti tie s given by Eqs. (20 l4b) and 2 .15b) are the

same for both coordinate systems. In addition, one finds that

1 + '6./L J

cos w-. =

J V~2 + (1 + '6. /L) 2 J

~ *.(1 - BoiL) J J

".. 1 -! ~ 2 'i'j

59

APPENDIX Co MODAL METHOD OF ANALYSIS

In the modal method of analysis, the radial and tangential dis­

placements of the arch analogue are expressed by the following e~uations:

where

w. = J

~ (r) -\r) f (t) 1 c wJ r r=

v. = J

-(r) th w . = radial displacement of joint j for' the r mode of vibration J

-(r) th v. = tangential displacement of joint j for the r mode of vibration J

f (t) = r

th dynamic amplification factor for the r mode

(r) th c = participation factor for the r mode

n = number of degrees of freedom of the system.

The participation factors are given by the e~uation,

(r) c . =

where w is the circular na~J~~l frequency corresponding to the rth mode, mo r J

is the concentrated mass at joint j, and W. and V. are as defined in Appendix J J

B. For a uniform all~around external pressure of intensity p, V. is equal to _ J

zero and W. is equal to -pL cos ~2Q The amplification factor f (t) is a time J r

function which depends on the nature of the forcing function. For a step

pulse of infinite duration,

f (t) = (1 - cos w t) = (1 - cos 2~ Tt ) r r

r

60

The bending moments, M.~ and the axial forces, N., are determined J J

in an analogous manner by use of the following equations~

N. J

M. = J

n

where N(r) is the axial force in bar j for the rth natural mode of vibration

and M.(r) is the corresponding bending momento J

The values of the symmetrical modes of vibration and the corresponding

natural periods for the arch described in Section 40201 are listed in Table

Gel. These values were determined from the equations given in reference [3].

Also included in this table are the values of the participation factors, c(r),

for the particular modal solution mentioned in Section 402020 Th; modal

response coefficients presented in Table 4.2 for w.~ NoJ and M. are the products

of the participation factors and the correSPOndingJVal~es of w~(r), N~r), or -(r) J J M. j respectively.

J

61

TABLE ~4 .. 1 N.ATURAL PERIOm OF A !fWO-HINGED UNIFORM CIRCULAR ARCH

f/ L :: 0 .. 2, L / r := 100, z:: 1.2 o 0

Tabula.ted values represent the ratio T_/! , where T is the rth natural period and To is given by ibe°ex;pre •• 1oK

Vii f;4

T = 2W ~ = 0.04555 --2-. 0 EA EI

Order AntisymmetricaJ. Symmetrical of Modes Modes

Period

1 4.996 2.225 2 1 .. 204 1~066 3 0.592 0.784 4 0.444 0.465 5 0.389 0.342

6 0 .. 313 0 .. 297 7 0.164 0 .. 240 8 0.104 0.126 9 0.080 -0.090

10 0.069 0.073 11 0.064 0.066

WI w2 "'3 ( 1 ) = w4 p R2/AE w5 0

w6

~ ~I (~) = N4 Po N5 N6

~I(p ~) =

~ 0

M6

TABLE 4.2 MODAL SOLUTION FOR A TWO-HINGED UNIFORM CIRCULAR ARCH Uniform All-Around Rectangular Pressure Pulse of Infinite Duration

f/L = 0.2, L /r = 100, z = 12 o 0

0.3568 -0.4113 -0.3045 -0.0344 -0.0065 -0.0006 0:4601 -1.1729 -0.1949 0.0170 0.0093 0.0011 0.2270 () -1.5008 () 0.1483 () 0.0242 () -0.0066 () -0.0015

1 ()

-0.2270 f1 t + -1.3086 f2 t + 0.2020 f3 t + -0.0313 f4 t + -0.0001 f5 t + 0.0019 f6 t + -0.6588 -0.7406 -0.1116 -0.0100 0.0065 -0.0021 -0.8346 -0.4385 -0.3310 0.0343 -0.0094 0.0022

-0.0494 -0.8955 -0.0613 0.0000 0.0006 0.0001 -0.0506 -0.8843 -0.0623 -0.0015 -0.0001 0.0000 -0 .. 0482 () -0.8674 () -0.0731 () .. 0 .. 0042 () -0.0004 () 0 .. 00001 () -0.0426 f1 t + -0.8601 f2 t + -0.0833 f3 t + -0.0035 f4 t + -0.0003 f5 t + -0.0001 f6 t + -0.0362 -0.8678 -0 .. 0802 -0.0029 ... 0 .. 0011 -0,,0001 -0.0320 -0.8776 -0.0714 -0.0052 -0.0008 -0.0001

-0.2125 -0.2087 0.3530 0.0733 0.0191 0.0Ql9 -0.2824 0.1983 0.1991 -0.0318 -0.0271 -0.0036 -0.1862 ) 0.5248 ) -0.2459 ) -0.0536 0.0192 0.0052

1 0.0165 f 1(t + 0.3171 f 2(t + -0.3173 f 3(t + 0.0657 f 4(t) + -0.0000 f 5(t) + -0.0063 f 6(t) + ••• 0 .. 2117 -0.2210 0.0907 0.0197 -0.0192 0.0071 0.2914 -0.5131 0.3637 -0.0157 . 0.0272 -0.0013

f (t) = amplification factor for rth mode, a function of the pulse shape r

0\ ro

T.A'llLE 4 .. 3 MAXIMUM VALUES OF RESPONSE AND TIMES OF OCCURRENCE FOR A TWO-HINGED UNIFORM CIRCULAR .ARcH . Uniform All-Around Tr1angular Pressure Pulse W1 th In! tial Peak

r/L s 0.2, L /r = 100, P /p = 1.0, td/T = 2.0, z = 12, 6t/T = 0.01 o 0 0 cr 0 0

Joint-or Bar w t -1L. :L .1L t a t a t

j R2/AE 1'0 poR To PaRr T p RIA T p RIA To Po 0 0 0 0- ,

c/r = 1 c/r = 2

1 ... 1.076 0.41 -1 .. 712 0·50 0.762 2 .. 08 -1.814 0.44 -2.401 1 .. 59 2 -1.959 0.47 -1·700 0.50 . 0.585 2·77 ~2.180 0.47 -2 .. 703 0.45 3 -2.384 0·53 -1 .. 684 0.50 0 .. 934 1.61 -2.222 0.54 -2.813 1.60 4- -2.'277 0.58 -1.678 0·50 0.856 1.64 -1.890 0.61 -2 .. 635 1 .. 63

I

5 -1.93~ 0.52 ... 1.679 0·50 0 .. 457 1.12 ~~.e52 0·52 -2 .. 030 0·53 6 -1.874 0.49 -1.679 0 .. 51 -:6:136 1.66 -2.068 1 .. 64 -3.191 1.65

TABLE 4.4 STMIC EFFECm IN A TWO-HmGED mlIFORM CIRCULAR ARCH FOh A UNIFORM ALL-AROUND PRESSURE

r/L = 0.2,. L /r = 100, z == 12 o 0

Joint w. v ,

or Bar R2/AE R2/AE ..1L .J!...... poR pRr 'j Po Po 0

1 -0.400 . -G. 101 ..().995 0.025 2 -0.780 -0.152 -0.994 0.047 3 ":1.1l0 -0.159 -0.994 0.063 4 -1.363 -0.128 -0·994 Q,,016 5 -1·523 -0.071 -0·994 0.083 6 -1.518 0.000 -0.994 0.086

~

TABLE 4.5 EFFECT OF TIME INTERVAL OF INTEGRM'ION ON MAXIMUM VALUES OF RESPONSE AND TrnES OF OCCURRENCE Two -Hinged Uniform Circular Arcbj Uniform All-Around Triangular Pressure Pulse With Ini tis! Peak

r/L = 0.2, L /r = 100, P /p = 1.0, td/T c 2.0 o 0 0 cr 0

Join"t or Bar w t N t M t fJ t a- t

j R2/AE T P R T pRr T P RIA T PoRIA T Po 0 0 0 0 0 0 0 0

c/r:: 1 c/r = 2 (a) z :c 10, 6.t/To = OJ095

1 ' -1·311 0.42 -1. lED 0.50 -0.884 1.69 ... 1.946 0.44 -2.615 1.67 2 "'2.211 0·50 -1 .. 701 0.50 0.565 0.52 -2.258 0·51 -2 .. 822 0·51 3 -20326 0·56 -1 .. 686 0 .. 50 : .1~O72 (1.69 -1.965 0·58 -2 .. 948 1.66 4· -1.999 0.52 -1.683 0.50 , ___ ,0.228 0.94 ... 1.802 0·52 -1.924 0·52 5 -1.914 q·50 ... 1.683 0.52 .. 1.149 2.57 -1'.865- 1 .. 66 -2.893 1.68

(b) z = 12, 6.t/T :I 0.005 0

,. 1 -1 .. 076 0.42 -1·713 0 .. 5c1 0.763 2.08 -1.814 0.44 - -2.401 1.59 2 -1.9,9 0.41 -1.699 0.50 -~-o" 586 2.77 -2.181 0.46 -2 .. 702 0.45 3 -20385 0.54 -1.684 0.50 0.934 1.61 -2.222 0.54 -2 .. 812 1.60 4 .. 20278 0 .. 58 -1.678 0.50 0.857 - 1.64 ... 1.891 0.60 -2 .. 636 1 .. 62 5 -1.936 0 .. 52 -1.619 0.50 0.451 1.12 -1.852 0·52 -2 .. 031 0.53 6 -1 .. 815 0.48 -1.619 0.51 -1.131 1.66 -2.07G 1.64 -3.200 1.64

(c) z = 12, /j.t/T = 0.02 0

-1.076 0.42 0 .. 761 2.08 -1.814 0.44 * 1.58 1 -1·713 '0·50 -2"212,, 2 -1 .. 957 0.48 -1.699 .0 .. 50 (0.581 2.78 -2.181 0.46 ':'2 .. 561-'" 0.46

-3 -2 .. 384 0.54 -1.684 0.59 0.934 1.62 -2.221 0.54 -2 .. 6,1 0.54 4 -2 .. 277 0.58 -1.679 0·50 0.856 1.64 -1.890 0.60 -20406 1 .. 62-5 -1 .. 937 0·52 -l.679 0.50 0.458 1.12 -1.851 0·5~ -1 .. 979~ 0.54 6 -1.874 0.48 -1.671 0.52 -1 .. 133 1.66 -2.062 1.64 -2.,883 1.64

'-------~---- ~--- ---------~-~- ---~-.-----~----- '----------~.- .. - ... --~-.--- -~----- --- ---- ---

*stresB values in this column are tor c/r =~o ..

0'\ ,fr

.TABLE 4.5 (Continued)

Joint w t N t M t or 2 T p R T pRr T

Bar J p ... R lAE 0 0 0 0 0 0

, (d) z = 16, tiT = 0.005

0

1 -0 .. 802 2 .. 66 -1 .. 709 0.50 -0 .. 546 1 .. 59 2 -1 .. 531 0.44 -1..101 0.50 0.610 1.94 3 -2.090 0 .. 48 -1 .. 681 0.50 0 .. 560 2.59 4 ... 2 .. 374 0.53 -1 .. 611 0 .. 50 -0 .. 912 1.10 5 -2 .. 391 0.57 -1 .. 614 0.50 0.923 1.54 6 -2.144 0.56 -1.615 0'.50 0.296 1..48 7 -1 .. 863 0.50 -1 .. 615 0.50 0 .. 123 1.12 8 -1.810 0.46 -1 .. 615 0.50 1 .. 020 1.14

(e) z = 16, tiT = 0.01 0

1 -0.801 0.40 . -1 .. 709 0.50 0.547 1.94 2 -1 .. 531 0.44 -1 .. 700 0.50 0.609 1.94 3 -2 .. 090 0.48 ... 1 .. 687 0.50 0 .. 561 2.59 4 -~,,374 0.53 ":'1 .. 67$ 0 .. 50' ... 0 .. 912 1.10 5 -2 .. 391 0~57 -1 .. 674 0.50 . 0.924 1.54 6 -2 .. 144 0.56 -1.674 0.50 . 0.296 1.47 1 -1 .. 863 0.50 ~1.615 0.50 0 .. 723 1.12 8 -1.810 0 .. 46 -1.616 0.50 1.018 1.13

(r) z = 20, tiT = 0.005 0

1 .... 0.638 0.40 -1.708. 0.50 -0.549 1.51 2 -1.250 0.42 -1 .. 703 0.50 -0.596 1.52 3 .-1,,775 0 .. 46 -1.692 0.50 0.638 2.56 4 -2.158 0 .. 49 -1 .. 682 0.50 0 .. 555 0.46 5 -2.366 0.53 -1.615 0.50 -0.884 1.08 6 -2.435 0.56 -1 .. 673 0.50 -0.838 1.10 7 -2.288 0 .. 56 -1.671 0.50 0.532 1.49 8 -2.010 0.54 -1 .. 672 0.50 0.340 1.02 9 ... 1 .. 810 0 .. 48 -1 .. 673 0.50 0 .. 851 1.12

10 -1.759 0.46 -1.674 0.50 1.152 1.12

* Stress values in this column are :for c/r = 13 .

(J' t (j t P RIA T p RIA T

0 0 0 0

c/r = 1 c/r ::r: 2

.... 1.840 0 .. 56 -2 .. 141 1 .. 60 -1 .. 994 0 .. 44 -2 .. 393 0.40 -2.208 0 .. 48 -2 .. 745 . 0 .. 41 -2.199 0 .. 52 -2 .. 161 1 .. 59 -2.087 0.60 ... 2 .. 814 1 .. 54 -1 .. 191 0.44 -1.964 0 .. 43 -1.894 0 .. 54 -2 .. 200 1.56 -2.148 0.58 -2 .. 921 1 .. 58

-1.840 0.56 -2 .. 135 1 .. 60 -1.994 0.44 -2 .. 393 0.41 ... 2 .. 208 0.48 -2 .. 145 0 .. 41 -2.198 0 .. 53 -2 .. 166 1 .. 59 -2.086 0.61 -2.817 1 .. 55 ... 1 .. 791 0.44 -1 .. 964 0 .. 43

'· ... 1.893 0 .. 54 -2 .. 205 1 .. 56 -~.149 0.58 -2.919 1,58

-1.849 0.54 -1 .. 986 * 0 .. 56 -1.864 0.44 -2.015 0 .. 40 ... 2.089 0 .. 46 -2 .. 418 0.44 -2.220 0 .. 48 -2 .. 623 0.48 -2.161 0.52 ... 2 .. 573 0 .. 60 -2.185 0 .. 58 -2 .. 653 0 .. 60 -1.944 0.51 -2,,226 0 .. 60 -1 .. 754 0 .. 46 -1 .. 855 0 .. 42 ... 2.001 0 .. 55 -2.298 0 .. 59 -1 .. 319 0 .. 60 2 .. 116 1.12

----- -- ------- ---- - ----- - ------ ._- -~--

0\ Vt

TABLE 4. 6 COMPARISON' OF SOLUTIONS BY NUMERICAL nrrEGRNrION PROCEDURE AND MODAL METHOD Two-Hinged Uniform Circular Arch; Uniform All-Around Rectangular Pressure Pulse of In1:iD.ite Duration

r/L = 0.2, L Ir == 100, z::: 12 o 0

W' N .1L --

F?/ AE PoR pRr 1-'0 0

t/To l/4-point crown crown 1/4-point crown

Numerical Modal Numerical Modal Numerical Modal Numerical. Modal Numerical Modal Method Method Method Method Method Method Method Method Method Method

0.1 -0.191 -0.191 -0.190 -0.191 -0.190 -0.190 -0.004 -0.003 0.000 -0.001 0.2 -0.708 ~0.7a8 ' -0 .. 692 ...Q.692 -0.633 -0 .. 634 -0.017 -0.017 -0.001 -0 .. 001 0 .. 3 -1.424 -1.424 -1 .. 331 -1.331 -1.200 -1.200 0 .. 056 0.056 0 .. 003 0 .. 003 0 .. 4 -2.135 -2.135 -1.878 -1.878 -1.675 -1.675· 0.296 0.297 -0.009 -0.009 0.5 -2·552 -2.551 -2.128 -~.128 -1.888 -1.888 0.518 0.518 -0 .. 138 "'().139

0.6 -2.476 -2.473 -2.046 -2.046 -1.811 ·1.~10 0·532 0.530 -0·399 -0.400 . 0·7 -1.940 -1.935 -1.842 -1.845 -1.433 -1.430 0.406 0 .. 404 ... 0.477 -0·" 475 0 .. 8 -1.llO -1.105 -1·770 -1·715 -0.9.46 -0.944 0.191 g~l88 ... 0.097 .... 0.090 0.9 -0.201 -0.195 -1.858 -1.864 -0.485 ... 0.482 -o~116 ... 0.181 0.487 0.493 1.0 0.507 0.511 -2.024 -2.029 ... 0 .. 216 -0.217 -0.618 -0.622 0.911 0.914

1.1 0.726 . 0·727 -2.226 -2.230 . -0.232 -0.235 -0 .. 876 -0.871 1.120 1.122 1 .. 2 (0·330 0.326 -2.383 -2.4ll . -0 .. 472 -0 .. 474 -0 .. 762 -0·159 . 1.055 ' L053 1.3 -0·552 -0·558 -2 .. 402 -2,,400 -0.899 -0 .. 901 -0 .. 320 -0.316' . 0.643 0.638 1.4 -1.586 -1·592 -2 .. 'Z{6 -2·273 -1·332 -1.~4 0.231 0.235' 0 .. 058 0.054 1.5 -2 .. 392 -2.395 -2.061 -2.0511 -1.680 -lit 8~ 0.646 ,·0.646 -0.480 -0.483

1 .. 6 -2.693 -2.691 -1.832 . -1.829 -1.811 -1.816 0.756 0·753 -0.186 .~:It 1·7 -2.426 -2.421 -1.678 -i. 677 -1.617 -1.675 0.588 . 0.584 -0.619

1.8 -1·731 -1.724 ~.564 -1.$62 -1.340 '-9..334 0 .. 290 0,,287 -0 .. 054 -0.045 1.9 -0.843 -0.835 ... 1·320 -1·311 .... 0 .. 853 -0.848 -0.078 -0,,082 ·~o. 423 0.423 2.0 ....().·O72 ...Q.068 -0.902 -0 .. 890 -0 .. 415 -0.415 ... 0 .. 420 -0 .. 421 ~o" 531 0 .. 524

- - ----- - _._---_._--- --- ~---.--.----- --~---

L ____

~

TABLE 4" 6~ ( Continued)

i

'W .1L M

R2/AE Po poR PoRr

tiT 1/4 ... point crown crown 1/4-point crown 0

Numerical ~ 'MOd.u Numerical. Modal Numerical Modal Numerical Modal Numerical Modal. Method ' Method Method Method Method Method Method Method Method Method

2.1 0 .. 248 0.246 -0.477 -0.466 -0.136 -0.134 -0.459 -0.453 0.392 0 .. 385 2.2 ' -0.006 -0.014 -0 .. 233 -0.226 -0.127 -0.128 -0.140 -0.131 0.152 0.146 2.3 ..0.667 -0.675 -0.253 -0.253 -0.419 -0.421 0.237 0.240 -0.173 -0.171 2.4 -1.466 -1.473 -0 .. 547 -0.554 -0.894 -0.902 0.469 0.467 -0 .. 493 -0 .. 493 205 -Q,,161 -2.164 -1.075 -1.088 -1.446 ' -1.454 ,JO.561 0·557 -0.603 ... 0·591

2.6 -2.54l -2.539 -1.6tt-8 -1.6>92 -1.820 -1.821 O.5M, 0·511 -0.435 -0.425 2.7 -2 .. 497 ... 2.490 -2 .. 124 -2.134 -1 .. 912 -1.909 0.389 0.386 -0 .. 137 -0.129 ~ 2.8 -2.042 -2.031 -2 .. 248 -2.251 -1 .. 677 -1.668 0.249 0.248 0 .. 123 0.126 2 .. 9 -1.280 -1.267 -2 .. 044 -2.0~3 " ... 1.184 -1.179 0.050 0.045 0:238 0·235 3.0 ~.438 -0.426 -1.698 -1.697 -0.647 .-0.642 .. 0.190 -0.193 0 .. 222 9.218

'-----~ ________ . ____ L-_ -.~--.---- ~~- -~--- I

TABLE 4. 7 EFFECT OF WAD DURATION ON MAXIMUM VALUES OF RESPONSE AND TD1ES OF OCCURRENCE Two-Hinged Uniform Circular Arch; Uniform All-Around Triangular Pressure Pulse With Initial. Peak

f/L :I: 0 .. 2, L /r = 100, p /p =: 1.0, ,Z = 12, 6t/T = 0.01 o . 0 0 cr 0

Joint or Bar w t N t M t a t a

j R2/AE T P R T p Rr T PoRIA T PoRiA Po 0 0 0 0 0 0

c/r = 1 ' c/r :: 1{3

(a) td/TO = 0.25

1 ~O.623 ~ 3.00 -0 .. 675 0.34 -0.480 1.41 1.038 1 .. 93 1 .. 374 2 1.069 3.00 -0 .. 610 0.34 -0 .. 447 2.98 (1.083 } 3.00 1 .. 406 3 -1 .. 296 1.42 -0.669 0·33 0 .. 700 1.43 -1 .. 196 1 .. 42 -1.107 4 1.127 1.91 -0 .. 672 0.34 -0.552 1.87 1.05l 1 .. 90 1 .. 445 5 -Q.q9l- 2.54 -0.674 0.34 -0 .. 342 1.41 -0 .. 831 1 .. 44 .-1.078 6 -0 .. 820 2.61 -0 .. 672 0.35 -0 .. 757 1 .. 47 -1.248 1 .. 44 -1 .. 799

(b) td/To = 0.5

1 ~O.958) . 3000" -1 .. 128 0.42 -- 0 .. 783 1.95 1.125 2.00 2.271 2 1.587} 3.00 -1 .. 119 0.42 (-0.655) 3·00. -1·513 0.40 ( 1.972) 3 -2 .. 195 1.51 -1 .. 1l1 0.42 '1.150 1 .. 51 -1.975 1 .. 50 ... 2.815 4 1.818 2.00 . -1 .. 110 0.42 0.195 1.55 1 .. 607 1 .. 99 -2.181 5 "'1.444 2.63 -1.1l2 0 .. 41 -0·597 1·52 -1.437 1,,51 -1.873 6 -1.534 2.69 -1.110 0.41 ";'1.320 1 .. 56 -2.149 1 .. 53 "'3·110

(c) td/TO = 0.75

1 -0.880 0 .. 31 -1.360 0.46 0.853 2.01 1.887 2.06 2.486 2 -1.612 1.59 -1 .. 347 0.lt6 (-0.658 ) 3·00 -1,,178 0.43 ~2.112

3 -2.390 1.57 -1.337 0.45 1 .. 183 1.56 -2.106 1 .. 57 -2.972 4 1.930 2.07 -1.336 0.45 0.841 1 .. 59 -1.764 1 .. 58 -2 .. 378 5 . -1.682 2 .. 71 -1·332 0.45 -0 .. 629 1.57 -1.578 1.56 -2 .. 038 6 -1.816 2·15 -10326 0.45 -1 .. 472 1 .. 61 ... 2.415 1·59 -3 .. 489

--

Values in parentheses do not repn'i8~t. :8. true: maximum. They are the largest computed values tier interval investigated.. '

t To

1 .. 92 2 .. 99 1 .. 42 1.89 1.45 1.45 m

1 .. 99 3·00 . 1·50 1 .. 53 1·51 1.54

2.05 0.42 1 .. 57 1.58 1.57 1 .. 60

TABLE 4 .. 7 ( Continued)

Joint or Bar w t N t M t

j PoR2/AE T pR T pRr T 0 0 0 0 0

Cd). :f1d/T. ~ '1 , . 0 -

1 -0.953 0.39 -1.492 0.48 0.749 2 .. 02 2 -1.720 0.44 -1 .. 480 0 .. 48 (-0" 5721 3.00 , -2.062 0·50 -1 .. 466 0.47 1.0lB 1.58 4 -1.935 0 .. 54 ~ -1.463 0.47

I 0.735 1 .. 60

5 -1.679 0 .. 48 -1.461 0.47 -0.534 1 .. 59 6 -1.645 0 .. 46 -1.455 0.48

I

-1·Zf7 1.63

( eo) f, dl To ~ 1. 5

1 -1.033 0 .. 41 -1.636 0.49 0.813 2.04 2 -1.876 0.46 -1.623 0.49 0 .. 532 0.40 3 -2.272 0·52 -1.608 0.49 O~920 1.59 4 -2 .. 157 0·56 -1 .. 602 0.49 . 0·799 1 .. 62 5 -1.846 0·51 -1.603 0.49 0 .. 451 1.11 6 -1·795 0.48 -1.601 0.50 -1.150 1.64

(t) .td/T~ ~ 4

1 -1.145 0.43 -1.833 0·52 0 .. 831 2.13 2 -2.091 0.48 -1.819 0·52 0 .. ~74 0.42 3 -2.564 0·55 -1 .. 804 0·52 I 0.957 1.64 4 -2.576 1,,64 -1.800 0 .. 52 I 0 .. 952 1.67 I

5 -2.079 0 .. 54 -1.801 0·52 0 .. 464 1.14 6 -1·999 0·50 -1.802 0·52 I

-1.ll7 1.69

C1 t PJflA T

0

c/r' = 1

1.60'7 2.09 -1.929 0.44 -1.93:2 0 .. 51 -1.580 0 .. 57 -1 .. 601 0 .. 49 -2.109 1.62

-1.736 0.43 -2.093 0.46 -2 .. 121 0·53 -1. 78~~ 0·59 -1. 76~) 0·51 -1 .. 843 1 .. 62

I

-2.1.6J4- 1.63 -2 .. 320 0.48 -2 .. 381 0 .. 55 -2 .. 25~5 1.65 -1.989 0.54 -2 .. 426 1 .. 67

~

P·.JVA 0

c/r :3 if3

2 .. ll6 -2.281 -2.559 -2.077 -1.744 -3.043

2.185 -2.463 -2·517 -2.085 -1.889 -2 .. 680

-2.745 -2 .. 715 -2.984 -2 .. 945 -2·129 -3·239

t T,

0

2.08 0 .. 43 1 .. 59 1 .. 60 1 .. 60 1.62

2 .. 07·· 0.44 0.54 0.64 0·52 1 .. 63

1.63 0.47 1.63 1 .. 66 0 .. 54 1 .. 67

0\ \0

TABLE 4. 8 COMPARISON OF APPROXIMATE AND EXACT MAXIMUM STRESSES Two-Hinged Uniform Circular Arch; Unifonn All-Around Triangular Pressure Pulse With Initial Peak

r/L = 0.2, L /r :: 100, p /p :c 1.0, Z:sz 12 o 0 0 cr

I

c/r; sa: .~ c/r = 13 td

a a a (J (J

amax max 10

max 'fo max max % max '/0 Location PoRIA p RIA P RIA PoRIA p B/A T p RIA Error 0 Error 0 Error 0 Error

0 o . Eq. (a) Eq. (b) Eq. (a) Eq. i(b)

0 .. 25 1/4 point -1 .. 196 1·372 14.7 0.970 -18.9 -1 .. 707 1.884 .10.3 1 .. 386 -18 .. 8 crown -1.248 1.429 14 .. 5 1 .. 012 "'18 .. 9 -1·799 1.983 10.2 1 .. 473 -18.1

b: 5d 1;4 point -1 .. 975 2.260 14 .. 4 1.598 -19.1 -2.815 3·102 10.2 2 .. 280 -19.0 crown -2.149 2.430 13·1 1.725 -19·1 -3.110 3.396 9.2 2.542 . -18.3

0·75 ·1/4 point -2 .. 106 2.509 19.1 1.771 -15.6 -2·972 3·315 13.6 2.441 -17·9 crown -2 .. 415 2.798 15.9 1.981 -18.0 -3.489 3.386 ll.l 2.874 -17 .. 6 a

1.00 1/4 point -1 .. 932 20473 28.0 1.176 ... 8.1 -2 .. 559 3.218 25.8 2 .. 502 :~2.2

cro'Wll -2 .. 109 2 .. 732 29.5 1.936 -8.2 -3.043 3.667 20.5 2 .. 641 -13 .. 0

1·50 1/4 point -2 .. 121 2 .. 521 18.9 1.841 -12 .. 9 -2·511 30194 26.9 2 .. 259 -10:3 crown ... 1.843 . 2·751 49.3 .1.971 6.9 -2.680 3·593 34.1 2.556 -4.6

2.00 1/4 point -2.222 2.613 11.6 1.921 -13 .. 5 -2 .. 631 3 .. 295 25·2 2 .. 330 -1l.4 crown -2 .. 070 2 .. 816 36.0 2.028 .. 2 .. 0 -2.~3 3.639 26.2 2 .. 581 -10 .. 5

4.00 1/4 point -2 .. 381 2 .. 759 15.9 2 .. 040 -14.3 ... 2.984 3.460 16.0 2 .. 448 -18.0 crown -2.426 2.919 ro.!i 2.120 ';"12 .. 6 -3.239 3·137 15.4 2.644 -18 .. 4

71

TABLE 4.9 EFFECTS OF PEAK PRESSURE AND INITIAL OUT-oF-ROUNDNESS

Quantity

wJ

Po R2/AE

3l pR..

0

i pRr

0

a

~ 0

elr ~- 1

C1

i-hx 0

c/r = V3

ON MAXDruM V AWES OF RESPONSE Two-Hinged Circular Arch; Uniform All-Around Triangular Pressure Pulse With In! t1al Peek

tiL = 0.2, L Ir ~ 100, td/~ = 2.0, z = 12 o· 0 - 0

P /p = 0 .. 5 o cr ..., /p :::: 1

o cr po/Per = 2 Joint

V W or Bar

w W 'W m m m 2!! = 0 m

j L= 0 r;-= 0 L::: 0.001 L r;-::: 0.001

0 0 0 0 0

1 -1 .. 040 -1 .. 076 -1.136 -1 .. 235 -1.508 2 -1.892 -1.959 -2.003 -2.131 -2.337 3 "'2 .. 323 -2.385 ... 2 .. 613 -2.535 -2.913 4 -2 .. 270 ... 2.278 ... 2.469 -2.277 -2.720 5 -1.948 -1 .. 936 -1.971 .... 1.917 -1.950 6 -1.873 -1.875 ... 1.874 -1.881 ... 1.881

1 -1.698 -1. ~13 -1.726 -1.743 -1.756 2 -10685 -1.699 -1·713 -1·729 -1.742 3 ... 1.669 -1.684 -1.696 -1.714 -1·725 4 -1 .. 664 -1 .. 678 -1 .. 687 -1·707 -1.716 5 ... 1 .. 666 -1.679 -1 .. 685 -1·703 -1.708 6 -1 .. 668 -1.679 -1.681 -1.699 -1·700

1 0.743 Op763 0 .. 832 0.957 1.018 2 0 .. 539 0.586 0 .. 667 0.832 0 .. 936 3 -0.893 0.934 1.060 1.123 1.278 4 0 .. 777 0.857 0.967 1 .. 005 1.148 5 0·525 0.457 0.495 -0 .. 454 ... 0.528 6 1.065 -1.137 -1.135 ... 1·329 -1.326

1 -1.780 -1.814 ... 1.832 -1.896 -1.914 2 -2.128 -2.181 - -2.205 -2.310 -2.337 3 -2.181 -2.222 -2.250 -2.317 -2.349 4 -1.894 -1.891 -1.917 ~1.852 -1.968 5 "'1.822 -1.852 -1.855 "'1.921 -1.925 6 -1.993 -2.070 .... 2.066 .... 2.167 -2.164

* -1 2.336 -2 .. 2l2 -2.336 -2.449 .... 2.593 2 "2.620 -2 .. 561 -2.592 -2.751 -2.786 3 -2·711 -2.631 ... 2.783 -2.794 -3.058 4 -2·527 -2.406 -2.599 -2 .. 553 -2.807 5 -1.979 -1.979 -1.991 -2.095 -2.107 6 -3.003 -2.883 -2.890 -3·139 -3.134

* . . Stress values in this column are for c/r = 2.

TABLE 4 .l~ EFFEC'r OF ARCH DIMENSIONS ON MAXIMUM V AIDES OF RESPONSE AND TIMES OF OCCURRENCE Two-Hinged Uniform Circular Arch; Uniform All-Around Trilimgular Pressure Pulse With Initial Peak

p /p = 1 .. 0, td/T :: 2.0), Z == -12 o cr 0

Joint or J3ar w t N t .lL t a t a

J PoR2/.~ T p R T pEr T PoRIA To P RIA 0 0 0 0 0 0

c/r == 1 c/r :: if3 (a) tiL: 0 .. 2, L /r == 50, 6t/T == 0.01 o 0 _0

1 -I .. 23)+ 1.45 -1.130 0.-54 1.166 1.45 -1 .. 780 0.54 -2.270 2 -1·191 1.45 -1.112 0.54 1.646 1.44 -2.133 0·50 -3·011 3 -2 .. 209 0.49 <-l.695 0.55 1.138 1.44 -2.296 0.48 ... 2 .. 788 4 -2 .. 45:2 0·55 -1.686 0.55 0.403 0.44 -2.042 0·53 -2.312 5 -2 .. 673 0.64 -1.684 0.55 1 .. 091 ]~.94 -1.948 0.65 2.314

, 6 -2.841 - 0.69 -1.681 0.54 1.569 1.94 -2 .. 041 0.68 3 .. 163

(b) f/L = 0.2, L- /r = 200, ~~tLi' = 0.01 o o· "- - 0

1 -1.453 0.43 : -1,,136 -. 0.49 -0.891 ~~. 73 -2 .. 044 .0.45 -2.31.6 2 -2.169 0·51 -1.126 0.49 .. 0.749 ~~. 72 -2 .. 126 0·52 -2 .. 438 3 -2.064 0·53 -1.717 0.49 0 .. 622 ~!.10 -1 .. 120 0·52 -1.819 4 -1:916 0·50 -1·710 0.49 0 .. 893 ~~ .. 23 ... 1.763 0·51 -2 .. 338 5 -1.909 0·50 -1 .. 105 0.50 0 .. 269 1.39 -1.110 0·50 -1 .. 115 6 -2.443 1.63 -1 .. 103 0.50 1.;185 1 .. 13 -1.'898 1 .. 64 ... 2.681

(c) r/L .== 0.1, L /r = 100, b.~t/T = 0 .. 004 000

1 -1.160 1.42 -1.134 0.54 1 .. 225 1.41 -1-.811 0.54 -2.J1JO

t I

T 0

1.47 1.45 0.47 0·50 1.94 1.94

0..43 0.54 2 .. 66 1.63 0 .. 50 1.67

1 .. 43 2 -1.670 0.48 -1·73l. 0.55 1.665 1.40 -2 .. 149 0.49 -2 .. 964 -1.41 3 -2.226 0.50- -1.729 0.55 1.091 1 .. 41 -2 .. 291. 0.48 -2 .. 764 0.46 I

4 -2·529 0.56 -1.729 0.55 0.393 01.43 -2 .. 100 -0.55 -2 .. 372 0·54 I

5 -2.823 0.66 -1.129 0.55 1.198 1.89 -2 .. 059 0.66 ... 2.418 0·70 6 -3·01~) 0.10 -1.731. 0.55 1.101 1.88 -2.171 0·70 3 .. 1.05 1 .. 88

"-

~ M

T.A:BLE 14 .. 10 (Continued)

Joint or Bar w t N t M t C1 t a t ----

poR/A PoRIA j P R2/AE T pR T PoRr To T T 0 0 0 0 0

0

c/r: 1 c/r = V3 (d) -r/L = 0.5, L /r = 100, b.t/T = 0.02

o 0 0'

1 -1.686 0.46 -1.799 0·50 0.605 2 .. 48' -2.148 0.48 -2.434 0.46 I

2 -2.ll.'7 0·52 -1·755 0.50 0.539 2.80 -1.931 0 .. 52 -2.090 0.54 I

3 -1.926 0·50 -1.118 0.48 -0.802 2.74 -1·739 0·50 -2.068 2.68 4 -1.863 1·52 -1.682 0·50 -0.461 2.12 -1.692 0.50 -1.623 1.52 5 -1.858 0·50 -1.659 0·50 0·538 2.86 -1.664 0.50 -1.669 0.50 6 -1. 8~S3 0.50 -1.656 0·50 0·159 2.28 -1.663 0.50 1.851 2.18 -~ -.~-- .. -~------.~ L ____

-~~---.-- ~- ------

\;i

TABLE C .. 1 SYMMEl'RICAL MODES OF VIBRATION FOR A TWO- HnlGED UNIFORM CIRCULAR­ARCH AND- PARTICIPATION FACTORS FOR MODAL SOLUl'ION

r/L = 0 .. 2, L /r :::: 100, z :::: 12 o 0

r = 1 r :::: 2 r == 3 r = 4

T/To 2.225 1.066 0.784 0.465

- - - - - - - -j Wj v. w. Vj Wj Yj Wj Y

j J J

1 -1.179 0.096 -0.630 -0.134 -2.212- 0,,084 .. 2.098 0,,133 2 -1.520 0 .. 288 .... 1.643 - -0.162 .... 1.415 0.257 1 .. 041 0.188 3 -0.750 0.453 -2 .. 299 -.0 .. 080 1 .. 077 0 .. 210 1 .. 479 -0 .. 004 4 0.750 0.470 -2 .. 004 0.026 1 .. 467 ... 0.028 -1.913 ... 0 .. 004 5 2 .. 176 0.300 -1 .. 134 0 .. 056 -0,,854 -0,,141 -0 .. 614 0 .. 134 6 2 .. 757 0 .. 000 --0 .. 672 0 .. 000 .. 2 .. 404 -0.000 2 .. 092 0.000

(r) I--

c .... 0 .. 3027 0 .. 6528 0 .. 1377 0 .. 0164 R2/AE _Po

r = 5 r=6 r = 7 r = 8

T/To 0,,342 0 .. 297 0 .. 240 0 .. 126

- - - - - - - -j Wj Vj Wj Vj Wj Vj w- Vj j

1 -1 .. 517 0~U4 0 .. 551 -0 .. 048 ... 0.512 -1 .. 070 0 .. 132 1 .. 865 2 2 .. 152 0.071 -1 .. 078 -0.015 ... 0.197 -1 .. 826 -0 .. 126 1.853 3 -1.540 0 .. 019 1 .. 528 1° .. 045 -0 .. 015 -2 .. 101 -0 .. 241 ... 0.001 4 -0.017 0.108 -1.862 -0.014 0 .. 261 .... 1 .. 815 .... 0 .. 120 -1 .. 860 5 1.500 ... 0.019 2.094 -0.018 0 .. 432 -1.047 0 .. 121 -1 .. 851 6 ... 2.182 O.qoo -2.149 0.000 -0.505 0.000 0.242 0.000

(r) -.

c 0.0043 -0.0010 -0.00091 0.00013 P R2/AE-o _

r = 9 r = 10 r = 11

Tr/To 0.090 0.073 0.066

- - - - - -j Wj . Vj Wj Vj . 'W'j Vj

1 -0.005 ... 2.154 ... 0.040 1 ... 865 -0.034 1 .. 077 2 0 .. 144 0 .. 006 ... 0.043 -1 .. 870 0 .. 020 -1.868 3 0.000 2.154 0 .. 085 0.003 0.000 2.158 4 -0.143 -0.003 -0.043 1.867 -0.020 -1 .. 869 5 0.000 ... 2.154 -0.042 -1 .. 868 0 .. 035 1.079 6 0.143 0.000 0.085 0 .. 000 -0 .. 040 0.000

\ (r ), -c - _

-0 .. 000039 0.000015 0.000006 R2/AE Po

75

x

FIG. 2.1 MODEL CONSIDEBED IN AlIALlSIS

76

(b)

Original . Position

FIG.. 2 .. 2 FREE BODY DIAGRAMS FOR A TlPICAL BAR AND JOINT

c'

y b'

"J-I

x

FIG. 2.3 DIAGRAM USED m DETERMINDla DISPLACEMENT-DEFORMATION RELATIONSHIPS

T7

FIG. 2.4 BILINEAR STRESS ... DEFORMATION RELATIONSHIP

FIG. 2. 5 DEFORMED JOINT

78

FIG. 3.1 COORDINATE SlSTEM FOR CIRCt1I.M ARCH M:>DEL

Po

FIG.. 3 .. 2 TRIANGULAR-8RAPED MOVING PBESSURE PULSE

79 r-- -- -- -- -- ---- -- -- -- --I!IOUT IRE 11111 . l

_ ... _..........lIR(~c: ubL.I-a~1 INPUT PROBLEM I ~O "':~~W:l C:SJ:rS r Pft nO' Pml81Dll I ~ j~ L2 ::n~ ~ __ Of ~I~ ___ .~ :D~NC __ J

r- -- ------ ----I ROUTIIE (01) 1 SUMour IN[ (12) I I EVAllJA TE PRUSURE COEfF I C lENT - I

I FORENO =~: INTERVAl I I i I I

SUBROUTINE (10) SUBROUTINE (11) I

CALCULATE ~ CALCULATE DEFORMATIONS AXIAL FORCES

L ______ ~~S--I

ROUTINE (20)

fOft DID Of TUE INTERVAL

FIG. 3.:; FLOW DIAGRAM OF COMPUTER PROGRAM FOR CIRCULAR ARCH SUBJECTED TO A UNIFORM ALL-AROUND PRESSURE

I ASSUME FIRST TRIAL

VAWES fOR

I YJ' iiJ I I • ~ FOR Ell!! Of TIME !!!TE!I¥AL;~

I · I~TEGRATE fOR I ~J' ~J and £.1' ~.1

I I

r-""""""'" --- WIE (01 ,-- ---l ,. SUBRWT tIlE (12) SUIIIWT fIlE 1111 .:1

I CALCULATE"';;: CAlCULAT£ I D~~fONI DEFDRMATIOIS

I ,. I I SUMOUTIE (13)

I CALCULATE rrMUD II '" CALCULATE I

I FLAtuD AX IAl FOICEI AM) MMN'f"S

~aME~~tE~) _____ ---1

r-·iMiNEH1)~;- ----, I ... 'f1:11 TEST If J l J.... I TIME • 0 I I ,. NO I

I

r - - - -l I NO TEST f(Jt LY£Ia__ I lOUT IE (22) 1 COIVERG£ta: r-

I I"'Mour IE (21)> I 1

ExJ:~~~l,k.·1 US IIIG IMPROVED

I 1 I I ACCELERAT lOIS I p j' ~ 0 INTEGRATE f(Jt'

~ j' ~ j and e J' 11 J I I I I I

I CAlCULATE IMPROVED I NO I

ACCELERATIONS I . --'" j TEST If I e i elf I I I T tHE • 0 ~ J' J L . ,,'fEa . I I I FlU., I ______ ;.. _ :J

L- Of TIME INTERVAL J r-r--- -- -- -- -- -- ROOTINE (MU---- -- -- -- -I-- -- .....,

1 TlSI rEST IF AIIOTIO NO STOP I MOIlB4 FOLLOllS

1 YEa

r1 TEST fOR ENO' ~ TDT If RESULTS

I Of PROBLEM ME TO E PRIIlED.....- IU.MIME US)

AT THII TlfiIE CAlCUUTE fiNAL

IIADYAJU TO NEXTI ~~ 1"111 STRESSES IN FLANGES

, fOR END Of TlPE INTERVAL lu.ourarE (1-.) TIPE 1fft'9VAL

l AUNT L. CALCULATE fiNAL RESULTS 1~ .... I-----fAX fAl fOACES

AND MQMEJlTS fOR END CJ" .

T IfE IITOYAl .

'l' I SUBRMINE (1)

CAlCULATE fiNAL I DEFORMAT IONS fOft EM> Of

TIME IIIYEJIY A L I

SUMour INE (12) I '----- CALCULATE fINAL ~ fLAICf DEfORNT lOIS I

faft Ell) Of T IPE IrrERVAl

_______________ --l

FIG. :;.4 FLOW DIAGRAM OF COMPUTER PROGRAM FOR ARCH SUBJECTED TO A TRiANGULAR MOVING PRESSURE

P

Po

81

FIG. 4.1 TRIANGULAR PRESSURE PULSE WITH INITIAL PEAK

First Antis,-tr1cal Mode, T l1li 4.996 T o

Second ADtisymmetr1cal MOde, T. 1.204 T o

Third Ant1s:ymmetricaJ. Mode:l T'. 0 .. 592 T . 0

(a) Anti8~tric8J. Modes

'FIG. 4.2 NATURAL MOlES OF VIBRATION ~ A TWO-HI1I1ED,tmIFORM CIRCULAR ARCH f/L • 0.2, L II" l1li 100, z • 12

o 0 •

First Symmetrical Mode 1 T = 2.225 To

Second Symmetr1caJ. MOde, T = 1.066 T o

Third Symmetrical Mode, T = 0.184 T , 0

(b) Symmetrical 'MOdes

FIG. 4.2 (Continued)

84

(a) Displacement Configurations

I ,

/~.

"" /'

i ~ ~r\. VI at if 4 'Point' f+-j

\

W a~, crown 1\ / // , \ \ I,

'\ -.,

r/ \ II ~ V \ ) ''"'''''-~ / \ V I 1\ 1\ IjI ./ , , rF- ""'- ~v~ 1/' \ ~ ~ 1\ / " '" ,~ If' -......- '\ ~i

, . .1 v / .I ~"

If' ~I V I ..........

, t'-V J 1\. ../ .. 2

'. _"tI ~'

'-3

o 0.5 1.0 '1.5

t/To

2 .. 0 2.5 3 .. 0

(b) Radial Displacement, w

FIG. 4.3 TYPICAL RESPONSE CURVES FOB .AR ABCH UmER A UNIFORM ALL-AROUND PBESSURE PULSE

tIL m 0.2, L /r :z 100, p Ip l1li 1 .. 0, td/T • 2 .. 0 o 0 (j' cr 0

---- -r

\ / ~~V,J)f -1~-4~~~4-~v~~~+-~~-+~~~+-~~~-r~~~~~

l\

\ I , V !\

I

I j 11 v- -. ~" t,

l/ .......

1\ .\

\. j --... 1

! I

0.5

r ~ -+--+-.+--+----+--- .-.

j

: ! i !

t/To

( c) Axial Force I N

! I I I

! I --

I I

/ (M at crown

I i .-i. 1- M at 1/4 point

I r\ r-, !It'l . I . V

I 1\ 1/

\If

I \ / \

~ \ --,

1.0

/

~ J

~- I \ I ~ /~

1/ \ /' . \

~I \ l

1.5

t/To

I~ , I

\ 1I -~

I ~\ I \ ~

J /

I'-'

(d) Bending Moment, M

FIG. 4 .. 3 (Continued)

'1--

~

1\ ~,. ~.

;1 , 1---

1\ I'

I~ I ~

'.. ;' \ "-~' '" /

2.5

I---

f'"

J V

~-- --f--

f---- --f--I

- f--

r---~ ""'~ y<, II ......

I

3.0

o

-1

-,

o

-1

o

86

I

A'

",,-l\ .- I '\ "j \

/ "'- / \ ~ 1\ ~ .............

/ '\ \.

"- '-/'

M at 1/12 point

r " / '\ ./ \.

'\ '\

M at 1/6 point

/~ r-..... '" ~V "'~

/ \

. M at 1/4 point .

0.5

/ )

v

./ ",.-

/

\ \

\ \

'-""

1.0

\ , \

. \

"""~

J 7 /

j

/

'-...: ./

...........

"-

/ J

/ /

1.5

tiT o

.,."...

~

I \ / V \ /

J \ v /

...,

/1"" \. .,;""- "'" / \

./ ,;' '\ /

"'- ..",.-."

1\ \ / ""I\.. ~ / '~ \ If r"..

i\ / \ Ii

\. J "-'

2 .. 0 2.5

FIG.. 4. 4 RESPONSE CURVES OF BENDDG MOMENT IN AN ARCH FOR A UNIFORM ALL-AROUND PRESSURE PULSE

t/L = 0.2, L /r D 100, P /p = 1.0, td/T = 2.0 o· 0 0 cr 0

~

\

\ ,

"'

3.0

... ,

.. , 1

... 1

o

I-

/ ~.

) \ v \

~ r-. v \ \ \ ~

M at 1/3 point .. -

AI" ~

/v

/v r-.....

'" V V

M at 5/12 point

I if

/ /

I

~\ I \ / 1\ /r

....

M at croY.n r , r

0.5 1..0

87

j

v

"

~ \

I I

I .."

/ '"

'\ r\.

f'.-

\

\ \

1\ \ \

i\ \

1.5 tiT o

h-

l\ \

1\

\ \

./ ~ V

J

I I ~

J /

~

FIGir . 4-.4- (Continued)

.~ .......

/ '1\ 1/ \

-7 I, J 1/ I'--~

\

\ V P

..Iv .......... l\.. .A>

"\ r--... V v

...... """" ~ ~

~

/ \ If ~

1\ ... ~ \ /

\ / i\ '(

)

'\ 1/ "- ./

2 .. 0 2 .. 5 s.o

""'" ... 1

2

"' -1

-3 o

88

r I I

L- ~ ~ LA ~~ ~ ~ 16--~~~: J~ ~~ j / "'~ Ar ~

, ,,(/ :; \ I IJ ,! I ~\ /; - 20 bars \ I

\ I ~~ .

j '1 -1 " ) \ I i .~ /; 't -12 bars, 1,\ I

~, ./

\ I ~'-- L) 7 -J

" -" / : -~

0 .. 5 1.0 1 .. 5 2.0 2 .. 5 3 .. 0

t/TO

(a) Radial Displacement, 'W, at 1/4 point

J-l--~i---t--- ~-l. 1

! ' i I ! I . il!

I --~

20 bars-~ / -16 bars I .,' I ' I I , ! .>v/~ ~ fi-~o bars -""I __ I~ I I ~/y .... .. :\,\ ..........

\ i I

! r ). ..... V········ ,- I'.~ \ I I I' .. "" ~ ' "

! -1 . . ~~~"" '\~ 12 bars i .'. ,. -+-.+-t .J .... .\ ' I ~ ~ ... ~ ~~++I

\ i I~ __ ....

\ -j-fR" ~'. /~ .. ..;

-1 .::':':' .....

I \. 1 ___ i.... I !

'~~ll ! , I I I i I I

! T T ! L " I !

! I I I i I i i !

! ,--

1 , i ,

0.5 1 .. 0 1..5 t/To

2 .. 0 2.5

(b) Radial Displacement, w, at crown

FIG;, 4. 5 EFFECT OF NUMBER OF :BARS ON RESPOISE r/L::: 0.2, L /r IllS 100, P Ip a 1 .. 0, td/T == 2.0 o 0 0 cr 0

1

I

I ~

~~ ~~> 6

I ~ -' I

a.o

! 1 .-

.~ ~ &. I 1 i

1

~1"' '" ~ .. " ti.

o

! ;2 ~ ... , !\\! I I : )r ~ T i II 1~ , Ii : I-L-+-

1/ \ f/ I i ~~.~ v--Wbars 1 J

0 I ) \ , 1 ~ 20 bars.~. : ; ! Vii

~' I: I i\ !. , ~ ..

i II : I . '\\ .... :! Lit:" 1 .' I •

\ " ++\ -L._ .. ~

[1' .. i '.' : I 1 I; 12 bars : J I 1'\'. il)/

~ J .. --~-

~\ ...... 1 j~1f' I I ,./1, 'i

-' ./ I' Jaf i t· ~b~;~-V ~~:lIJ~ -C-!

I , Ij i ~""I! I I I I 1 ! I I\. : +-1 '_ ' '. -t-._J-\' I

- -

-+- !-+- ' , t-+ -- .--~.-.- ·t··- .• ---i t-t- ' , i-t-

f---I--- r- . ,. J- I I i ~.+-H- \ 1/

_.- -+_.-t---r- -- -'1-+-4 1\ I ; 1 I I i i ........

( r--' ---t"·" --t·_- , : r -+---r- -- .~--I--r-- --VI' i

. I I I iii ; i I : I ; i : I

~l

_ .• o 0.5 1.0 1..5

tiT

2.0 2.5 3·0

o

( C) Axial Force , If, at crown

i •.. -I' ----t--_··-.---

I '

--i---~ ~ at -1 I .

~I~o : ~ 1~~+_+_~4_~_r_r-r_+~~~--~~--~~~+_~~--~~~~~~~

H-+­-+-t-t ~-+---i--.-+----+--.-l-, -t.-t---+---l~+---l--+-~~

tiT o

(d) Bending Moments, M

FIG. 4.5 (Continued)

2

..........

-1

i--

· ... 3

o

'" ~ ~ ~

~ .. ,,, 10- .....

~--1-- .. - f-_.- r-

0 .. 5

90

(a) D1splacement Configurations

~ 'I

-,""' w at --" "\ croW'n,-~ / i',

// / " ~ --If ......- 1\ V ~ v '\ I"-..~ ..,.V l'"'-

I -, /

1\ /

\ / "

w ~t .1/4 point-~ f"-~

1.0 1.5

t/To

2 .. 0

(b) Radial Displacement, w

-

~/ .. .~-

~ ,/, \ ~

~, " /' V "- ,/

~" ""'" b>L~ V ,-"'"

2.5

FIG. 4.6 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--t IT = 0.25 d 0

f/L :I 0.2, L Ir : 100, po/p -..:: .. 1.0 o ~ . cr

~ ....

V

3 .. 0

9l

I

_ v--+- N at crown /

/ ~!\ I I/~ . ~!\--+- .. --- ---~-T-I1~ 1---4--4--.f--~+-_+-+-1 +--t- ~. r I I

,~ i ! II I :! \; I I 1 Ii! \ 1\ I _! II ~ r:».

0 0 .""- ; I I "I I \. / "I ! I \I' [I!

. 'l / -+ k- ... ! --t--+-+ r.,.-r)( I I

... ,

... 2

2

-1

1'\! /V:+ ! t-+,~~,.+I_t-t; +-+tl! +-\~---Vl-i-~ ~--t-:'~.--l,;L--I---!-+-+---+-+------1-";'-'-+----'--- ---t- -,- -f--r- I --'~~t-I

I "1-. :, ! +-r-ii -~. I; I ; Ii ,'+-i+-~ I i I I ii' / I i I

I i

I

1

--n' i __ ~+-ttt: I· i ++-+! t:l+-'--, . I I I 1- I ' t i

l--l--~-I---l--+--+--+---lT-. -+--- - I ! -- IT 1 - -r J--- --=.::=: I--~+--+-+-+--+--+--+--+--+--t--:r-- I I I -I I r i i -- I

0

I 1

~

i :

:

i I

I i i I 'I ' 'I I !

0.5 1 .. 0 1 .. 5 2.0 2.5

t/To

( c) Axial Force, N

I I 1 I 1 1 I I

I I I ~l T -;-! =rn-+- I I I I---+-- --t- --I---~-- . - -'-I

I i I I

I ! - --w----

+-+---I

i i M at crown -t---\

-LL! +-iY I I. ~_ V i"",

,/'1 "~~ 7 I .......... ~" i f'\.. T

...... ~ , i "-

I I I I

! I !

0,,5

~-I ,- I

r\! \. .-i\ I I I I

LI ... -'

i 1.0

~ ~t:;f3 ~ / .

1'-~-, --+ -,

I I

~ i\

I

~ :

I

~ V

1.5

t/To

I I I

~M at 1/4 pOint

c4~--t--

.-

,/ \ / ~ \!/ \ /

'f\, ~K. / i'.. ... /' ..... \'....

I

-+ 1 I

I !

2.0

( d) Bending Moment, X

FIG. 4.6 (continued)

--! ! I - --

'. ---- .--.~ ., -- _ ...... 1---

1'" V

I--V

-

--I--

2.5

3.0

--- --

~r

---+-I I

--t---~---l---I I

---1--- t--+---I I I

~~~ " .... ""'-l- ,_ ....

--

-

3.0

........

-1

·· .... 3

o

~

"- /, ~ l../ I -, /

,...-

0 .. 5

(a) Displacement Configurations

,

1/ .... 1'\ ~-/

--

/ ~ i\ w at crown-1\ .i 'I-

/ \ -" J ..... " ~~ \ V 7

~ F- V ~' \ r-...... 1\ V V:I "\

/ ,

'. / \ . !

\ :/ \ / \.

~ V w at 1/4 point~ ~ .. J -..... ... '"

1.0 1 :5

t/To

2 .. 0

(b) Radial Displacement, w

--

1\ ~ 1\ / \ \ ~' \ '-

" '\ ",/ / -, ..-" " -

2,,5

FIG" 4 .. 7 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-P.BOUlID PRESSURE PULSE--td/To = 0.50

t/L == 0.2, L /r = 100, p /p :: 1.0 o 0 0 cr

/ i

l/ J

I I

/ f7

J l7

3 .. 0

~I:oo

-1

2

I~·o X 0 ;p.

... 1

93

T

I I

I /

! iJ i

'\ I / , i\ Ii I I I \ I I il \ ! I i \ v I I ," II " if

!\

'\ / t---+--t\--+-+--t---l---4--t--+---+---+---f-;-+\~+ -1/ _L+-- - \ / , V !\

! --.... I' 'tI I IF\."

"-~ I I

i l---l---I----!---i----+--+--+--+--+-~-+--+-___1.---+___~f___+__+- -.-J.--l----l---l---f-- f-- . - ---+--+~ I

i

! i I I

r I

I I

i 10- ... J

~/ " I /', \. ;1

I .......... ~ )'. '\ / \

........

i I

I

:

i

0.5

i

t/To

( c) Axial Force, N

! I

-- f--I

/ ~ M at 1/4 point

J~

/ ~ ! \ Il-,

1\/ ,\ ,

\ ,I f\ '. 1_' \ "- ..... ,

1 .. 0

/'~ --< r\ \

I\. j \ I ~~

1.5

t/To

. .

/

r", I \

\ I ,

\1 J\ I \ / \,

"

V ",",,"

-

(d) Bending Moment, M

FIG .. 4.7 (Continued)

-M at crown --

/' ..... '"

/ " 1\ ~'

~ )

/. r\ V '~ L/

-

2.5

f--

f--~f--I :,

/: " ~

" .. '. io. ...

-,

3.0

..........

o

, , ,

r\ , ~ V

" :,/ ---

0.5

94

(a) Displacement Configurations

}I .,

\ w at crown-~. \

J

/ \ ......... ./ ..

lv V I~ 1--00.

l{ 1\

l// \ , \

1/' \

I I I t I ~. j

w a.t 1/4 point - r-/ \ v/ ~ .........

1.0 1.5

t/To

r\ ,," \

.J~ ~ f--~ /

Ii

J

I I

I

/ l/

2.0

(b) Radial Displacement, w

1-....

~ --

~ \ \ I \ l .- . ,

I ~.

\ i\ J

\ \ I/i -- --H--1\

1\ \. _ A I .'''\1\ ~i/ ! I

j V

r'\ VI f'.-.'

e

2.5 3 .. 0

FIG. 4.8 RESPONSE CURVES FOR AN .A..RCH UNDER A UNIFORM ALL -AROUND PRESSURE PULSE--td/To = 0·75

f/L = 0.2, L Ir = 100, p /p = 1.0 o 0 . 0 cr

95

I i I

...... , /~:

I I '/ "'r\ I / 1\' I i

/ \: v--I-N at crown / , / V- I ,- 'lI_ N 'tj if 1\ I -T-' (_ .. -t --+--; -'--j '--l-I-' I

I J f- --+-+--+ L--

f; --: I :\ i i I: ! ! i ~l ! I ,

i

~ i I

L.-l--L : ; , ! . il i +-~-~J-'E ' i it-! I

+E' ~ I f--+---. -, --t- .

I I ~-I Ii! : , i 1 +1 b1=~_Lt_·+- +.+ c- .' I -t- -t--'i-'- .- - ---

~+- -v+-I

I I ; 1 I . I ! I --+--+-- r- -- 1--- -+--+_ ... _.-f--+ --t---;--i\ ! I : I Ii+- ; ) iii i.~ ii, i

\ V ! ' --t-. -i- ....... V -Tf . 1\ :) I I ! : I j ~ I I

I f\l I I I if- _ 1 'III*! -+'-i-iU - <--'--'k.,...V ! T· I . , I I

i ! *1-. '-t i--',----+- ~-t-

! I I I I

I I i I I . . I ; i 1

i I I

~ I r 1 I I -I i : ; I ! ;

1 I I -2 o 0 .. 5 1.0 t.5 2.0 2 .. 5 3.0

t/To

(c) Axial Force, N

2 i : I

1 ! I :i= I I

I I +-! - ---r

J f--t-I I -- -.- r---+ .. _-_I_t-tt- . i i I

! - .. ~ _.- -1-f-- i---

----+--'-1--- - ~ --if M at 1/4 point --to -- r-"-1---

M at crown-\ I i I I ~~ . I i I , i ! ! \ ~ -t-t'- -M- ---1- . -1.+-, . ~-

il -- -- -- .- 1--

i i ! 1 I -j--.. ....; ---i It .... - .---- --+--i--,-- +---

L~~ ~-... ~ / __ 1_ ~·t/L ., / '"' ~-T

1.--

't'f~ .. - I--t--~~- ~--f---

I J/ '. j I I

:\" \ T~ --~--.-I I 1/' i '\ .I \~ " \ l I !

I I', l/\ /\. / r, IX VI -I,J -1 " /

. I II !\ I V 1\ " } I I r\. ~-\; I I'.. v \ ! i/ i\ "' 7 "" ~v -+-t r--

: I '\ I , I ,-

I 1'- i ') I

I '..,J./ I !\ I

! I \ 11 i .- -",-i---i 1 !\ ) I

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

-1

I I

-2 j

o 0 .. 5 1 .. 0 1 .. 5 2.0 2.5 3.0

tiT 0

(d) Bending Moment, M

FIG. 4.8 (Continued)

---

........

--3

o

w a.t 1/4 point -

I\. ) , ,

~' ~ I

'~ VI ~ ... / 1-

\ / , ~.

0.5

(a) Displacement Configurations

---. \

I

1/ ~

if"-

"'" "I \

\ '"-- '-"'"

-

1.0

w at crown-

~

\ ,

"".....-V~

r\

\ ~

,

"

1.5 •

t/To

/

/ II

~ '/ ~

T,

I I

/ /

2 .. 0

(b) Radial Displacement, 'W

~ V, '\ \

. --\\ \ I

I f'\ ~

rr I /1

-\~\ / J ,

~ V ~17 r"-v

2.5 3 .. 0

FIG. 4.9 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--t~To = 1.0

tiL ::: 0.2, L Ir = 100, p /p = 1.0 o 0 0 cr

1,,\

-1

... 2 o

2

! !

I

i I i

-1

[

i

97

1 I ! ! I

i v-~ I ~ r\ )

/ \: - [I ,

lr I-N a.t crown I 1I 1\1 11 ~-+-+- ~\+--=

r:---f--' . ! I

>-_~~ ____ L-..-+--/! \ ' , I i !I I I I ~ I i

I I VI t- ,

I t- ' f-1 i I ~ , I I I I i I i ! I I : l\ I 'I I

I : ! J : ! '_\1 i ~ : i '\ ~ i ! V-I I I J I I ·i-·-\ -- r--.-t-- f-T'-r-: --- -,--;+-[\1 1 !

, /i i I ; :. 1 1 I

T- ---~._r---. -....l.-- t--t--4-. -. -+- t--_ .. ·t ---+ ... ....J __ +--\ I ! !I 1 ' I I I I I: 1 I ' 1 I -~-tt I ~'if ! , I _\ :

~---T- _.- ·r-..L.-·- -l=1-T-T- -T--l-r ~ \1 i ! I ' 1 l ! i '+-1 i\. I :JA

1

I

\1 i/ I' I -I"""" I ! I! I 1 i

1. ! V I ! i I J 4- 1 I I T I· i- ' -r--r--r-

~ /i I ! -t i ! I i -+-_1_- 1 I , I _.-t- ._ •.. 1 ---t--'--!'-~ i !

i 1 I i i I I . I

: 1 I ! , --1---4-+ I !' ! I ! I

I i I , I

'-j '-I I ! ! i I I i I 1 i : i 0 .. 5 1.0 1.5 2.0 2.5 3.0

t/To

(c) Axial Farce, N

! 1

! I f---+-+ : i I I --.-

I , I , I

L+ _-1.. +- - ---r----

I I 1

I-~ t '-r' i J. i 1

.. - M at 1/4 point I--1-, i M at crovn-'" V

! , I

",.--1--, .-~-t- ~

L' f', v/! I 1\/

I'~ 1/\ '\ v

"-V

I

0.5

,

~ V "<1 I --t-.. - -'--1 t---:t\1~~

>-. -1'-I ~-+-'iJ- f-- ~ --

1- ~\-l--f-- 1--- f-_. 1-- "-j\-- ---'I

\ / ~, ~I ,1/ '\

J

v'\ I I 1\ " \ 1 , /

, i

\ Ii \ ~ ' .... ~ ..

\ II I \ I· t .I' i

'''"",' I \ I \ II

.I F'"""

j f 1 .. 0 1 .. 5

t/To 2.0

( d) Bending Moment, M

FIG.. 4.9 (Continued)

- - - -- -- 1---_. _._- --I _.+-+-+-

-- f--- .--- 1-"-- ._-r K _/ "'-I' .... / v ...... K ...... j

V ~. 1'", '\ /

--T -

i 1-.

.-1----r- -I

-~-t

-+ i

2.5 3 .. 0

""""

· ... 8

o

~ "

(a) Displacement Configurations

w at 1/4 point. ---

I ,

I

\ Vi~ , ~ ~ / I

~ '-V I

"- I " l-

0.5

// "'\

I)' \ \

I

- I'--. _ ....

1.0

w at crown-,

~\

\ V

/

/ 1\ \

.......

1.5

t/To

V

)

,I ./

v'" ~ L V

1/ ;,

2,,0

(b) Radial Displacement, :w

""...>c ........ ~ \\ r\

" \ \

\ \ ~\ i\ I

' ..... ./ )CO......

...

2.5

FIG. 4. lO RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL -AROUND PRESSURE PULSE--td/To = 1·5

r/L = 0.2, L /r = 100, P Ip :: 1.0 o 0 0'. cr

7

I J : , V

,/'

3 .. 0

99

r

I I l,...-..

v-- N at crown; 7 '\

\ 1/ : ! !':v f--1~~~~~~4-+-~~-4-+~~~+-+-~~-+~~4-~+-~~

~~i~~-'~~~/~~-+-~!-+l-~~+-+-~-+-+-~+-+~~

2

-1

jl \ II , I

~~~~,~~~~~i/~~-~i-r-

I I

i·

t/To

(c) Axial Force, N

! 1 I

I i i . . -.

I

r ,...M at crown: . -+-I I

I 11

~ M at 1/4 POin~_. I I ""'-" /. :" ..

- f-- r---

I j

1 , .,"'" ...... ~

/ " v' I',

\ ~ ./

I

i

i

0.5

I{ \ / ~

/ , / \ I I \ 1/ '1/

I,

'\I 'i \J /~\ I' J\

If \ 1,1 ~ I \. /1 \ J

I \\.. 1/' I ,

I - 1\ II '\f.-..i

1.0 1 .. S.

t/To ( d) Bending Moment, M

FIG. 4.10 (Continued)

""\ ~, 10--

~ - .... - ;...--

\ -,

'" J \ I 'K ~. /

.J \ V "

1./' r\ J '\~ V

1

2.5

--I--

V ~

~ ..... .......

3 .. 0

2

-1

100

(a) Displacement Configurations

1 f

w at 1/4 pOint- -, '\ y/

...... "' \. / \ I

--. ---

/ '\ ~ V / ~ , I

J

""'" ~ " \ I 1/1" \ "-

--~ I

I--

o

"\ ~ 1-----

\ --

~ -_.-

V I'... ./ '~

"- / -

.. 0.5

\

~ \ r-.. ,

Vv' """ j ~ -, --T I

1/ w at crown ~

1.0

..,.'" \ ..... V'" \

"

1.5

t/To

/ -

I I l,../

, /

~' /

2.0

(b) Radial Displacement, w

I \

~ \ ~

\ , if'

~- .. /l

I 2 .. 5

FIG. 4.11 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--td/To =4.0

t/L = 0.2, L /r = 100, P /p = 1.0 o 0 0 cr

I

. If

/1

1 3 .. 0

101

I I

I

I I

~I:O°D-,LI i' I l-'Lt-f i : I IV +-ii-~r-ml I i~1 cT+ I,' i /. - J-1~ i I iJt=tit, -- j(~~_ _ i i

I i ~ i i: !~r-!-- - I I-X---' -, i ' I :\1 ry ~--+JlH-+-+-t-+---;jf-t-i--t---1I---rI-rl- ~-- - V - I I ~ i j

.... , \l / I I I I \ I~ ! i_I \l V \ ,-r-f--+-t- ~! J t I! -t I L - -~I! -+-1,\ -~\.l--: -+-+-J-It'l-r-; --t-r-+-y..--::, I "R -or r-- -~H ----f -tf--=f--i --

, , I 1 -H I ~ +-r-r--- ---t-i- --+-l--~i---tl--4!,'-+~/-+:'vf-+-t--t--t--t-':-I! -II, I -+ __ ~ i l , i -H--- _-+-_

~~~,--t-i~~-+-+--j--t---t--rl--jl-r--l Ii: I i j J 1 -2 0 0.5 1.0 1.5 2.0 2.5 3.0

t/To

(c) AxitU Force, N

J i \ !: . ..!-+-+---1--+-+-+-t--r--r--r--- 1---- ---+------11 1--~~-+-+-r~4-~-T-TI-rI:

-+-'-+--+ __ .l--I--i~--+--t-+_-+--+-t--t--+--t--t--r--r--r -- --f---- -·-f----

I : ---+--1-:-t---4--+--+-t--r---t

M a.t crown -l\i ,...k M a.t 1/4 point +--1---1---+--+---1 i--t----t--+--+--t--r-! ~ V II !

~~~-+-+-+~~~4-+-+-~-rl-r~~i-t-t-t-t-I1~--~i----~--II ~-+~4-+-~-+~4-+-~-1-~~T_rl_T_j-II~,l'-1 I

1 .. 0 1.5

t/To

( d) Bending Moment, M

2.0

FIG. 4.11 (Con.tinued)

2.5 3.0

102

.. at 1/4 pOint-~ .-~. P..!

2

1

o

2

o

/.-( ,,"

/)

/ /

/ V (V /

1/ / V /1 if V o

I I

/ I :/

li,/''''' lIT Ii W

D~

I I I

V o

" .... --.. \~ V ......

.. ~

~ ,."..I. ~

" ~P'"

1--.,( r---

~ "' I

2

~d/To

----

'-at crown .

(a) Maximum Radial Displacements, w max

~ '\

...... ~, i---" ....... ..,(

~ )"'""" ""'-~

-"

I

at 1/4 pOint-

. .,.--...

.~

.. _....c )-'

,~ V ?

2

td/To

--r\

L

r\ ~_\ ,.::::::"'..--I-"'""

- at crown

(b) Maximum Stresses, C1ma,x' for c/r = 1

.. I

.-

FIG. 4.12 RESPONSE SPECTRA FOR AN-ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE

f/L = 0.2, L /r D 100, P /p = 1.0 o 0 0 cr

....... )

2 .. 0

1.5

)

7 0.5

o V o

1.5

j 7

0.5

7 o o

)

7 I

V . .,.,

/

V ',r..

10;

.... ~ ~ ~ >---

2

td/To

( C) Maximum Axi al Force, N ma.x' a.t crown

,v ~~ at crown-1\ A

7J ~ ......

7 i',

'" "-..... +-.. ,.

~ -""- .. 1---.. tp... ....... 1--4 \' ~- """" ....... -- I=--

(d)

'L I- at 1/4 pOint

2

td/To Maximum Moments, M

max

FIG. 4 .12 (Continued)

~- - ...... ........... )

2

-2

-9

2

-1

-3

"',,-"

:

o

'" ~ \

o .

104

IP/P =0.5-/G .....

\ ~ ~ VI """ " o cr \ \

! ~ 1\\ / :/ \ \ I I ,[\ II J \ I

/ /f \\ / / \ \ v

~ I I

\ V / If

\ I I

A

~ ~ II I '\ / [\--

0 .. 5

\~\ i~ ~

r- po/pcr/= 2.0

I I I I I 1.0 1 .. 5

t/'tO

V IJ

1.1 " V ~I I

",-1/

2.0

( a) Badial Displacement, 'W, at 1/4 point

• t p /p :: 0.5::' ~y ~ ~ : 0 cr

P /p :r: 2.0~ ~ V /' o cr ,

;>- ...

'"" V I / ./

"- ...... ~ ---~ ........ / / -.. - - '/

1'- __ ~

'" / /' \ - ./

'" I/~ V '" t""-~ V ~ ;

"' ~ I./v

0.5 1.0 1.5 ~ 2.0 t/'fo

(b) Radial Displacement, w, at crown

\ \

~

2.5

~ l\. ~

2 .. 5

J v

J

V I

,/ 1/ l\. ,I --: ....... ~

3.0

\ . I

\ , .... _/ ) '-.-/

a.o

FIG. 4.13 EFFECT OF PEAK PRESSURE ON RESPONSE FOR A UNIFORM ALL-AROUND PRESSURE PU'IJ3E

f/Lo = 0.2, Lo/r = 100, td/To = 2.0

'\

... 2 o

2

I

-1

i

105

I I

P /p := 0.5 I--.. .I ~ K .... ~ 1 o er ; V :/ ' \,

i-'. 1\ 1\ .. l--t- .j

I /~ ~"""I\. 1/ I i \ \ f I VI \\ Y II ! L I i 1\ " i I J [I 1\ "\. 11: I I I

~~ \ ! I iI-I . J-f- t-o_

L +--: I\! I 1// '1\ \

I-yt-_· r-_o -

j i lL.k ~-~-+ I \ I

~ l-l 'II I -\ ~ .. !: 11 - -L ~:\ 11 I - r---

I I

~! , It i \' ~ /..,'( ! I / / \1 Ii I I"'

..... :7 I

~' ~ /' "- : \

rt I r~ ,-I- po/Per

I I I =-2.0 ~

\1 It' i 1 -~ I : r-- ---

\. .I~' !

I -, ! I i i ....... ~ ....... I

! I i ! ! ,

I I I

i I j ! 01 I I !

0 .. 5 1 .. 0 1,,5 2.0 2.5 3.0

t/To

(c) AxLal Force, N, at crown

I I 1 I

: I r- .- --.- ---,

. -+-_. 1 -r-1

i po/Per == 0 .. 5-h I po/Per :: 2 .. 0 r---

~ ......... .)? ! I

J ~ I J

I I , ..........

I'\. V I \ )

~ ../

I \ J

'\, .;"

i

0.5

I ~ I / \

1- -/ ~

I

!I I

.' ,I

I

1 .. 0

,

-- r--- -

" \ , ~.

'\ , \ \

1 .. 5

t/fo

I I \ --~- f---

II \ --r--- --- f-.-

1.1: ~ , -\

i " '\

;--, I "

, ~ 1\

11 V \ \ J ' \

/ I \ / !

J

~ / ~ 1 ' ... ~'

(d) Bending Moment, M, at crown

FIG. 4.1; (Continued)

f--o_t---~-- ~--1---

l- --r-

~-- .-

V ~ V IV

/1/ \ V )' ~~ k: /~

-- -+- , ,

2.5 3 .. 0

-1

. -3

r---

"'"

-

o

106

(a) Displacement Configurations

1j1 p ........

w a.t 1/4 point :- --..... ..... 1-.. V \

~V \ / ./'" ~ / V '" \ :1 v., \\ I A I ~.

\ 1 if \ \- K" i / \ /.

~ / J'I \ " ~V / / '\ \ \ \ ~ .J- -

\. X~ I I \ .-

i\ A 7 ...........

" / , \ ~ri 7

, if' i _.

..... .;;;;,;

--.

~ _ ...• / / , .... [..1 \ '-V " / ;' I \. I

\'--' -' I

~ V w at crown-U "' K --" !\ ......

0.5

. -

1.0

....... .".,

1.5

t/To

\~ 'iT at 3/4 point f-- .

2 .. 0 2.5

(b) Radial Displacement, w

i

I / I /1)

~r.t '~Vi

I ! /'

3 .. 0

FIG. 4.14 EFFECT OF INITIAL OUT -OF -ROUNDNESS ON RESPONSE FOR A UNIFORM ALL-AROUND PRESSURE PULSE

f/L = 0.2, L Ir : 1001 P /p = 1.0, td/T = 2.0, W /L = 0.001 o 0 0 cr 0 rrf 0

107

I I

! V " I / !\

/" I- N a.t crown! 1/ r\ : i·! .. f-- r-- ·-t·_- -r---t--I---

I;"'" ~, ! / j - I . I I i ! !

! if \ I II I i 1\ ! ! j I

" I

/ i\ ! I I) i \ 1 ' I ! :

m~ -_.-+_.t-

1\ \ 'j ; 1\ i 1 I ~ '=r 1/ \j . I -:- \ f---t--r--

\~it-! -- r-ttr+- I I

.~ : / I I -t-r'J, ---I . ! i

\ 7 I :, "I ,/ I I

" ./ i i , I I I I I J --

I~I / ! I i I i : I --f--r---H--+- 1--. - f--- - -

\ II 1 I

I I ! , -+ I

I' :;.:7 I I ! 1 I I I

i !

I I .. 2 o 0.5 1 .. 0 1.5 2.0 2.5 3,,0

t/To

( c) Axial Force I N

2 ! r i I

! I ! I

~l t·-----·-_· I --'-- - .. - ~.- -~- ---. --I I

I

-+-- -t-.- _.- --I M a.t crOWIl.-1\ A: It a.t ;/4 point 4- ,.- - --,---f--- 1--

I \ V I I' j 1 .- L ~ i ' I ! I )1-._ ' ,

r .. -. t .. --.. -f-- -.. ,.---l- ' ~J] ~/l\l\t ._-~+I V-t-- h V "",,: ! I

I_t-_~ - . l ____ ___ . r---I--f---~ 1--;"- ....

~ / /i / 1\ I I ! : !~ ....... I , 'I \ 1\ ,..

~~1',~ -I--r-l

£ ........ ~~, if

1--- t--- l- - V, " \ !/ ' -~

It~-

--- ----~. v< ""'" I ~! \' XLI I i\J 1\-' ./ ~" " r\ ) .....

i I J 'f\ / I\~ Ii t ~ \ !/\ .+ tL " I 1\ \

I \ /1" )1\ I II 1\ '\ I /, \" J I ~ ! "- ) ~ /J!I \ J

\ ........ V V \ v.. ____ -

V I - \\ /1 r\ / \ / '-\ '-" I

I ~.,/ \ / -.... • J

/ ! M at 1/4 point- ,../

I ~ J

-1

; , I : I ; I

-2 o 0.5 1.0 1.5 2.0 2 .. 5 S .. O

t/To

(d) Bending Moment, M

FIG. 4.14 (Continued)

. -3

...-

loB

(a) Displacement Configurations

Ir ~w at crown :.-

V N / ~ 1-"

w a.t 1/4 point ~ / 1\ / -, I IP\ 1--. r-. ~ ~ .... "-~/ ~ "'\ V

, ,\ L I /

'r\. . / 1/\ J; , ,I \ I· I \

.~

0

.'i \ 1K., 1/ -.

~ / / \ .j I .. ' /

~ / V ",,- /~ \ V. ....... ,/ '\ "- \

'\,,,- /1 I '\ V ........... '\ f-/ J ~ r---VV

0.5 1.0 1.5 2 .. 0 2.5

t/To •

(b) Radial Displacement, w

FIG. 4.15 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSUBE PULSE - -L / r = 50 o

f/L = 0.2, P /p = 1.0, tAfT = 2.0 o 0 cr u. 0

\ \

~ \

3,,0

109

! i I i. -+--+--+--+-+-+-+-+--I----1f--~-+__.+__+! -+-:-+--+--+--+--+I---!---!I----f

! Ih I

t/To

(c) Axial Force, N

0.5 1.0 1 .. 5 2.0 2.5 3.0

t/To (d) Bending Moment, M

FIG. 4.15 (Continued)

,1

" -1

o

110

(a) Displacement Configurations

w at 1/4 point -~ 1(1 IC, ~ "" ~

'" 7 V \, l\ 1,,-

/ /" ,..'

~~ . -

.I / \~

~ / 1// \ '\ " / '~ '" / j

i\ J I \ '~ p. ...... "" .......

L~

j \ V I \

J 1/ If

~ f-. 1/

7/~ 1;;'

~ V ~.

0.5

w at crowo-

1.0

r\ )

V " -

1.5

t/To

il J

~ V

2 .. 0

(b) Radial Displacement, w

, / ~

2 .. 5

FIG. 4.16 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--L /r = 200 o .

r/L = 0 2, P /p = 1.0, t~/T = 2.0 o 0 cr Yo 0

J

1/ r

3 .. 0

-2

2

-1

III

-: N at crown- r

" h j !

\ V \. -I '\ ') I ,

I \ I 1\.

" I / 1\ 1/ \ 1/ ! \ I ---r--

\ J I \ / ~, / \ !j ,

J 1\

\ I , I J

"-:./

i o 0.5

! I

i .-I

"

I

I

!

-! M at 1/4 point-

! i ,,'

"'" .;

i

I I

o 0.5

I \ J ~ I ~

i

I i ;

1.0 1.5

t/To

(c) Axial Force, N

I ! i '

I M at crown 1'\

1\, ~l .... ,

\

'\ ..... ~ \

" ~,

......... ~

1.0

/ V

J

/ II /1

/ ,.

~-

1'-I .....

1.5

t/To

Y ""'\ ~

I \ \

'..,;>- ......

1//

2.0

\ \ ' .. \

~

\

\' .. \ ,

2.0

( d) Bending Moment, M

FIG. 4.16 (Continued)

--I""

" ./

:

J \ 'L \ /

\ V \ / ~ '"./'

2.5 3.0

-- --.

?,~ -' f' "I~ I( ~ ..

J'l \ /rl ,

/1

/ I

j

2.5 3 .. 0

112

(a) Displacement Conf'1gurat1ons

4l

2 / w at crown I /'" r....

Ir R / " \ -1 V W at 1/4 po1nt~ V ~/\ I',

_l-._ \ , ... J

> ... .. .......

r--

1

2

3

...... 0

r- ..... ~/ \ ;1 " / ,/ \ ~ / /\ i\,l '\ V / 1\ ~- Yi

.'-

\ 1/ .,~ V\ I'

I J /' ..,,' ~ ~ /1 / "-- .. ~. ~ / ~ /1 I \ / J ;

\ ~ ... V '-~ i

l':~ ./ V -

i i

O .. S 1.0 1.5 2.0 2.5

(b) ~ial Displacement" w

FIG. 4.17 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--f/L = 0.1

L Ir = 100, P /p = 1.O~ td/T = 2.0 o 0 cr 0

"

\

\

3 .. 0

'" I

\ I , I /

1\ V \ J

\ / \ i/

~ / r----II

... 2 i o 0 .. 5

2

I

/ I J I V I

j..-. J 1'1'1 1'\

'" II // 1 i 1\.

,til / .\ \

" II'

i\_ I

"-V \ r\ \

-1 M at 1/4 point....)

:

i

!

. ...2 I o 0 .. 5

113

'-I V "\.

/ ....... I ,

r--.... ~ If i ~ v \ ,

i

N a.t crown

i i I

1 .. 0 1 .. 5 2.0

t/To

(c) Axial. Force, N

! ! A- M at crown I /'~

~ VI t \

I ,

\ /- po. ... J ~ \ It·

\. I \ I

\ / \ II \ { i\ -

!\ r \ V X i\ I \ II \ 1

/ ~ \ i ./ 1\ j \ [7

lJ~ \. I [\ /1

\ II ' .... If

.~ .)

1.0 1.5

t/To 2.0

(d) Bending Moment, K

FIG. 4.17 (Continued)

I . I

A'~

V ~

/ I \ -J . \

V r\

J \ i7 \ J

\.. V

I

2 .. 5 3.0

----

,... .. \.

'/ -y\ ·1 " 1\

1/ ~\ ;/ \ \/ \ j

, X , I

[1\ ~ I -, J\ /

~ I , /

\ 7 "'"'

\ 1/

I 1\ 7 \ V

2.5

2

-3

114

(a) Displacement Configurations

I I

I I

1 I

w at 1/4 point - 1\ ! i

w at crown- ~, I ! I

I\t 1- ..... i 1 ./ A " I

,/ """ ~~ I 7 ~", ; 7' j /

io--"-I 1/ j

........

~ l{~ ,,'" 1"-

'" " -~~ \ ; ".I.-... ~~ / ~ , I

"~ ~ (' " " '-.... i-p' T " ! :/ , L;? I i

\ )'i' I ! ~! Vi \. . ~ , ~ ~ ?' I ! I

I I i I !

0 .. 5 1 .. 0 1,,5 2.0 2.5

tiT o

(b) Radial. Displacement, w

FIG. 4.18 RESPONSE CURVES FOR AN ARCH UNDER A UNIFORM ALL-AROUND PRESSURE PULSE--f/L = 0.5 o

L Ir = 100, p /p = 1.0, td/T . = 2.0 0' 0 cr 0

I . .....-//-

7

i I T

" I

i I

S .. O

115

I I +--+--+--l--+--+--+_+--+--Iif--+-_+-+--+--+--+--+--+-+-+

I--+--~- --- -- '-- --f---

! I 'r '~ I I. 1..- 1 it

t/To

( c) Axial Force I N

;

! I·! 1 t: 1- --r ~---4-+--+--+-+--+---+--+--+---+--+ -- -- +----+--+. ---I-- ---.;... - ----f-- +--1----1

i .--+----+---I-~+_-+-- -- - --'-+--I--+-.---f i: I T--: ----. M at cro"m=~-I- -1--,--- --f--f--+--I--I

I! I I! ' '\1 T

j' -+--~---t-I- -- --t- -~--r' -1--- r-- -t--~ --~ - --- --1--1---

, 'I I / N I I

~+--~-+--+--+--+---+--+--t.---~---;-- - r-rt= M a.t 1/4 point --~....... I -- -- -- i" -1 --! --

I----t-+--l--+--+--~__+_, -+--+-~-~;..r·~f( ~~ri-I Lj'F-- ~:>~\ --J--1- ·t----~-+--+--+-+--+--+i ---+--::~-t--+-" -- \--- - - T--+-

I L ....... ".. I, i" ) \ \ i : j

!

0 .. 5 1 .. 0

! 11 I ' -----t--+~-! f--t-+'---+--+--l

I I

1 .. 5

t/To

--·t-·i--f-I---1-'--- --+---+~--I--1-'-- --e-

i

2.0 2.5 3.0

( d) Bending Moment, M

FIG. 4 .18 (Continued)

a C\J. Pt°

CH 0

J ~

I ~

,. -

2 -

0

-2

-at

-s

I

w at

--

ll6

Position of Load at tiT .:: 1 .. 0 o

(a) Displacement Configurations

I

3/4 point ":i .." .. -- . ...... I" . ...;

L . - w at ", " -vat crown

---I- --- \. I, I 1--- r;- i--

~ v at 3/4 point -:-< po.-- ~ " 6.e"'" ........ ~v ~~ .-1-' 1- -I-- ;-...... "

i

crown

~f-{; V

/1 1 ~. I.,...o-C Fo"- r-- - r--+ - - i-- --

• I~ ~ r-... _' t-.- ........ (' -/. --~ 1---, '" 1/4 point / / .~ ~ v at

r--...... / /' " " r-...... /'" v 1' ....

..... '" ....... '" :/

" "", ...... v i,.-/

' ... ............ ...... .",- -t

'- - ."

"' _ ...

",' t ! ...... ........ ! ,

...... 1-.-1-- ..""'('

~

- w at 1/4- point : -

I

-.-

r\ ""'"

.,;..,J. '-./

V "-.1/ "-

~'

...... - - ' ' -r~r------..;;: r--:--...--=.·i -. _ . "- r---.~

"- . !

"

r--... .......

!

! ! ! ! ! t

-aH I I I I II I I I I I I I I I I I I I I I I I I I I I I o 0.5 1.0 1.5 2.0 2.5 9.0

. t/!o (b) Radial and ~t.1aJ. Displacements, w aDd v

FIG. 5.1 TYPICAL RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MJVING PRESSURE

f/Lo :r:II 0.2, -Lo/r ::: .1;00, po/Per z 1.0, tt/To := 1.0, td/tt == 1.0 "'

,..../'11".

I ,

I ! i -

i ~ ~

~. '",

i I I

~ ~~ i i I

, !

I

I ; . -,

-.

i o 0 .. 5

i

i !

0.5

111

I I I

~.f •• 1./4 JOint .. -L I

·N at crown-1\ I

. 1 ~ ,,~? y .-::;;;,~ -~ .. ./ 1"""" ::::;;:..

I

~.;- I , ./.

::::.. ~ V-i . ...,

! ;- .", I I I I I'

I r\'i 11 at. '5/4 .:po1D.t i : i._ ~-1 i ; I ! i

I

1.0

I

I 1.0

!

l 1.5

t/'lo

i I

( c) Axial rorce, B

i 1.5

t/To

2. 0

2 .. 0

(d) Bending Moment, M

FIG. 5.1 (Continued)

I

I I

_\ ~ ~ ~

~. r..... ~ ~ ..... ~ ~

i j -

I i I

25 8.0

2.5 3.0

llo

r r\. ./ ~ "'- ........ / "'1\.

J ,~ I '\ V ..........

" '-o .JV .... , V - ........

~ ...... .......... """'-b.. --~",

- M at 1/12 :point I I I I I I

i """,,--.....

/'f' ........

'- -!ooro..

/ , 1/ ~ j

.",",

'" / '"'- ...",.,... --.." 'f "'" /. " .......... ........... ../

" ---I M at 1/6 point

I I I I I I

I ! :

~ ........... "-~/ "-

/ , /

b' ~'" ~ ~

/ - " ~ V " I ,'\ j

r-.......

'" V " r--.... o

~ M at 1/4 point , r'\

I I I II r ,

I

./ i -F-- ......... i'oo...

/ ~~

/ \' i/ I\.

J r--r--~ ~ ..........

'" V '" o

[,-, V I \~ ',,-

- X a.t 1/5 point . "-I--

i I I I I I -1 o 0 .. 5 1.0 2.0 2 .. 5 .' 8,,0 1.5 t/To

FIG. 5.2 RESPONSE CURVES OF BENDING mMENT IN AN ARCH FOR A TRIANGULAR MOVING PRESSURE

f/L = 0.2, L Ir = lbo, p /p = 1.0, t~/T = 1.0, td/tt = 1.0 o 0' 0 cr v.. 0

.... 1

1

-1

o

... ,

o

ll9

./ ~ " V '\ / ~

/ \ V'

~ .I ~V ',-

/ ~

'\ V ,~ , V '" '-" ~ ~~ ~ , ia...-

~ M at 5/a JOint I I I I I I I

--V '\ / f\.

I "'l~

V r\. ~ / ~

\ / ",,-~'- /1/ "t\.. I

\ ~

./v ~f'\.

I« at crwn "- / -....... -'"

I I I

~, / ~

.... " \ ./

~ ......... i--"" "'~ '\. ..... v' " "-~ V

10'

" ~ 1/ ...... -i--"" r-- • at 7/a po1D't

I I I I r

t...,....-o

-~ ~ AV

~ ~ P' ...........

~ --V

~ "'.".,.. "~ V

f" / f- )I at 2/., point!

,~ If I-- ..J ...... F"""

I I I I I I

o 0.5 1.0 1.5 2.0 2 .. 5

t/'fo

FIG. 5.2 (Continued)

'" ~

........

1/

J '7

j

V

J

V /1

I

a.o

1

1

o

"

I I

I i

!

1-_._..1 Ii at I I

I 1 I

l-

I' i --..

I !

!

I

f---

-1 i I

f-- . ...;. Kat I I

o

I--N at -1 I 1

o

120

I -.

I ! I

i V"'" I"'" I ---f---

I V I TI I I

l- - ---17t-I --

.--- ._- --- --- .- .- _ ... - - .. - --.1-~ ~ I /. '.

I

" I I

) ; i '\ V I

~- V I ! I

i\ I V i I

I i I \ , / i i I i I

I I r\ I'

I / ~ +-- -1--1--\1 ! I J-- _l_~ __ - --1- p:y ..... V

--~-1--

,/4 point· -.--: ' I Iii I I j

~ I

i J

i 1 . -- --t--. -. -

~-!- -- -_. -

i-i--

i 1

! -~ ~

~ ;

--1-- --- i----- . ----1---':"-- -- - - -- -- -- -.--I

: J I /

\ I--I----t- i

I -J -- i , '{

--- I---l-~ J t .-.. . _-,/ i\ I . i I I :

\ .... ! I I J . ...,.. I

I i j'\ I /~ 5/6 point 'r

1 I~PP I I I I i

I i I I !

I I

1 ; I

! I I

i I

I ...... ~ I

I

I 1'-0... i i

~~ ./N I . ! ) i

~ ..,/'" I i~1 ! -IV i

!

I i ~ ..-V

ll/12 point I I I [ 1 I

0.5

! I

1.0

I""'-

1.5 t/To

r---.. ~ V

FIG. 5.2 (Continued)

2.0

i

i ."H..... I i/V

" -- -.-~ I- . +---- --- -. -

I I-- --l---- ---- ---f-V- ,

J, ------i V-;

/ --- --I-- --_. '-- ---f- -- --

- -.

_ . ._--'--- ._- --.-

I

v ~ r........ V ......... ,

!/ J

V I

/ -

2.5 3.0

, ! (\l. it p.0

'-4 0 2

J 0

:J

I ... 2

-it

~ .. , -8 {J

W at 3/4

121

Pas! tion of Load at tiT == 1.0 o

point r\., -1. __

.)... - . , - -tv at crown~ v ........ v • 'to.. -"'I.'\.

I --

-,... .... .J'

.... _. - .... -... I, -,,7 ~~ ----~ ,,{ .... ~ .".. . - at ....

~ ~ v crown . -.. K --I"- ....... r-.., ~ '>i 11"1'"

'" D.

r.... '" ~

/ V. .. i,.o/ I', '"

~ t'-..... ~ ~...;... -1- ~

"' r-::::= .... r--... "'- -~ ...

i--1-, "'- .... w at 1/4 point

8.5 1.0 1.5 2.0 2.5 t/!o

(b) Radial and ~tia.l nteplacements, 11 aDd v

FIG. 5.:; RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE ... -t It. 0.5

d t r/L • 0.2, L Ir • 100, P /p = 1.0, t./T • 1.0 o 0' 0 cr ~ 0

......."'-...... 100...

....... 1'-... ..... .... --.

! - '-

I

3 .. 0

132

i ,I

J ! 1 ~~~~~~~-~-+-+-+-+-+~-r-r~~~~~~~~~~~~~

2

o

-2

i I !

i : i I ! 1'-1': I! i!, I , " ! i-,-+i---1'----~!__-f__+-+_1 -+-jl--r: '-'--t--+

i -r'-r- '--+---,; -+--~ -nri--+-t-- ~t-+---:--~

1 i 1 . ,i I I iii

o

I : F--..L..

i I

! i I-'-r i I - -T

I

i

I I

....... i ,,:,- .. ! I I i

I

I I I !

I.

i

~-++-i I

0

0.5 ' .. 0 1.5

t/TO

2 .. 0

(C) Axial Force, I

I ! r 1 I

: ! I I

1 1 I 1 i 1

I ; ! i

-;tM ~t I' I t-- I I 1/4 point I I

! I I / ! i I t i - ~

! ' 1[-r ! ~' ~M at crown : J.,-i- ',i 1

/1 '~ YI ! ~ I i 1 ...... j : ....

!I I

. I +-+-- , \! J,...{-" 1/ V"I ' i i i i, I i I

; . -I .~.

; V 1 " ...... !_L -.. ~i ..J.. .<,! " / i ,-Ii \ "\, ,/t' ! I I ~ ,,1"" : I

v+- I--~--t-- i , "' r-.... I ; I I 1 __ 1·/,

2,,5 3 .. 0

! I

I !

! ! 1

-

I i I 1

1 1 -- . - .

I

; -I~ ! I"

I ; , I ~ ~ ,

'\ /'-~ : i i\ ~ 1 :X~j / . 1 ;

I " ~,/ \ I ,v j 1 "'~- i

7 1'-4--+ ...... i ! : I I i

1

i I \1 : V i i ~ I I I I I I tTY--,

I i • i • i , i 'I " ' ~

I

I ! I i "4.1\ I ! , I i I I i I' i liM at ;/4 point J. ! I i i ! i I 1 I I I i

I I

! I I I i i

I I ! I ! I I I

----L.J.. I ! i I I : 1 i i ! ! : 1 I 1 I 1 I I

1 1

I I I

1 I I

1 : I I---t-, 1 !

I I I

I -I i I I I

0.5 1.0 1.5 t/To

2.0 2.5 3.0

(d) Bending Moment, M

FIG 0 5.3 ( Continued)

• .- '-./

"", - .... , ............

'" ..........

i w at 3/4 point l/ "

I ~ "

! .. t\ap;;

0 2 PI

~ 0

I)" " / .... .... .. ...

'" "f' • 1\ j ~ ",

v .... lJ' ..... ,,' / .... ,v a.t crown / ". \.

I' I"'"'"' -\. i..--" ...... ' .' /", .... /" " ... ,/ ", \ ,,' "'" .···v V ..... ,

"" J 0

;1 -2

~ fo"':-- ........... / ~'. " ~ " / '\. V.

"'" w at crown V ): '. ", I'... Iv' / v 1\ '.

r, "< V / -, " /

1 .... -, "- )", .I' '\. -, ~ ,/ \.

1-, ........... ~ V " 1\ I \ ! ~. ;' [\

'\ '"

"

-6 " .I' '--"

:'1 at 1/4 point 11f " ..... /

" if'

,/ 'II

.........

-8 -... t-_ ,/ ...... '- ~,. --t-- --

-10 0 0 .. 5 1 .. 0 1.5 2 .. 0 2.5 1.0 8 .. 5

tl!o (a.) Bad1al and Tangential Dlapl.8ICeaenta, w aDd v

FIG. 5.4 RESPONSE CURVES FOR AN ABCH SUBJECTED TO A TRIANGULAR KlVDQ PRESStmE--tJtt - 2~O tiL = 0.2, L Ir = 100, p Ip = 1.0, t~/T - 1.0 o (f of cr " 0

~ \.)I

o -• t:o

--~ -1 0 L ____ ,_,_

tIT mill! 1.0 o

~ ~ . -......:: ~. ~ ,. ~~

0 .. 5

~ -- --'--. 1.0

t/'lo !at 200

Posl tionB ;of 'Load

(b) D:l8p]ftIIe!.Jt_

B at 1/4 po1nt- .J::. - ....... ........ - r-..

) ?/ .~

~ :--. r-......

1.~ .~ -N at crown

-

- ~

~.

.." ~ -" .. :::::-: ::==:- "'::: ~ ~ I I I

~ 7' t- -..... • -~I-" - I at ,/Ii, :point

~ --1.5 2 .. 0 2.5 8 .. 0

t/'le

( c) .A:1el. I'cree,. :I

~G. 5.4' (Continued)

>c - ~~ .:::::

8.5

A-' I\) ~

2 1 1 I 1 1 I I , I I , I

M at 1/4 po1nt-~ v ..... "'- i-"" ,-....... p...- / ", It ..- -- -,.' - 1'-~' 1', .' /'

V V ............ " ./ I-- "-JL/

1

.' V ...... 1---

'" 1' ....... I, "

l ~

. K, / / M at crown ~ ~

" V 'I' / I /

~ '1\ .1....- V I ~ "

I'" \ " / \ \ / r" .. '..'l.

o

·~o f\.. lL 1\ l\ \/ L

r-- \V "r--. / \. J r-..... / '\ V \ - f\

\ L ~ .,.V \ , ~

Vl -1

1\ f- M at 3/4.point· J \ /v-

,"""" ./ L

1/ , "

1" ..... .1/ L. --...... .' .-... 2

....... t--.• ."......

I

-90 0.5 1 .. 0 1.5 2 .. 0 2 .. 5 3 .. 0 1.5 t~o

(4) Bm11.~)1

FIG. 5.4 (Continued)

126

I ~ ---,~

~ at ,/4 POi~t ~ v'/ ".."

IV" ," /~ at 1/4 point-~ ....... "'"

/ ...............

/ I.---'

./' ..... "",,"" ~-

/" ~ Mma,x at croW'll ~ I -~ ,

{'/ I

/.~ ~ 11 ..",..,..,.

V ~ ~.V

~ .~ t-li· at crown . J

;' 1# max

I( V IY

o 0 1.0 1 .. 5 td/tt

<a) Maximum it>ment,' ~" and Maximum .Axial Force, .max

/ // W-.. , .'

o 0

.~" I I I /'"

Wmax at 1/4 point -~/ [// /"

/ ."" V

.// ..".' V / "' .........

~'. . .

./ Vmax at 3/4 point -

/ I -=-'-" , ., v at crown ~ b.---:::' ~"""" • max .F

-------~ .... .....

A f'

/;?' 1// /

~

/ ... ..... ..... ' 0··-

0.5

/Y ..... ~ ./""

.... .. '

.. ' .,. ~.,

1.0

td/tt

.. ' b V at crow-max

I I

i

I 2.0

(b) .Maximum Radial and Tangential Disp~acements, Wmax and Vmax

FIG.. 5. 5 RESPONSE SPECTRA FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE

tiL :a 0.2, L Ir = 100, P Ip == 1.0, t IT = 1.0 o 0 0 cr t 0

I I I

10

8

6

! It N--. P:(O PI

~ 2

J 0

!i

~--.; '" ,~

r:~=t==F=~=t==~~=i==t=JL=t~~i==t==~~=t==t=~=i==t=jt~ / ~~ L w at 1/4 pol,nt ---r'""C1-r/-jI/H---t-+-+-l-.l~ _: w at 3/4 point ~ .-F"= I..... / rv' ~./ " -~' v /' / -

, . / -

rTi-t-+-+-+-+----L--1 ~~ ' ..... - .. ' ............ ". I\. 1'/ -rll-t-t-l-+-+-.~-k.-7~.1"~··1 v at croWll /'" .. ......~ v~"'" /~;--... 17 i1at CrowD-r---+----L~-

.,'1/ ..... "

. _. . r-.. ~~~V;;=h~/t==t±=±=-fk/"-'f~=t~~l=t==t±~= 1'-" ", , ==t=~=t~~~=+~~~== ~~~=l==t==t~==t=~=4==t=~~~iz~/I==j~~'~"~~' ~ _ ,,"- V '."",. " Ii

I: ,,'- / , I .... ' ". 1\.'

\ .... " V! )/ '. " " !\ ......... .....V" .~"

, l./ /" '. "-I'. /"\ .... ,--" . . ... _ " l/ • '. _. ,1-0" "\.. '. -- , ................ ~

-,,..._.... '. -"\

-6

-8

-10 0 0.5 1..0

.~ -. -" --

'--' ..... '-' ,-- >--. - --:::

1.5 2 .. 0

t/To 2 .. 5

I------, --J .. -- -t.:- -1- ...., - --- ---

3.0 3.5 't.o

(a) lted1al and ~al. Displ acements I v ana. v

FIG .. 5.6 lRE8PONSE CURVES FOR AN ARCH SUBJECTED TO A TRIArr.ruLAR )l)VDG PRESSURE ... _p Ip ,. 0.001 c/. or r/Lo ::: 0.2" Le/r :&!!::lOO, tt/~~o::: 1.0, td/tt == 2 .. 0

~.

~1tt4 8) Pfo

-1 0

t~

. .....; ~ ~"' -....:: ~ l'

~ ~ ""IIIIi

~ ~

0.5

---t--':.;::= . . t-- J ...........

1.0

tIT = 2 .. 0 o

Positions of Load

(b) Displacement Configura.tions

I 1 I I I at 1/4 »oint '- ./ -l~ ~ ----

~ ...

.... - 1-,.;:::;: V ~

~ .... I-" • "- 0"'( ,/

~ 11 at ,A point ~ jt:

I-'

, I I' 1 I 1.5 2.0 2 .. 5

tiT o

( c) Axial Force, N

FIG. 5.6 (Continued)

--~ &-t- • r-. L"'""'"'" -r--+-- _ 1-1. .~

N at crow-~

3.0 3 .. 5

..... 1--F--

'-~ r--

11 .. 0

I-' I\)

CP

~~o

2

1

o

-1

-2

/ .... , ./ 'I\.

l--II----l.----\--t--t--t---t----r--+-+--+-+-+-+-+---t----t-------t--r---r- M at 3/4 point :"~", .... '\

.".~~' V

I

..-1--'...... :~. V'" ~, ~/ '~V~~ .' j I'

v~ ') -"r-- ~~' ".... V V I\. /L L r\. 1 .......... ,,.,,./ -... .... 1 ,)1 at crQwn ~~

L-L-L-~~~+-+-+-+-+-+f~~~~i-~~III\.~_t~_t_r:f~'.r_+--r-r. V .' V 1\ LO, --+--t--t----I----A--f--t--t--r-ri ~ / - ._, / \ I' \ ,/

...... I \ "/ ,\/ r'\ j ,/ I' "'\ V

1\ )~ 1,\ "-

\ LV \ iJ '\. " ~l--"lL / '~ v"

/ --, 1\ \.

\ / "

1\ L , ". L --l-."L2~~~ ~-+--f-_t__t_-r-rl

'\. ./". -f- . M at l/~ po,~n~-v , <I>

~ \

1\ " /'

~to.... / I ..... 1'-_""

-3 ., 0.5 1.0 1 .. 5 2.0

t/To 2 .. 5 3 .. 0 3 .. 5 Di.O

(4) BfmjI1ng J.tcmaent" M

FlO. 5.6 (Continued)

I-' (\) \0

130

Poalton of Load a.t tiT = 1 .. 0 o

(a) Disp.l.acement Configurations

FIG. 5.1 RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE - -p Ip == 2.0 o cr

f/Lo::: 0.2, LJr = 100, tt/To == 1.0, tJtt:: 1.0

-.;;; ~ 1' .....

-1

o

2

i "I

o ....... , I

-2

I

131

I

-* 1/4 JOint-~? .:..;:

--~

'" ~' ~ ~. ~ - :r' -p...~ ~ ...".. -~ ~ --~ I ~ ......... '~ 1-'--

~ i ~ ~~~ ~

.. ...,..,..;. ~ I·" )/J4 JeiDt -~ / - -. -"'\

i i :

0.5

i

I I i

i

/1-'/' l /

" ~' :

1/ .I

'"

1.0

N at c~1

1.5 -

t/To

2.0

( c) Axial Force.. Ii

I

-- M at l/~point L

I ' .....

"'-I""

V \ ,'-' " .. V r--- ....

1--,~/ '\

1 IL " - r...... ..

I

/ I 1;] " -'-

-"- .V ,! I V! ,/i \

.. Ai I ! ""-

r\ 'l>( ~[\ l I I

'\.. ~IL /i l\. "t M ~t . crown.; I

I

1 \ \ I

I I

I :\ I . " V , "

I •

1 i) I '/ I

. M at 3/4- point J/ '~ /' ---t -'

1 I I I '-./ I

1 I ! I

j l ! 1

I

1 I

1

0.5 1.0 1.5 2.0 t/To

(d) Bending Moment, H

FIG. ~.7 (Continued)

i

2.5

" f--

" ./ "-I-"

-, "-\ /

!\/ .. / ~

./

/ I

i

2 .. 5

~k-~ ~\ ~

I

I 8.0

--,. "'- ./

"~ ............ / " .' "\

L -........... L/

I !

I !

3.0

8

, t ~o It

PI

~ 2

r-t- ~ 1-- --t----- f-- ...- - f-. '-

W at 3/4 point V - ...... ~ , "r--.

17' f'- v r--. I /' '" I L,,' .... .... . ... '. / 1\

", "tI, " \ V \. I " / .... I /'

I " v a.t crown \ / I

I ,',' x

I 0

~

I .... 2

-It

~ -6

/ ->4( -.'- /, /

V ,.' :;" '- '" .~ / -K' ~ " / ," J'

...... " II ' '" .- .-'

" I" ~ I

~, '" I-w at crown / [;/ \ '0, -v ~ / /' " -" . I I,

"- r\. IL L

" " .-I' \ 1', I\. // '.

'" '\ j' I "- f\. /r/

......... '\ IL j \

" I'\. / / _11 "'-" "- 1/ V 1\

w at 1/4 point V ...........

" / / \ i" .....

" v V ",

"""t----1-- .... --t-' '\. -8

._ ......

---' 1--- --_ ... -

... 10 .. - - . - - -

tflg (a.) Rai!ial and ~al DiQJ~, V and v

FIG. 5.8 RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE - -tt/T 0 ::: 2.0 r/L ::: 0.2, L /r = 100, P /p = 1.0, td/tt::: 0.5 o 0 0 cr

...., ~

.~o

tAn = 1 .. 0" 'I&.0 . . tfto - 2.0

BluJ1tiaDe Olt Load

(b) D1Ql __ letIt_

I I I I I J-. .1 I I ! I I I I I 0. --;:~ I I I

;--~. ..e.. . . '", I 1 I~ I :.. ... ~.-:::.P"'" '. ~\ 1./1

J I A \\ I' I H at crown-~·· [\. ,,\ . . . . ~ \~ /' .

11 I ~ Rat'/4]101Dt-v "~~r·~ __ _ _ I I 1 I J I I I I I q 1 I I 11 I I I I I I rrrrrr I JJ -10 II I

3 .. 5 o o 2.5 3.0 0.5 1.0 1 .. 5 ~ 2.

o (e) Axial JJorce, )I

FIG. 5.8 (Continued)

...., \).I \.)I

I

I

f-- /-- --- -- .I--.~

2 I I -

i 1M a.t crown v '7

-+- ! / I --r /

M at 1/4 point- \ I_ I :,~ ....... 1// 1\

/ -'r--'" ~~ .'~'/ -\ ,

1

// V I'\. /

V V ., /', I -"

j ,/ '. / \

-I!o 0 rt- ./ ~ .1 ---... r-- ....... \. \ !-... .' --

1\ )( \ V

/ 1\ V r- M at 3/4 point

1\ V Y \ J \

-1

\

~ ./ \

~ , i 1\

1\ ,if ._-

-2

'. ['\ .

I' j 0_5 LO La; , n

t/'lc (4) Bend'. MMlleD't, M

FIG. 5.8 (pontinued)

!

-- .

...... \.

\ 1\ \ \

1\ -./

-~\' / \

~. I \ " .. ,/ \

."x I

/

j

./' I

I

J . J

I' ., e;

L /

I I'

/ II

r"\ Ii I \ I· / "":",ll

L I

\. \

1\ \ ~ \

\ \

~ n

I

/

t-,

'"

/

",

\

" \ , --1--

I /

J

.J

'.:I 11::

I-' \J.I -t="

, !

(\A •. .. Pfo

~ 0 2

J 0

r-V t--

~

I "'2

~

!C .. , -8 0

.' ..

at crown -~ -

1-'" .. , .... .. , .' .-..

r"--, --i'... !, ...........

'\ -"-

"-

" 1' .... --

135

Position of Load at. tIT ::: 1.0 o

£' w at 3/4 point -/ ~ .. w at crown V ... \1

h

"-v.: ., .... . ... ' . .,;V" ...

'"

~ ' ... ~ ", .... . .,..... ~ ... -. ./ --- ..... ~ ~:"

..-~ . r-- '/ - r-- ,1- 1'-_ ...... ~.

z " / ..... 1'-_ -'"

~/

./ ~~

''''- '-W at 1/4 po~t

-

I' - -~' " ./'

,~ .. .:.: , ....

"

I

0.5 1.0 1.5 2.0 2.5 t/!o

(b) hdi81 and ~tial Displacements, 'W aDd v

FIG. 5.9 RESPONSE CURVES FOR AN ARCH SUBJECTED TO A T'RIAIGULAR MOVING PRESSURE ... -tt/T 0 == 0 .. 5

f/Lo • 0.2, LJr::t 100, pJpcr = 1.0, tJtt. 2.0

--

-1""_ r--,/ ~

......

" '" .... ~

" , -.-

a.o

136

~~~~~+-~~r-r-r-r-r-r-~'-+-+-+-+-+-+-+I-+l-+I+1 I i 1 I,! I - ,-~ !

~--+!--+--t---+------ B I at c~ --m !. - li ar JA1JI01f .~+-I '-\"\-.t-T----;.'-'-----+r-_--+-+---_lt---_-I

i ! -LJq'f\ i iiltrh\4~\,~+___t-+----j H-- ~--L-L- I .. 'ill j ! ,~~ : i.1 I ' \~~

, Iii ,: i "f. l' ,,'- I';i !-t- s" ,I o ~~,~~~--~r-+_+_~~~.J~r-+i-+I~~'~.~~-,+_+_~~n~·~~+_+_~'~~~~_~ii~ l'~~' . ____ I 1-- //; -t!-+! '\ il/. I '\~-+-I '~ " i, i/4' , \\ " I I t ---j---r ~\ -:-+ ~r;- -~-+-t- -- - ,i\ ~ ~--+- ~-- --- - , . I --t--

~~. 1 ' I I I I ! ~ .. V7 I I I I +-+ ~ I _. __ -1. - -r' -----t--+- 'it I ' --J.-f-.... 1 - - '

, i I ~ I I he '-I!'! I' •

Iii 'I': _ 1 , I __ __ _ :I at '1-" point ~ I j i ,+' I __ ~~ j : I '~,,;!/f j l ' I. i !,! i iT !

.. , " Ii, ~. ';/',/" iii ' - !!: Ii: -1 i I I ! ' ,; :~ I ,," ,I' '--i- ' ,I i--t-tT- :'-- , i -- r-'l---r--]---r---+-- '! :---r- 1-1--- r-- i -:~-r-

o 0 .. 5 1 .. 0 1.5

t/To

( c) AxiaJ. Porce, I'

2 .. 0 2 .. 5 i.O

l I 1 J .[!! I! i ! I 1 J ! : 2 ~~+_+-~~_r_+_+~~--~~+_+-~~_r_+_+~~--~~+_+_~~_r_r__l ~_:II ii, ,!I! !Ii

: ' I! i;; I i ~ ~ 1 i ! --+---+--t-i-- -+--+-1 --r---t-- r--+-+---r--t-i I -t I I ,I I il i,'; i i 1 I j I

i I I I e-lltrM at 1/4 point: ! I i I_ Ii! Iii/I 1 i I, I iii I , : I 1

1 1 > If ' \" , I ! I 1 i i ~ _± __ , I J.-.. I I ~" i I

I \ : i-~-'_ ' i +' ! "\ 1 V' : l' JI i , ... / ". " I, I,' "'·-T'. ' I I I ./ --~- I ;, !i '\ v' t4-I I I j I, i I' I /, : '\ ! /' '\: I __ -+-_ .-;L _ I ; I .' \ ./ ~ .. l' '/ 1 i....-r--. \. i/ . i :

! ' : / ; Ii: ~;: i 1\ /1 I Ai "\. ': I

!T---L ---' I / +' -~ !~' i j /1 -Jl_ ./ : / 1 !,' / ,\:, :.1 \1 1'1 ! I :". ~ .,.... !: I './ o ~ , , ", J. ' '" j i y:

; ..... :-:. ~'r-!'~A' : 1"l./ \.'~I',~: .. L/: ~';--T--r------;-i- 'J-... ',i 1/\ ! 'v' - ~~-.Y ! ,t-

; ! '-L ;! ;'\ : " L. ~J I i ./ 1 I ---;--1 I 1\ f----+-t---t- ,-- ---T',' '--------~___t--'o--+---:-----r , \ v --t-- --, ! ' '" I ! : i' ,,'. ':, --L . \ r\. ,/; I I ' '. ' ...

~' 1-'+-+-'-' -t---.. ~- .. + I' .: ; -; , - I : ,_/ j: ; :~-+--'I ,! I :; i i j-"1Io. . 1- .L !

- 1 :--r- j --ll i : -'1~T-i;' I "- M at crown-~: i -; '--t-.. 1 , 'I ~ :! :" . .A1/ 1 l! ! I !! ! I ;: H-- I !, ,i! i i; 11'-~-M at ;/4 po~t-~- --:t i ;

! I -t-r r-T : i 1 ! i I T- - 1 1 i 1 !. , IT I : r---L-+-+-+-,- -+---L.--l.---L----t---+---i---f.-- J I i __ ~_,+- i I ! _

iii! !' I! ::; I if I i! iii I I I '-t ,I "I" " 1 Ii' '+--t----l---~I - ~---tl-i--- r-----t--~I- -, I I I, -r--r--o-----:---+--I ' : i :! :" j ,! I I ;,"

-2 : -4'-; !,!; I! I 'II , i I I i r--'r--, --+- -t--i- I j 'r--t'~,' '-- !--!--+-+-

j ,! II I t I .; J I I : i I : ! :, ! ++i 'I " I I Ii

1-_____ --+-_-,_,-+-,' -I ___ ~- -~Ii_-1-i-- ! iii -+,', --1i,~~ j I I! Iii Ii, I

o 0.5 1.0 1.5 t/To

i.O

(d) BendiDg Moment, K

FIG. 5.9 (Continued)

2 .. 5 3.0

, ! OJ. ,.

AO

~ 0 2

J 0

~ ~

I -2

~

lC ... , -8 0

....... .1/ .. '

~ .. ,

.-II. ' , v

.... ~ ~ ...... ~

'" "-.......... "-

1' ..... .......

....... ~

w a.t 1/4 poillt

137

Position of Load at tIT :; 1 .. 0 o

w at • crown ~ -- - ..;> - ~ . - .... . -.. ............ L /' 1\

':/' ~ 1.-i-' h'-, \ JI ". / .... \ at crow- I " J~. " 1/ I

. ... I . I " .

I V -' '\

~ / !I

" V ,,/

1"- V i' )- ~

...,..., ,,'

r-_

i\ \ \

"

'\.

l' ......... 1-0.. ~ w at ·3/4 point

r-- ... ........ ,..,,"

~ . ...

". \

~ "- r".... ...

.... ",... . -:

o.s 1.0 1.5 J.O 2.5 t/'to

(b) Radial and ~ti8.:tm8placement81 w and v

FIG. 5.10 RESPONSE ctJRVF.S. FOR· AN ARCH SUBJECTED TO A TRIANGULAR K.'lVING PRE~URE--Ljr • 50

tiL • 0.2, P Ip = 1.0, tt/T = 1.0, td/tt = 1.0 o 0 cr . 0

--./ --...

I' /'

I '" ., . ... . 'r

I / ~

... -~ ~ v

3.0

1)8

I--l--~I--f---If-----l---!---!--+--+-=--+--+--+--+I --+--+-+-/~V~/-+·;l.\~rt-,'T-o-\-\ +--+ ~ at ,p' J01nt-1 H \ ~

-, ~~~~~~,i~i~~i ____ r-

,1.J'I, I T I ~~ -- I, I-" ii, 1 1\ I I -It ' I

o 0.5 1.0 1.5 2.0 2.5 3 .. 0

t/TO

(C) Axial Force" I

I , ! i! 1

2~~~~~~~~~+-~+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-~+-~~

! ! :~! ! , M at crown / \

I ~ l/ \ Y iii !, ~ :L'~, I 1 7: ! i : ! o~~~~~~~~~+-~+---~+-+-~~~~+-+-~+-+-+-+-+-+-~

1- \. .1/ \ I \; / i :! :: i! \ ! ' " r( t II I : i ! i -- II--r-r-t--~j'"--r--r ! ! \! j,' i : : i +-+ I \' k.. I I I 1

I i ~/ I\! i_: ; I -tJ I'! / ' 11~ 1 ilL+--r-- ! \ i J-L .. -L: 1 I I ! '-V 1,,-/ 1\ ! I.-~+--LL-+-

[ ~ ....... 1 I ; I ! I . I-- t' i '\! : .... h. I ! i i -1 i i : ,'\: i i II i I r 1-1 if \ i : : I

I ! I \ "I' I I ". I! .;;.. I ,,:+-1---I : \ __ L I -y" +_.' -W---+- I J

~+--+--+--+--+--+--+--+-~.i'-· \ +--.J-i -+-: -+-,' .Tt-- M at 1/4 point : i--+-'-r- - ~ I ~1

iii \ I i ,/j' I I lr--:-t-1 ,-- ~~ ++--ti-~-+--+------+--'~-'-ii li-t--K-- !.~ i !-+-t--;--rr--r-- ·~--~:\t+f-

-2 i I Iii! \,'.1: ir- M a~ 3/4 point : I \1 7: ! I" ; ! ,I I I I ! I, ! ,-v I ill I ,I I I

i I I I I I I ! ~+--.+--"'"--+--+---+-1-l--+- --~-+--+-..,f--+--+--+--+-+--+---+--+--+--+--+--+--+--I----l I 1 ~! I i I j: I -] ,

0 .. 5 1.0 1.5 2.0 t/To

2.5 g.o

(d) Bending Moment, H

FIG. 5.10 (Continued)

, ! (\I~ -Plo

CH 0 2

J 0

~

I ... 2

--S ... , ... 8 0

~ 'II at

It-':':

... """ ~ ...

~,

Position of Load at. tiT :: 1.0 o

• j~. - . --I ......

'" fIJ

~

.'

-t'- .~ .. .........

" I 3/4 :Point ~ 1/ v at crow \ -,

lY \. . " .. ..... . ... .... '" .-' ....

/' .... " . . .. J" .'

••• :1" ..... ..:: --r-.......

~ w at crown

L7 ~ ~ \ ""-.. /'

" ........... r-..., ......... /' '

'. I', -- 1--1--, ."...

"- " I-" ./

...... , /' ~'-w at 1/4 point ......... .... '

"

. ....

/

0.5 1.0 1.5 2.0 2.5 t/'to

(b) Radial. and ~tial Di~ementsJt 11 aDd v

no" 5 .. ll RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVIHG PRESSURE--LJr lIB 200

f/Lo B 0.2" pJpcr :: 1.0, tJTo = 1 .. 0, td/tt. 1 .. 0

......

" ···0 0" " • .:-::-

.....-p..--

./~

, ....... - .....

a .. o

o

2

I

I-- M a.t

I

r-L .........

I

o

I

I

0.5

140

I

I !

I !

1 .. 0 1.5

t/TO

I

i i

2.0

(C) Ax1aJ. Force" 11

! ! :

i I

!

!

! I i i 1/4 pOi?t-'"\ i / 'I- /" r-.... ........ 4..,.

I

I I

1 I i I i i

I I i I ! i 2.5 8.0

!

I

i r f-M at crown I :

! .( ! i 'I

)r I , / '\~ .... -t-i, I

I I

-f-........ ," : I

I

0.5

I / : " / I i I V I I I 1

! J " i i I

/" ! I I i i

! /1 I' ~ ___ 10 I

':::::..L---I... t-- ~ ! ! / I I ..... - _. ..,............ i i

I ! ~1'.~ / I i i ,/

i !-1 i ! ~v I I I I : I'.. I

I I I I

I I .............. 1. I i I ; I

! I ....... .... '-I i

I I i 3/4Point~ i 1 ~M at

I i 1 I I I I i I 1

I !

i ;

! I ! ---t-

! I I

I I -I

I I 1.0 1.5

t/'lo 2.0

(d) Bending Moment, M

FIG. 5.11 (Continued)

'\ "..l/ ' ! \.

,\1 / I I

/

./ ~ I ·I/!· • I

~ ......

" /i ",' i ,/" ~ I , -, ,71 I

1 i 17 I

I I i

',/' I i i !

I

1 i 1 I 1 i .-

! I I

j 1 I i

I i I I i

i

2.5 3.0

,

\

~ w at

2

J u ' .... ,

t:.

It I is

-'0

3/4 point

v '/

- -'., ." "-

~, " ......... "-",

" r"-

0.5

141

Positian of Load at tiT • 1.0 o

(a) Displacement Configurations

1\ w at crawn~ ~

~ ./ r.... .~ .... / ~ '\"-

....... ",..' f .... --/

. 0·· . .. , .... .... . ... r-- . \ J "f ,~~ _'\ \

v at J / ,

'" '" crown

I" :'\ / /" .

:'\.. / "-'\ / /

'\. v it

..........

\ \

'" ..... V .t.r W at 1/4 point 1' .... " I--~ /

........ -1,,0

~/

1.5

t/'fo

2,,0

.~ .. ...... _ .... "

. ....

'\. \

.......... r\. l.,....

"' ""-

2,,5

(b) Radial. and ~ent1al Displacements, y, and T

FIG. 5.12 RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE--r/Lo a 0.1

L Ir = 100, p /p • 1.0, tt/T = 1.0, td/tt. 1.0 ,0' 0 cr 0

",..- - -f ... ••• 0 ." -;

/ "j. ...

/"" J

V V

3 .. 0

142

I J~

II! ., ! ~

I , ) \. r-~ • at ,/4 J01at -:I at 1/4 J01J!t-~ Vfi ~ V

I I~' IJ I \\ 1

I I 7 ·i\ i -. I I

1 II \ I

~ ~ I I ~~\ ./.~ ! ! ~

1 N at crown-~ 7i \ }/C1 .... I - . __ .L-

i 'I~~ i~ Vi ~\ if:" i I" ~ II' ~ . II I

I i " ~ :/1 ! I ," .d i ; i I i ~ ~ i i : ~~ p-, ~ ,

I ----i I I j i i i I

! i : i I i o 0,,5 1.0 2.0 2.5

(c)

! 1

2 I : I I

i i i I \ I

! i I I ! M at crown-~-I \

I I i /" "\ 1-1 i

I II I if , r 1-- t

--

/ t--+--f---+--+--+--+--I-----1---------f--t---;w'---+--?Jl,. ...... ...... . ... - .. - I II I .. - ~-... .

j I I -' /t--i! I i\ .- . 1\

i !) ! i:/ i \ v:~- ~ , j l ~ ~v·-v / I ! I/: I !.-- \ ! \ Ii/I--: '. I lI" 1\

/ 1 !.Y, I 1'\", / ! \ (\ II I i /j i I Ii: \ " : \ -j I

I' i I

... 1 i

i I

I i , I

... 2 I

,

I v' \ Ii! I 1 ; T /', i / i :\ II 1 \ or-~! -,,~~/~\~. ~+~~/;-. ~l-~i~~--~i~-+~I\~I~~\+-+-+-~/-+-+~~'-"~~~'~;' -~ii ~

I \ A -i " - iii 'J' - ........ , / iii t---f--.l, / I -+--! , - -+-+-+--+--+--f----.4

~ I~! I ::! / -...., 1/ i I I

1\ i- i i : I I I \ I \ I !! ill I \l ~ i I ! .r i I 'VI \ I ,tt"\ I i I [/ i : I·! ; 1 .V i I: ! 't' \ i I ! if ! I ; \: '. i:t±'-' , ~ i i ./i

1 ~ '~~ _ M at 1/4 point --~---~. t I ! I ~ ! J I I I I -L-.+- i\ i / i

1--+--+-_-+--+-+-'+-: -+I--+-i, ----+!._ \ _~l~ ! i ! I -.L \ I It 1"_, iii! 'Aa I I I I I \;!/: I'"

I i I I "' t : "-.:.. M at 3/4 point I : 1--f--~I-----1f---~----l-~!--tI--+!-· ---r~i - ~ I -

I !

I ! iii 0 0.5 1.0 1.5 2.D

t/To 2.5

(d) Bending Moment, M

FIG. 5.12 (Continued)

,

' ..........

v at

... ;;.... ~ -' .......

, .......

"

0.5

143

Position of toad att/! • 1.0 o

~ -

- ~t/T = 2.0 o

-.... - -crown ........ ......

./ ./

............

"

~ .. ' .... /'fl

" ~., ,,-'"

.' ~ 0" f- .... "--y at ,

-r--'--

w at '--'T-' -_.

, w at 1/4 )?Oint .............

' ... .........

1.0

"'"- -1.5

t/'lo

.... --

3/4- point

--~ ;>........,

crown ./

,~ .. -......

2.0

0_ - .--... .

"' ............ .............

--2.5

(b) lls41aJ. and ~1al D!splacemerrta, w and v

FIG. 5.13 RESPONSE CURVES FOR AN ARCH SUBJECTED TO A TRIANGULAR MOVING" PRESSURE- ... :r/L :as 0 .. 5 . 0

L' fro. 100, Po/Per = 1.0, tt/To == 1.0, td./tt == 1.0

_0- ~ O-

" .... .. .

........ 1-'" ~

--

3 .. 0

2

o

-1

~I' f" ..........

" \ '''' ...

" I

1 o

I

I

I

I

1

144

I

i : B at 1A J01lrt -~

" ~ \ "

" ~ ~ .... ~ '-. ~ V

i i

0,,5

t;;.-= ~- -- --/" / ~ ~. ~~

~/ ~

" .t-. :'" r--~ N at crown

i I "I I

1-r-J--.~ i I I

I i i

1 .. 0 1.5

t/To

I

I

~ ~

.. ~ ~r- ......... ~,

~ .~

'T ~! . r-~ • at )/4 lOiDt

I

i i

I i 2.0

(c) Axial Force; I'

I 1

,

I I I I

! I I

i i I i 1

1 I M at ~/4 point-1\ i Mat crown-~

I 1 ~ I . ~/

-.... . .............. : L I I I

! I '''l''-_ .L 1/ .... : I -1-- 1--

I I

v~ V,I F .' k ~ ~ ~/ III!..

....... -V I

1 I I

2 .. 5 8,,0

!

!

I

~ I--t-.....

I" ....... J'.....

I I _-4:./, i 1 I / ...............

i ~ i .........;. -- ....... J.../ i - r- ............. .......... i -~ / -.. ~ T ---r I r-t--4-1 I I ---r-r-- ; I

-...;: ~ V .-r -......

1 ; I ~ . I I ! I

I ........ . _ ..... ..........

........... I I I . ~ / I -r-

1 [ I / ! i I i

I I i 1 3/4 po1~t- ...J I

I I I ! : M at : I i I I i

! I ! i : I I

i I 1 I ! I

i j

i I I I I. I

I i

I ! J j --

-H-I ! ! i I ! I

Iii ! 1 ! ! ! ! i i I

I i I ! I

I I I

I i ! ! !

j I i I i ! i I

o 0.5 1.0 1.5 2.0 t/To

2.5 g.o

(d) Bend.iDg Moment, M

FIG. 5.13 (Continued)

145

. . ... , ~'--- Ev :: 1 .. 0 p R/Mi ~ w 0 L~' ~~ __ ~ __ ~~ __ ~/~/~ --.:. - - - - elastic L-=-~::~- Ii-' -;.' -~--I---+---r----,vL/l--~ t----+---+---+---+-----I---.:.... E at crow (bot tom) "!===='~\'-F.~-r---+ '---I

Or-~---~~~~~--~~~~--~_+--~~--~~--~~ ~~~ /~

~c at '1/4 point (top) /1\ \ ~ ... 2 ' .. 'i • ~ " ;I' -- --

\ X

1.0

~ 2~_+--+-~--~~--~--~_+-~~~4-~--~--~~

/'" .-v

, ~r--+--+--+--~~--~--~~~-,-+--~~--~--~_~

'" - - - t-- -:ri--

""' ~ 1---+---+----+---+----4-- W at .1/4 point ===r-7_+-----+------l_-I

I I I' I

t/To (b) Radial Displacement, w

FIG. 5 .. 14 CURVES FOR DELASTIC RESPONSE OF AN ARCH SUBJECTED TO A TRIANGULAR MOVING PRESSURE--Ey =: 1 .. 0 poR/AE

r/L = 0 .. 2, L /r = 100, P /p :: l~O, tt/T = 1.0, t~/tt =: 1.0 o 0 0 cr 0 y.

146

Ir If ,~ -O.~ - .

I I I I . I I ........ ~ ~. • at ~J01Dt\

....... ~ K. ..... ~ ~

~f): !::=.°l ... ~ ~ ~ .. -N at crown-!-i

-, a 0.5 1 .. 0

M at 1/4 pOint- f\. ~

\ --/" ;- ... - 1-"

... " L .......

......... K / ./

.... ~. ~ - /~ ...... ; \. " V -" ... "

~ l/' f"o_ , V ..... I-r'

M at croWn-11 '" '7'-

... 1 / M at 3/4 point ~

If 0.,5 l..O

t~o (d) BeniUng Jtmrent; II

FIG Q 5.14 (Continued)

147

., -~ ~ ~ I- ' - •..•.. 1 ..

tt at Cl"'Oll'D (bottom) --.. .... ~ "'1' ,

"'T" "" " ~ ,.~ - ........ ~ -'\ 0;;;;. , r ""'l.. . - '..::::F-f" 'r 7 - :--:-::,- -::

\.\. 1 .1">-..,.. . T... I . I / « at ;/4 point (bottom)

\ -~

\" I

1\ \

'" . " - '-« at 1/4 point (top)

" ./ ?:.

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

--- ...... 1"-- -

o 0.5 1.0 1.5 2 .. 0 2 .. 5 3 .. 0 t/To

( a) .Strain, 6

, T T T T T 1

I r I I I I I

.. w at 3/4 point-r... ' -r- .. - '- - .-... "" . ( , .... .....

..... , . .;,; ~

2 .. : ,0'

I--- ...... ' v at crown

~ (\J 0

&X4 ~O

I"~' I I I •• I~ I

I 1

I ....... r-........ I J

.......... r-- Yat crown "- ~

CH -2 0

.......... L

"" """'-- -- -r--. '--~ -to---

(Q '\.. e ..... r, '.

~ '\.

!1 """ '. "' .p

I .. '\.

'" "-

i -10 r-I gc .,... t:l -12

" " r-- wat 1/4 p?int

'" 7 ~ 7

..... J .....

-- -- --, .. 1"-- ..- ----- -l

0.5 1 .. 0 , 0 5

t/To

2.0 2.5 s .. p

(b) Radial and Tangential Displacements, w and v

148

D.pt I I I I I I I I I

d ~ ... ~: / -......: "~ . 0.2

/1 ~~ \ 1//) 'N - I •

~ I ,'t' ~

" 0 rv ,-' ..:1= N at crown-

!/ ~ " ~

'-\ ~ " r- ", f'.: , C.t ~/ /J " \. 1/ ~» at ,/4 point i .2 f'.~ l' ..c:.

/1 I ~ ~ if at 1/4 point 1\,,' ~ , ! I /

" \'

.... 6..:>« ;- :;.,...-' \ .\

fi': . " A' L /,,' • lit ......

~ ............. ~ip /' !- -l-

r---- -I ... o .. Go 0 .. 5 1.0 1.5 2 .. 0 2.5 3"

t/TO ( c) Axial Force ~ N

-----

". i / ",,-..,"'\ , "

tj=J=r--+-+++-r-in M at 3/4 point;::p \ J

-o .... U---L-++++-t-in

0.5 1.0

t/To (d) :semj 138 Moment, M

5.l5 (,Continued) FIG.

2.0 2.5 9.0

TR-60- 53

No, Cys

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

5

3

DISTRIBUTION

HEADQUARTERS USAF

Hq USAF (AF8DC), Wash 25, DC

Hq USAF (AFOOA), Wash 25, DC

Hq USAF (AFOIE), Wash 25, DC

Hq USAF (AFDRT), Wash 25, DC

Hq USAF (AFCIN), Wash 25, DC

CSAF Dep IG for Insp (AFCDI-B-3), Norton AFB, Calif

MAJOR AIR CQMMANDS

ARDe, Andrews AFB, Wash 25, DC

(RDRW)

(RDZ)

AFBMD (WDAT), AF Unit Post Office, Los Angeles 45, Cali!

SAC (DINC), Offutt AFB, Nebr

TAC(TDA), LangleyAFB, Va

AMC (MCW7), Wright-Patterson AFB, Ohio

ADC (Ops Anlys), Ent AFB, Colorado Springs, Colo

AUL, Maxwell AFB, Ala

USAFA, United States Air Force Academy, Colo

ARDC CENTERS

WADD, Wright-Patterson AFB, Ohio

(Dir of Sys Mgt)

(WWAD)

(Structures Branch, Aircraft Laboratory)

AF Research Division (Technical Library), Bldg T- Y, Wash 25, DC

KIRTLAND AFB

AFSWC, Kirtland AFB, NMex

(SWNH)

( SWOI)

(SWRS)

TR-60- 53

No. Cys

1

1

1

1

1

1

1

1

1

6

1

1

1

1

DISTRIBUTION (con'd)

OTHER AIR FORCE AGENCIES

Los Angeles Air Procurement District, ATTN: Administrative Contracting Officer, 1700 Main Street, Santa Monica, Calif

For (RAND Nuclear Energy Division)

ARMY ACTIVITIES

Chief of Res.earch and Development, Department' of the Army, ATTN: Special Weapons and' Air Defense Division, Wash 25, DC

Director, Ballistic Research .Laboratories, ATTN: Library, Aberdeen Proving Ground, Md

Cornmanding Officer, US Army Engineers, Research and Development Laboratories, Ft Belvoir, V1i

Office of the Chief, Corps of Engineers, US Arrny, ATTN: Protective Construction Branch, Wash 25, DC

NA VY ACTIVITIES

Cornrnanding Officer and Director, Naval Civil Engineering Laboratory, Port Huenerne, Calif

Chief, Bureau of Yards and Docks, Departrnent of the Navy, Wash 25, DC

OTHER DOD ACTIVITIES

Chief, Defense Atomic Support Agency, ATTN: Docurnent Library Branch, Wash 25, DC

Commander, Field .command, Defense Atomic Support Agency, ATTN: FCAG3, Sandia Base, NMex

ASTIA (TIP DR), A rlington Hall Sta, Arlington 12, Va

Director, Advanced Research Projects Agency, Department of Defense, The Pentagon, Wash 25, DC

OTHER

American l'v1ochine and Foundry Company, ATTN: Mr. T. G. Morrison, 1 J 04 South Wabash Avenue, Chicago 5, III

University of Massachusetts, ATTN: Dr. Merit P. White, Dept of Civil Engineering, Amherst, Mass

Massachusetts Institute of Technology, .. Dept of Civil and Sanitary Engineering, ATTN: Dr. Robert J. Hansen, 77 Massachusetts Ave., Cambridge 39, Mass

TR-60-53

No. CYi

1

1

DISTRIBUTION (cont'd)

E. H. Plesset Associates, Inc., ATTN: Mr. Marc Peter, Jr., 6399 Wilshire Blvd., Los Angeles 48, Calif

ArIIlour Res earch Foundation, Illinois Institute of Technology, ATTN: Mr. Ray Sauer, 3422 South Dearborn ~treet, Chicago 16, III

Stanford Research Institute~ Menlo Park, Calif