pereirac.pdf

RICE UNIVERSITY

FREE VIBRATION OF CIRCULAR ARCHES

by

Carlos Antonio Lopes Pereira

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENT FOR THE DEGREE OF

MASTER OF SCIENCE

Thesis Director's Signature

September, 1968

i

ABSTRACT

FREE VIBRATION OF CIRCULAR ARCHES

by

Carlos Antonio Lopes Pereira

The lower natural frequencies and the associated modes of vibration of

uniform circular arches which are either fixed or simply supported at the ends

are computed and studied in this thesis.

A general theory which considers the effects of rotatory inertia and

shearing deformations as well as extensional and flexural deformations was

used. When both rotatory inertia and shearing deformations are neglected the

general theory reverts to a well-known classical theory which is based upon the

same assumptions as Flugge's theory for the bending of cylindrical shells. The

governing differential equations were solved numerically by a Holzer -type

iterative procedure combined with an initial value integration method. A Runge

Kutta integration technique was used. The method was programmed for an IBM

7040 computer, and all results were obtained on this computer.

A detailed study has been made of the lowest ten vibration frequencies and

modes for fixed arches for a wide range of slenderness ratios and for angles of

opening of 45°, 90° and 180°, using the classical theory. The effects of rotatory

inertia and shearing deformations have been investigated only for arches with an

angle of opening of 90°.

Simple approximate expressions for estimating the natural frequencies of

fixed -ended arches (classical theory) are developed and their accuracy evaluated

ii

by comparing them with the results obtained by exact analysis.

In addition, exact solutions of classical theory were made for hinged

circular arches with angles of opening of 45 ° and 180° and the accuracy of

approximate expressions for estimating the natural frequencies of hinged arches

proposed by Austin, Veletsos and Wung were evaluated for these angles of open

ing.

iii

CONTENTS

ABSTRACT

LIST OF TABLES

LIST OF FIGURES

I. INTRODUCTION

. 1. 1 Object and Scope

1. 2 Review of Literature

1. 3 Acknowledgement

1. 4 Nomenclature

II. GOVERNING EQUATIONS AND NUMERICAL SOLUTION

2. 1 Fundamental Equations

2. 2 Numerical Procedure

III. PRESENTATION OF RESULTS FOR FIXED ARCH

3. 1 Vibrational Behavior of Arch by Classical Theory

3. 2 Approximate Formulas

3. 3 Effects of Rotatory Inertia and Shearing Deformation

IV. PRESENTATION OF RESULTS FOR HINGED ARCH

4. 1 Additional Solutions and Comparison with Approximate Formulas

REFERENCES

TABLES 1 -15

FIGURES 1 -40

iv

TABLES

1. Convergence of the Numerical Method

2. Summary of Solutions of Classical Theory, Fixed-ended Uniform Circular Arches eo:: 90°

3. Summary of Solutions of Classical Theory, Fixed -ended Uniform Circular Arches 90 = 45°

4. Summary of Solutions of Classical Theory, Fixed -ended Uniform Circular Arches eo= 180°

5. Summary of Solutions of Classical Theory, Hinged-ended Uniform Circular Arches G0 =45°

6. Summary of Solutions of Classical Theory, Hinged -ended Uniform Circular Arches a,= 180°

7. Summary of Solutions for Theory Including Rotatory Inertia but Neglecting Shearing Deformation, Fixed -ended Uniform Circular Arches eo= 90°

8. Summary of Solutions for Theory Including Rotatory Inertia and Shear Deformation, Fixed -ended Uniform Circular Arches 60 = 90°, r = o. i

9. Summary of Percent Energy, Fixed-ended Uniform Circular Arches eo :. 90°

10. Summary of Percent Energy, Classical Theory, Fixed-ended Uniform Circular Arches 8o: 45°

11. Summary of Percent Energy, Classical Theory, Fixed -ended Uniform Circular Arches eo"' 180°

12. Summary of Percent Energy, Classical Theory, Hinged -ended Uniform Circular Arches eo= 45°

13. Summary of Percent Energy, Classical Theory, Hinged-ended Uniform Circular Arches $ 0 :180°

14. Summary of Solutions for Theory Including Rotatory Inertia and Shear Deformation, Fixed -ended Uniform Circular Arches eo== 90°, r = o. 3

15. Summary of Percent Energy, Theory Including Rotatory Inertia and Shear Deformation, Fixed -ended Uniform Circular Arches 90 :90", r == 0. ~ ,

v

FIGURES

1. Coordinate and Displacement Notations

2. Typical Plots of Determinant vs. Frequency - Fixed Ended Arches

3. Natural Frequencies for Fixed Ended Arches - Classical Theory

4. Predominantly Flexural Modes of Vibration - Fixed Ended Arches

5. Frequencies Associated with Predominantly Extensional Modes of Vibration - Fixed Ended Arches

6. a, b Predominantly Extensional Anti symmetrical Modes of Vibration -Fixed Ended Arches

7. a, b Predominantly Extensional Symmetrical Modes of Vibration - Fixed Ended Arches

8. Modes Corresponding to Frequencies of Nearly Equal Values - Fixed Ended Arches

9. a, b Antisymmetrical Modes Corresponding to Frequencies of Nearly Equal Values -Fixed Ended Arches

10. a,b Symmetrical Modes Corresponding to Frequencies of Nearly Equal Values -Fixed Ended Arches

11. Comparison of Approximate and Exact Solutions - Fixed Ended Arches with 60 =45°

12. Comparison of Approximate and Exact Solutions - Fixed Ended Arches with eo-=-90°

13. Comparison of Approximate and Exact Solutions - Fixed Ended Arches with eo= 180°

14. Effects of Shear Deformation and Rotatory Inertia for Antisymmetric Modes -Fixed Ended Arches

15. Effects of Shear Deformation and Rotatory Inertia for Symmetric Modes -Fixed Ended Arches

vi

16. a, bEffects of Shear Deformation and Rotatory Inertia for Antisymmetric Modes -Fixed Ended Arches

17. a, bEffects of Shear Deformation and Rotatory Inertia for Symmetric Modes -Fixed Ended Arches

18. Frequencies Associated with Predominantly Extensional Modes -Hinged Ended Arches

19. a, bPredominantly Extensional Antisymmetrical Modes of Vibration - Hinged Ended Arches

20.a, bPredominantly Extensional Symmetrical Modes of Vibration -Hinged Ended Arches

21. Comparison of Approximate and Exact Solutions -Hinged Ended with eo= 45°

22. Comparison of Approximate and Exact Solutions -Hinged Ended with eo= 180°

CHAPTER I.

INTRODUCTION

1. 1. Object and Scope

This study is concerned with the computation of the natural frequencies

and the associated modes of vibration of uniform circular arches which are

either fixed or simply supported at the ends.

Although the theory of vibrating arches is well established, only a limited

number of numerical solutions are available in the literature. The most com pre

hensive numerical study of the natural frequencies of circular arches appears to

have been the one conducted recently by Wung (1) at Rice University. This study

was concerned with hinged arches and irwolved the evaluation of the first eight

natural frequencies of vibration using both the classical theory and a more

general theory which considers the effects of shearing deformations and rotatory

inertia. On the basis of these data and by application of Rayleigh's method,

simple approximate expressions were proposed in Ref. (2) for estimating the

frequency values obtained from the classical theory. Wung' s numerical study

was limited to an arch with an angle of opening of 90°, and the accuracy of the

approximate expressions proposed in Ref. (2) could be evaluated only for this

particular arch.

The object of the present study is to extend the previous investigation

along the following lines;

1. To investigate the response of fixed ended arches using both the

2

classical and the more general theory,

2. To develop simple approximate expressions for estimating these

frequencies, and

3. To obtain additional data for simply-supported arches and to check

the accuracy of the approximate expressions proposed over a wider

range of parameters than considered before.

The nth circular frequency of an arch, p , may conveniently be expressed n

in the form.

b =_s_~IEI r n 5 ~ )' --;:;-

(1)

In this expression S denotes the curved length of the arch axis, EI the

flexural rigidity of its cross section, m the mass per unit of length, and en a

dimensionless coefficient.

When the effects of shearing deformations are neglected, en is a function

of the angle of opening of the arch, e0 , and its slenderness ratio, S/r, where r

denotes the radius of gyration of the arch cross section.

In Wung's study the first eight natural frequencies of hinged-ended arches

with e0 = 90° were evaluated for a range of S/r from 0 to 360. In the present

study the corresponding frequencies were also evaluated for hinged arches with

eo = 45° and eo = 180° using the classical theory.

A detailed study has been made of the lowest ten vibration frequencies and

modes for a fixed arch with S/r varying from 0 to 500, and for angles of opening

of 45°, 90° and 180°, using the classical theory. The effects of rotatory in-

ertia and shearing deformation have been investigated only for arches with e0= 90~

3

These solutions were obtained by applications of the numerical procedure and the

computer program used in Wung's study. This computer program was extended

to make possible the computation of the proportions of extensional and flexural

energies associated with the various modes of vibration. This additional infor-

mation proved very helpful in the interpretation of the numerical data.

A major part of this investigation has been concerned with the develop-

ment of simple approximate expressions for estimating the natural frequencies

of fixed -ended arches. In this effort the ordinary theory was used. These

equations are presented in the text and their accuracy is evaluated by comparing

them with the results obtained by exact analysis. In addition, the accuracy of

the approximate expressions proposed in Ref. (2) for hinged -ended arches is re-

evaluated in the light of the additional data that have been obtained, and certain

modifications are proposed.

1. 2. Review of Literature

A brief resume of two papers which are concerned With the vibrations of

fixed -ended circular arches is presented below. A more complete review is

presented by Wung (1).

Den Hartog (3) derived approximate formulas for the lowest extensional

and inextensional modes for hinged and fixed -ended circular arches, by using

the Rayleigh -Ritz method and by neglecting the rotatory inertia and shearing

deformation effects. For the first antisymmetrical inextensional mode for

fixed -ended arches, the following series were used,

(2. 1)

4

·where "w" is the radial displacement "v" is the is the tangential displacement,

" () " is the angular coordinate with the origin at the middle of the arch, " () 0 "

is the angle of opening, and "B 1" and "B2" are coefficients which are de-

termined by the Rayleigh-Ritz procedure. The coordinate and displacement

notation are shown in Fig. 1.

Den Hartog expressed the frequency as

(3)

where "1" is the chord between the ends of the arch and C 4 is a frequency

parameter, a function of the properties of the arch. Values of c4 are tabulated

for a wide range of parameters and are shown graphically.

For the first symmetrical extensional vibration mode, Den Hartog as-

sumed the following trial functions,

w = B ( f + cos 2 TT B) Bo

Ar= b . 2Tr8 1. St-n.. --

Go

(4. 1)

(4. 2)

where "B", "b1" and "b2" are arbitrary constants. With the series above, the

result for the frequency coefficient is

4. sin 2 Bo. z

Den Hartog shows that the lowest extensional mode of vibration can

have a lower period than the lowest inextensional mode.

(5)

Robert R. Archer (4) in 1959 presented a paper concerned with the in-

5 plane inextensional vibrations of an incomplete circular ring of small cross

section. In this work rotatory inertia and shearing deformation are neglected,

but the effects of damping are included.

The governing differential equation assumed was

(6)

where "K" ,_is the viscous damping coefficient.

This equation was solved by an analytic method and the four lowest :Ere-

quencies for 80 = 180°, 234°, 324° and 360° were tabulated.

1. 3 Acknowledgement

I am grateful to Professor W. J. Austin for his help and guidance of this

work. I wish to thank Professor A. S. Veletsos for his help, ideas and guidance

without which this work could not have been accomplished on time.

My sincere thanks to Rice University, Institute Militar de Engenharia,

Pontif:lcia Universidade Catolica do Rio de Janeiro and Panamerica Union for

their support during the time of this work.

1. 4 Nomenclature

Latin Symbols

A cross sectional area of arch,

a radius of centroidal axis of arch,

B, b1, b2 constants,

C dimensionless coefficient, equal to to .... ~ , n /nl£1:

0 determinant,

E Young's modulus,

6 F non-dimensional frequency parameter equal to

G shearing modulus,

I moment of inertia of cross section of arch,

K viscous damping coefficient,

k d . . 1 1 (: ) 2 non- 1menswna parameter, equa to :.._

1 length of chord between ends of arch,

m mass per unit length,

pn nth circular natural frequency of an arch,

Q shear,

R rotatory inertia index; equals to unity if rotatory inertia is included;

equals to zero if not,

r radius of gyration of cross section of arch,

s curved length of the centroidal axis of the arch,

s shear index; equals to unity if shear deformation is included; equals

to zero if not,

t time variable,

u shear variable, equal to a. j3

v tangential displacement,

w radial displacement,

Greek Symbols

a"- a parameter occurring in the expressions for the characteristic

functions, See Ref. (5),

{3 shear deformation,

r non-dimensional shear parameter, equal to r ~

7

constant of proportionality, equal to Q /GAp

An square root of the natural circular frequency of the nth mode of vi-

bration of a beam,

¢>h. characteristic function representing modes of vibration of a beam,

() angular coordinate,

00 angle of opening of the arch,

dimensionless coordinate, equal to ~ E;lc>

CHAPTER II.

GOVERNING EQUATIONS AND NUMERICAL SOLUTION

2. 1 Fundamental Equations

The governing differential equations for in-plane free vibrations of uni-

form circular arches, as given by Wung (1), are as follows:

w "" = ( ~ . ~ . ~ _ 2 _ !!: . F _ § . .£ J:w "+ ( g_ . .§. F( k + f- F) _ k + 1 - F). v.; + r .. ~t rr rr+-k k

+(R..t2_._F __ .1_ _ s.L -r R. F).nr' - f'+k k - r - (7. 1)

II ( nr = B:.. §.. ~k )·w"'+(B·~· Fk(f;fr+F) _I +-g.Fk)-w'+ 1+--F 1+--F

k k , + ( .@_. S. F h (!;- F) _ F _ R • F k). tv

- r -1 +T- F

(7. 2)

where ~ and ~ are coefficients equal to unity when rotatory inertia or shearing

deformations are included and equal to zero when rotatory inertia or shearing

deformations are not included, and in which

k = (; f r = 't- ..2:...

E

and 2 m.a.2 F = P·-

£A

(8)

(9)

(10)

In these equations A denotes the cross-sectional area, each prime denotes one

derivative with respect to fJ , and 'Y is a shear stiffness constant (See Nomen-

clature).

The boundary conditions for hinged ends are as follows:

(11.1)

tV = 0 (11. 2)

w"- s . ..!......N- 1 = 0. - r For fixed ends the boundary conditions are:

w :: 0

rv- = 0

9

(11. 3)

(12. 1)

(12. 2)

(12.3)

At midspan the conditions for symmetrical vibration about midspan are:

'WI : WIll : Ar : 0 (13)

and for anti-symmetrical modes,

w = w" = !IT" 1 = 0 • (14)

2. 2 Numerical Procedure

These equations were solved numerically by a Holzer-type iterative

procedure combined with an initial value integration method.

First, an arbitrary value is assumed for the natural frequency coef-

ficient F and all of the coefficients of the differential equations are evaluated

numerically. Then three sets of independent initial conditions are assumed at

the left support. Then for each set of boundary conditions the differential equa-

tions are integrated numerically using the Runge-Kutta's procedure. In the

table below are shown the three sets of initial conditions used for hinged ends

and the three sets for fixed ends.

Initial Conditions Set N w' w" ·w"' v' w v

1 0 0 0 0 0 0

.Hinged 2 0 0 0 1 0 0

3 0 0 S.1/f 0 0 1

10

Initial Conditions Set N

w ' w" w"' v' w v

1 0 0 1 0 0 0

Fixed 2 0 0 0 0 0 1

3 0 _§ rk 0 1 0 0 zrlt +ra+Fk

This procedure generates three sets of functions: w 1, v 1; w 2, v 2; and

w 3, v 3• The solution of the given problem is a linear combination of these three

functions if the assumed frequency coefficient F is a natural frequency. Thus,

(15. 1)

and

(15. 2)

where B 1 and B2 are constants to be determined by three boundary conditions at

midspan. Thus, for an anti -symmetrical vibration mode the equations which must

be satisfied at midspan are as follows:

w/'(~o) + B1 w;'(~) + 132 w~'(~o) == 0

rv-: ( ~o) + B1 IV"; ( ~·) + 62 1\r~ ( ~·) = 0

(16. 1)

(16. 2)

(16.3)

For a non -trivial solution of this set of three linear homogeneous equa-

tions to exist, it is necessary that the determinant of the coefficients equal zero.

w1( ~·) w2 ( ~o) w3 { ~o)

D = w/( ~o) w2' ( ~·} w~ (~·) (17)

1\r: ( ~·) ~I(- fJo) 2T ~; (~D)

11

For symmetrical vibration modes the required boundary conditions at

midspan are:

(18. 1)

(18. 2)

(18. 3)

and the determinant is as follows,

w;t~o) W~ ( ~o) w; ( ~o)

0 = w';'( ~o) wt(~) w~' ( ~o) (19)

~f ( ~0) N"z ( ~o) nr3 ( ~o)

If the assumed frequency coefficient F is not a characteristic value, the

determinant will not equal zero, and, then, other trial values of frequency co-

efficient are assumed, until the value of the determinant changes sign. A

natural frequency always exist between two assumed frequencies with values of

D of opposite sign. A linear interpolation based upon the value of the determi-

nant is used to close in on the natural frequency.

The criteria used for convergence to a solution are as follows: either

the absolute value of D must be less than 0. 0001 or two frequencies, F 1 and F 2,

for which the determinants are of opposite sign must be sufficiently close that:

< 0.00001 (20)

Typical variations of the determinant D with assumed frequency are

shown in Fig. 2.

After the natural frequency is found the corresponding values of B1 and

12

s2 can be found and the mode shape evaluated by superposition of the three

functions, as indicated.

The energy was computed using Simpson 1/3 Rule and the following ex-

pressions:

Energy of Flexure = FE = EAJ~( w" -r w- ..u ')2 de 2Q.

0

Energy of Extension = EE • ~ ~£(,.' + ., )' J •

Energy of Shear = SE = .!.A. . rf~2de 2a.

0

where "u" is a shear variable, equal to (see Eq. {20. i), Ref. 1),

JJ.. = _ 1 r w"' -r ( 4 -r F +..£..) w' + (L - F) N""J (t+rlll-F) rr r

and its first derivative is as follows (see Eq. {42. e), Ref. 1),

J!.' = + __ k_ . W 11 + (4 + J.t - F) . W k-rr ~+,...

(21. 1)

(21. 2)

(21. 3)

(22)

(23)

The values of "w" and "v" and their derivatives needed in the energy ex-

pression were computed by superposition of the basic three component solutions.

The data presented in this thesis were computed on an IDM 7040 computer.

Sixty divisions were used in the whole arch. The accuracy of the frequencies

computed with sixty divisions can be judged from the data shown in Table 1.

This table gives the frequencies computed in a particular case with 24, 32, 40

and 60 divisions in the complete arch. It can be seen that the lower four modes

are converged to about four significant figure accuracy and the higher modes to

about three significant figures when 60 divisions are used in the complete arch,

as in the solutions reported herein.

CHAPTER III.

PRESENTATION OF RESULTS FOR FIXED ARCH

3. 1. Vibrational Behavior of Arch by Classical Theory

Solutions of the classical theory were obtained for fixed ended uniform

circular arches with angles of opening of 45°, 90° and 180°. These solutions

are shown in Tables 2, 3 and 4 and Fig. 3, 11, 12 and 13. The percentages of

total energy in flexure and extension are given in Tables 9, 10 and 11.

The shape of the frequency-slenderness ratio curves is the same as

described in Ref. 2, i. e. :

a) With increasing S/r each frequency curve ultimately approaches

a horizontal asymptote which will be shown to be a natural frequency of

an inextensible arch,

b) In the lower ranges of Sjr, the frequency-slenderness ratio curves

are stepped. Between each inextensible frequency level, the frequency

curves lie approximately on certain diagonal lines. In the region of each

inextensible frequency level the frequency curves progress more -or

less horizontally from one diagonal line to the adjacent diagonal line on

the right as S/r and the frequency increase. If there is no diagonal line

to the right, then the frequency curve approaches asymptotically the in

extensible frequency level.

The following observations may be made comparing arches with the same

boundary conditions, but with different angles of openings. Refer to Figs. 11,

12, and 13.

14

a) The frequency coefficients, considering inextensional deformations,

do not depend on the slenderness ratio. They have a small decrease,

however, when the angle of opening is increased,

b) The diagonal corresponding to the first extensional mode varies

with the angle of opening; for symmetrical vibrations there is a large

variation; for anti -symmetrical vibrations the variation is not so large.

The higher extensional modes, for practical purposes, do not depend on

the angle of opening. The region between the first and second extensional

modes decreases when the angle of opening increases, although the frequen-

cy-slenderness ratio curve maintains the same shape, especially in the

symmetrical case.

By a study of Fig. 4 it is easily seen that for high values of S/r, where

the tangents to the frequency curves are almost horizontal, the shapes of the

radial displacement modes "w" are roughly the same as the lateral vibration

mode shapes of straight beams with the same boundary conditions, and the tan-

gential displacements "v" are very small. The strain energy in the arches is

completely or almost completely flexural for the vibration modes shown in

Fig. 4, as the following table illustrates.

All Solutions For 80 = 90°, 5/r = 500

cl'\ Percent Energy

Mode in Flexure in Extension

Anti -Symmetric Modes

1 55.8 100 0

2 193.0 100 0

3 409.7 100 0

4 705.5 100 0

15

Percent Energy

Mode en. in Flexure in Extension

Symmetric Modes

1 106.5 100 0

2 283.2 99 1

3 532.6 94 6

Subsequently it is shown that the frequency curves approach asymptotically

horizontal lines computed on the assumption of in extensionality of the arch axis.

Figure 5 shows the vibrational behavior for points close to the diagonal

lines corresponding to the first extensional mode. One can see that the number

of ripples in the wave shape corresponding to the radial displacement "w" in-

crease with S/r. But as these points come to a horizontal line corresponding to

the flexural behavior, the number of ripples is the same as for the correspond-

ing flexural mode. Note that the strain energy is primarily extensional at points

on the frequency curves near the center of the diagonal segments. The mode

shapes for these points are shown in Figs. 6 and 7.

As shown on Fig. 8, at the same level, all mode shapes for any slender-

ness ratio S/r, have the same number of "w" waves. Note that at these points

the energy is primarily flexural. On Figs. 9 and 10 are presented the details of

the mode shapes for the points marked on Fig. 8. From these one may conclude

that the only difference in the vibrational behavior at the same level is in the

tangential displacement "v" which decreases when S/r increases. For the sym

metrical shape in "w" it is seen that the magnitudes of "v" are small for any

value of S/r.

3. 2. Approximate Formulas

In the preceding section of this thesis it has been shown that the

16

frequency-slenderness ratio curves follow certain horizontal lines which are

related to pure inextensional vibrations of the arch and to certain diagonal lines

which are related to primarily extensional vibrations. In this section approxi-

mate formulas for these horizontal and diagonal lines are derived. With these

formulas one can quickly sketch the frequency-slenderness ratio curves for a

given arch of interest, and, hence, quickly approximate the lower natural fre-

quencies.

3. 2. 1. Flexural Modes

The Rayleigh-Ritz method is used to find the frequency. In a flexural

mode there is no extensional energy; this state corresponds to the following

constraint:

w + fl1" 1 = 0

The Lagrangian Function of the dynamical system is

L • ~: {[ '(,..' + w') - ~ (w +w")' Jds or

I

EA e£[ ( 2 2) ft( + -:0"2")t].J~ L = 2 a. . •.0

F N" + w _ w "' .,

in which, see Eqs. (8) and (10)

k = (:) 2

and F = t='~ 2

l'h Q,

EA

In these equations each dot denotes one derivative with respect to

(24. 1)

(25. 1)

(25. 2)

e ~ =- .

eo The assumed functions for "w" and "v" must satisfy the following

boundary conditions,

w(o) = w(f)::. o (26. 1)

r w'(o) = w 1 (1) = o

tv(o) = N"(l) = 0

and the inextensible condition

w + N" 1 = 'W + ~· = 0 eo

3. 2. I. 1. Anti -Symmetrical Modes

17

(26. 2)

(26.3)

(24. 2)

The radial displacement configuration for the nth mode is assumed to be

as follows,

m=.Z,4,G ... (27. I)

where B is a constant and cA11 is the characteristic function for the nth lateral n

vibrational mode of a fixed -ended beam. The properties of the <P"" functions

have been tabulated by Young and Felgar (5, 6).

The corresponding tangential displacement configuration is assumed as

follows

N"m = - 6., (~) [ ;~ <f~·· + 2 <><m] (27.2)

These displacement configurations satisfy the inextensibility condition. In the

above expression a.A1 is a parameter and Xm is a frequency coefficient corre-

sponding to the nth mode of vibration of a fixed -ended beam, given by,

4 ( ~M) : (28)

Values of am and X,... are tabulated by Young and Felgar (5).

Setting d L ::. 0 gives the following expression for the nth frequency. d6M

(29)

18

The first five frequencies are as follows.

eo First Second Third Fourth Fifth

Tr/4 60. 10 198.02 415.05 711.09 1086. 10 en 11"/2 55.83 192.71 409.34 705. 18 1080.07

Tf 43.28 173.91 387.93 682.45 1056.59

3. 2. 1. 2. Symmetrical Modes

Assume the following expressions for the displacement configuration.

w : B [ ~ - o<n. . ~ . ~1 ] It o<j 'An

S.. [ ••• o<'n f An ) 3 --L••• J - B X~ fn - cx1 ·1._~ ''11

(30. 1)

IV" = (30. 2)

for n = 3, 5, 7 ...

These equations satisfy the boundary conditions and the inextensibility

condition. The Lagrangian gives for a stationary value the following fre-

quencies,

where

B1 = [1 + (:;){~ t] (31. 2)

-2o<~[f- _2_ + -·-·( f6. f '- <><'I • .3.!. .<><,A,-·):: GJ]E~J' B2 = <><,. >.,.

(31. 3) C>(',., A.,. ...:,. :A,. 1 - ( ~:t

B3 = [1 + (o<"/(b_ f]teo f <><1 '>.n An (31. 4)

The first five frequencies are as follows.

eo First Second Third Fourth Fifth lf /4 110.03 286.38 542.56 877.93 1292.34

en 1T /2 106.86 282.58 538.43 873.61 1287.91

TT 95.38 268. 15 522.42 856. 69 1270.42

19

A very accurate approximation is obtained by neglecting some terms.

Thus

1 - f.82{ ~:/ (31. 5)

I + (..h..) 2 + < [ t.l>3 - __§_ J "" . ,., This formula yields the following numerical results.

eo First Second Third Fourth Fifth

1T/4 109.99 286. 68 542.93 878.33 1292.76 Cn

Tr/2 106.84 282.89 538.85 874.08 1288.42

Tf 95.07 268.32 522.91 857.39 1271. 26

3. 2. 2 Extensional Modes

The frequencies associated with the symmetric and anti-symmetric ex-

tensional modes of vibration can be approximated by the frequencies of the ex-

tensional modes of a complete circular ring, Ref. (7).

(32)

For the first symmetrical mode, n = 0, it is necessary to use an empiri-

cal coefficient as follows

(33)

The higher symmetric modes correspond to n = 2, 4, 6 .... The first,

second and third anti-symmetric modes correspond ton= 1, 3, 5 ....

Figures 11, 12 and 13 show the curves obtained by the approximate

formulas and by the exact solution.

3. 3 Effects of Rotatory Inertia and Shear Deformation

In Tables 7 and 8 and Figs. 14 to 17 are presented the solutions for

cases where the effects of rotatory inertia are included and shear deformations

20 are neglected, and the cases where both are included. It is apparent that both

effects, rotatory inertia and shear deformations, tend to decrease the values of

the natural frequencies. The energy distributions are given in Tables 9 and 15.

The data and the figures show that the effect of rotatory inertia is gener-

ally negligible when compared with the effect of shearing deformations. Both

tend to smooth the curves by smoothing the steps in the transition regions. The

diagonal lines corresponding to the first extensional mode for both symmetric

and anti -symmetric vibrations are particularly insensitive to rotatory inertia

and shearing deformation, as can be seen on Figs. 14 and 15.

These effects become more important for small values of the slender-

ness ratio S/r. The curves still are asymptotic to the natural frequency of an

inextensible mode.

In the table below are presented, for a value of S/r = 100, which lies in

the transition zone, the frequency coefficients and the energy percentages in

extension, flexure and shear for the symmetrical and anti -symmetrical modes

when the solutions include rotatory inertia and shearing deformation (r = 0. 1).

Cn Percent Energy

In Shear In Extension In Flexure

Anti-Symmetrical Modes First 52. 758 10 0 90 Second 168.132 22 1 77 Third 313. 757 23 32 45 Fourth 360.358 12 64 24 Fifth 515. 749 43 2 55

Symmetrical Modes First 90.483 13 20 67 Second 152.454 4 72 24 Third 251.061 26 6 68 Fourth 419.599 39 1 60 Fifth 606.271 42 13 45

21

For some values of S/r the frequency coefficient was computed consi-

dering the rotatory inertia and shearing deformation but using r = 0. 3. These

points appear in the Figs. 14 to 15 as triangles.

CHAPTER IV.

PRESENTATION OF RESULTS FOR HINGED ARCHES

4. 1. Additional Solutions and Comparison with Approximate Formulas

Wung (1) proposed approximate formulas similar to those described in

Chapter III of this thesis for two -hinged circular arches. These formulas

agreed fairly well with the solutions which he had available. However, only one

angle of opening, 90° , was studied in Wung's thesis. To supplement Wung's

work some solutions of two -hinged circular arches with angles of opening of

45° and 180° are presented herein. These solutions are for the classical theory.

The frequencies of two hinged arches with an angle of opening of 45 ° and

a wide range of slenderness ratios are given in Table 5 and the energy distribu

tions are presented in Table 12. The frequency-slenderness ratio curves are

shown in Fig. 21. The frequencies for angles of opening of 180° are presented

in Table 6, the energy distributions in Table 13, and the frequency-slenderness

ratio curves in Fig. 22.

The shapes of the frequency-slenderness ratio curves for the two -hinged

arches with angles of opening of 45° and 180° are very similar to the curves re

ported by Wung for 90°, and are essentially similar to the corresponding curves

for fixed arches contained herein. The same physical behavior also has been

observed. For example the mode shapes at various points along the lowest

diagonal line corresponding to extensional vibrations, shown in Figs. 18 through

20, are very similar to the mode shapes reported by Wung and are the counter

part of the mode shapes for fixed arches, as illustrated herein in Figs. 5

23 through 7.

Wung' s approximate formulas have been superceded by more accurate

formulas based upon Wung's data and the data reported herein. The formulas

for inextensional vibration frequencies, which are derived in the paper by

Austin, Veletsos and Wung (2), are plotted as dashed lines in Figs. 21 and 22.

It can be seen that the correspondence with the true curves is excellent; the

agreement with Wung' s data is also excellent. For completeness, these

formulas are given below.

For anti -symmetrical inextensional vibrations,

tn = 2' 4, 6 '

4 t} tm s Ill? E I

m2.TT4. [m2 -(-*)2]2

/YI2 + 3( ~J2

Equation (36) is a very accurate approximation to Eq. (35)

(34)

(35)

(36)

Equations (32) and (33) which previously have been applied to predict the

extensional vibrational frequencies of fixed arches apply as well for hinged arches,

as would be expected. The dashed diagonal lines of Figs. 21 and 22 are the

graphs of these equations. It can be seen that the agreement of these lines with

the exact frequency curves is excellent for the hinged arches with angles of open-

ing of 45° and 180° . The agreement is also excellent for hinged arches with an

24

angle of opening of 90°, although this is not shown herein. These equations fit

the exact solutions better than any formulas heretofore proposed.

25

REFERENCES

1. 'Vibration of Hinged Circular Arches" by Shyr-Jen Wung, M.S. Thesis, Rice University, 1967

2. "Natural Frequencies of Hinged Circular Arches" by W. J. Austin, A. S. Veletsos and S. J. Wung, unpublished manuscript, Department of Civil Engineering, Rice University, 1968

3. "The Lowest Natural Frequency of Circular Arcs" by Den Hartog, Phil. Mag. Series7, Vol. 5,1928, pp. 400

4. "Small Vibrations of Thin Incomplete Circular Rings" by R. R. Archer, Int. J. Mech. Sci. Vol. 1, 1960, pp. 45

5. "Tables of Characteristic Functions Representing Normal Modes of VibrationofaBeam"byD. YoungandR. P. Felgar, Eng. Research Series, N 44, The University of Texas, 1949

6. "Formulas for lntegrats Containing Characteristic Functions of a Vibrating Beam" by R. P. Felgar, Circular N 2 14, Bureau of Eng. Research, The University of Texas, 1950

7. "A Treatise on the Mathematical Theory of Elasticity" by A. E. H. Love, Dover, New York

Nuaber of Divisions in Co11p1ete Arches First

24 9'7.790

32 9'7.762

40 97.754

60 9'7.750

TABLE 1

CONVERGENCE OF THE NUMERICAL METHOD

FmD-ENDED UNIFORM CIRCULAR ARCHES - CLASSICAL THEORY

eo= 900 S/r = 102.101 a/r = 65.

FREQUENCY COEFFICIENT , p~•s4/EI SlMMETRIC MODES ANTI-SYMMETRIC MODES

See:ond Third Fourth Fifth First Second Third Fourth

158.451 298.339 554.824 660.194 55.739 191.808 344.344 423.438

158.411 29'7.621 551.204 659.964 55.732 191.591 344.175 421.769

158.399 29'7.406 549.986 659.885 55.730 191.527 344.123 421.244

158.392 29'7 .. 278 549.239 55.729 191.491 344.092 420.927

Fifth

720.993

714.201

711.709

710.056

N 0\

!

RATIO

S/r a/r First

12.5 7.96 26 • .350

25. 15.92

37.5 23.87

50. 31.83 64.983

75. 47.75 86.458

90. 57.30

100. 63.66 97.225

125o 79.58

150. 95.49 103.401

175. 111.41

TABLE 2

SUMMARY OF SOLUTIONS OF CLASSICAL THEORY

FIXED-ENDED UNIFORM CIRCULAR ARCHES - 9o = 900

FREQUENCY COEFFICIENT , p~ms4/EI

SYMMETRIC MODES ANTI-SYMMETRIC MODES Second Third Fourth Fifth First Second Third Fourth

78.773 118.894 158.494 234.760 37.458 62.154 118.588

116.927 161.715 292.724 319.038 53.192 87.393 195.463 2.39.463

239.383 297.105

122.193 290.438 .326.367 551.197 55.366 161.928 204.200 412.385

133.748 295.200 480.508 554.908 55.631 188.801 260.521 415.131

587.428

156.149 297.095 548.878 646.772 55.7.3.3 191.384 .337.922 420.057

551.086 802.454 392.855 448.960

212.924 304.592 552.102 882.812 55.783 192.448 404.449 520.719

4CJ7.017

Fifth

416o744

478.010

699.815

7CJ9.900

713.003

N -.._]

--------

RATIO

S/r a/r First.

200. 1Z7.32 105.021

250. 159.16 105.682

300. 190.99 106.019

350. 222.82 106.216

400. 254.65 106.339

450. 286.48 106.424

500o 318.31 106.483

TABLE 2 ( Continued )

FREQUENCY COEFFICIENT , p~•s4/EI SYMMETRIC MODES ANTI-SYMMETRIC MODES

Second Third Fourth Fifth First Second Third Fourth

256.760 329.385 554.333 886.612 55.812 192.730 408o044 665.642

Z/3.373 380.862 558.099 888.385 55.812 192.848 408.908 699.715

278.624 439.660 566.134 889.377 55.820 192.910 409.268 703.299

280.853 489.224 586.058 891.271 55.834 192.045 409.467 704.467

282.032 516.701 628.293 894.264 55.831 192.967 409.591 705.101

282.738 527.752 685.546 898.025 55.841 192.985 409.664 705.379

283.202 532.631 745.161 907.038 55.844 192.997 409.692 705.520

Fifth

734.9Z7 '

867.839

1015.440

1077.610

1077.620

1078.110 I

1078.690 I

1'.:)

00

~IO

S/r a/r

25. 31.a3

50. 63.66

75. 95.49

100. 1Z7.32

150. 190.99

200. 254.95

250. 318.31

.300. 381.97

350. 445.63

400. 509 • .30

450. 572.96

500. 636.62

TABLE 3


FIXED-ENDED UNIFORM CIRCULAR ARCHES- 9o = 45°

FREQUENCY COEFFICIENT , p~as4/EI SYMMETRIC MODES ANTI-SYMMETRIC MODES

First Second Third Fourth Fifth

Z7.329 119.757 158.355 296.832 315.665

.39o034 120o846 295.795 317.978 554.287

66.l44 124.657 298.022 553.520 633.195

88.484 135.308 298.9Z7 554.456 887.881

100.011 157.052 300.-394 554.831 890 .. 899

104.531 185.177

106.524 214.051 .3(17.268

107.567 2.39.964 315.569 556.745

259.243 330.777

108.85.3 . Z76.685 . .381.649 561.349 892.158

First Second

58.577

59.936 158.545

196.251

60.084 197.458

60.Z74 198.0.36

Third! Fourth Fifth

198.692

200.623 415.825 473.0]4

242.871 704.196

320.958 417.492 712.610

412.573 485.582 713.268

414.481 640.9Z7 716.536

708.2.30 808.965

710.558 965.767

1084.560

353.767 559.292

415.100 . 711.250 ' 1087.49 I'-' \0

RATIO

S/r a/r First

12.5 3.98

25. 7.96 60.414

:.J7 o5 11.94

50. 15.92 86.828

60o 19.10

75. 23.Vl 92.040

100. 31.83 93.557

125o 39.79

150o 47.75 94.537

175. 55.70

200. 63.66 94.861

TABLE 4


FIXED-ENDED UNIFORM CIRCULAR ARCHES - 90 = 180°

FREQUENCY COEFFICIENT , p~ms4/EI SYMMETRIC MODES ANTI-SYMMETRIC MODES


79.899 113.966 162.041 231.967 32.574 63.410 120.885 180.462

112.348 171.549 281.988 327.316 100.051 187.703 247.885

128.383 296.057 480.086 137.573 199.833 361.936

152.817 279.290 349.691 540.630 160.225 225.666 401.765

412.098 545.158

208.045 293.985 498.571 560.835 171.976 305.500 414.316

246.486 324.383 536.870 692.323 43.130 174o491 364o519 453.535

834.039 381o906 532.039

265.280 434.846 555.054 868.954 610.746

657.370

268.667 503.973 624o762 879.404 43.238 176.361 390ol09 673.336 -

~ -·-=-'=----~~~~~-~--~~

Fifth

210.503

417.291

495.077

588.725

688.721

701.149

708.733

727.754

780.125

861.150 w 0


RATIO FREQUENCY COEFFICIENT ,

S/r a/r SYMMETRIC MODES First Second Third Fourth Fifth First

250. 79.58 95.0C!7 269.892 518.491 740.4'51 895.041 43.250

300. 95.49 95.085 270.483 522.585 822.C!73 953.678 43.256

350. 111.49 95.133 2'70.823 524.445 846.800 1066.880 43.266

5oo. 159.16 859.349

p~as4/EI ANTI-SYMMETRIC MODES

Second Third Fourth

176.557 391.228 682.093

176.661 391.754 684.825

176.721 392.058 686.149

Fifth

1004.960

1050.810

1054.310

1065.360 I

CN ,.....

RATIO

S/r· a/r

5.9 7.5

7.9 10.

11.8 15.

19.6 25.

31.4 40.

39.3 50.

47.1 60.

62.8 so. 70.7 90.

94.3 120.

109.9 140. ----------- L_ __ ------

TABLE 5


HINGED-ENDED UNIFORM CIRCULAR ARCHES - EJo = 45°

FREQUENCY COEFFICIENT , p~ms4/EI SYMMETRIC MODES ANTI-SYM!VJEI'RIC 'MODES

First Second Third Fourth Fifth First Second Third Fourth

18.390 39.046 55.600

49.318 88.157 98.963 24.31.3 39.386 74.CY75 123.288

34.121 42.126 ll0.943 157.390

P17 .489 124.628 241.095 37.510 63.801 153.332 186.377

23.946 88.001 198.504 246.030 395.540 37.843 100.999 157.214 296.908

88.253 242.267 251.960 37o900 125.900 157.853 355.363

88.553 245.324 482.652 147 .4P17 161.179 354.221

44.459 89.416 11.5.TIO 397.398 155.604 203.498 354.504

49.442 90.046 245.770 446.462 155.912 228.370 354.638

63.041 93.405 482.417 156.202 302.944 355.685

70.046 97.757 482.611 345.145 364.000 - --------------------

Fifth

354.471

w N

RATIO

S/r a/r First

125.7 160. 70.761

157.1 200. 79.194

176.7 225.

196.4 250.

215.9 Z15.

235.6 300. 81.886

Z74.9 350.

314.2 400.

353.4 450. 82.678


FREQUENCY COEFFICIENT ,

SYMMETRIC MODES Second Third Fourth Fifth First

104.410

122.664

175.162

199.694 254.421 37.981

218.916 264.137

229.616 282.360

p~ms4jEI ANTI-sYMMETRIC MODES

Second Third Fotmth

352.289 4C17.353

353.362 5C17.001

568.899

620.004

628.344

629.376

156.388 629.938

Fifth

w w

RATIO

S/r a/r First

7.9 2o5 21.443

15.7 5. 41.481

23.6 7.5 56.425

31.4 10. 63.o61

47.1 15. 66.491

62.8 20. 67.3'79

78.5 25. 67.744

94.3 30. 67.931

125.7 40. 68.111

157.1 50.

188.5 6o.

TABLE 6


HINGED-ENDED UNIFORM CIRCULAR ARCHES - 6o = 1800

FREQUENCY COEFFICIENT t p~ms4/EI ::iYIVIMtt: 'HTC MODES ANTI-S~~RIC MODES

Second fhird Fourth Fifth First Second Third Fourth

48.714 80.544 101.o60 18.258 39.166 74.712

72.753 112o842 193.633 21.3'73 66.523 132.980 167.170

82o966 162.466 230.140 95.1.42 145.170 234.404

99.599 2rJ7.849 241.384 155.679

141.166 229.982 328o929 22.267 133.356 205.632 339.622

181.536 23'7.085 4Z1.782 135.894 264.857 345.867

208.864 256.494 462.662 136.754 310.647 366.006

218.163 293.454 466.877 137.161 326.163 416.034

222.536 377.865 474.156 137.530 331.912 534.232

223.740 439.379 506.080 333.298 596.8ll

224.267 454.344 583.691 786.469 333.881 605.794

Fifth

Z71.562

305.289

617.955

628.853

w ~

I I I

l

RATIO

S/r a/r First

235.6 75.

282.7 90.

314.2 100.

376.9 120.


FREQUENCY COEFFICIENT , p"ms4/EI SYMMETRIC MODES ANTI..SYMMETRIC MODES


224.649 458.732 7(]7.218 334.299 608.793

4h0.082 609.794

460.554 770.479 610.174

461.C!76 774.604

Fifth

959.978

963.463

965.815

w (Jl

TABLE 7

SUMMARY OF SOLUTIONS FOR THEORY INCLUDING ROTATORY INERTIA BUT NEGLECTING SHEARING DEFORMATION

FIXED-ENDED UNIFORM CIRCULAR ARCHES - eo = 900

RATIO FREQUENCY COEFFICIENT , p~.s4/EI S/r a/r SYMMETRIC MODES ANTI-SYMMETRIC MODES


25. 15.92 37.813 1(1:).772 160.009 248.748 314.634 52.066 86.120 175.165

50. 31.83 64 7'71 120.149 278.325 323.757 503.840 54.978 160.755 198.765 385.749

75. 47.75 86.196 132.928 288.822 479.0C/l 532.853 186.455 259.658 402.342

100. 63.66 96.947 155.764 293.438 536.167 645.710 55.635 189.925 337.(1:)7 413.078

150. 95.49 103.231 212.676 303.023 546.129 868.035 55.739 191.775 401.461 520.026

200. 127.32 104.918 256.342 328.744 550.919 878.001, 55.789 192.345 406.235 663.364

250. 159.16 105.614 272.930 380.576 555.949 882.542 55.796 192.603 407.735 696.295

300. 190.99 105o971 278.275 439.403 564.7(1:) 884.830 55.808 192.739 408.459 700.744

350o 222.82 106.180 280.586 488.8(1:) 585.250 888.575 55.826 192.819 408.874 702.618

400. 254.65 106.312 281.820 516.107 627.890 891.686 55.823 192.871 409.114 703.561

450. 286.48 106.403 282.570 527.187 685.285 896.465 55.837 192.907 4(1:).297 704.181

500. 318.31 106.466 283.063 532.158 744.893 905"605 ----~837_ 122.9_35 4_02~4~ 704.A87

Firth

328.168

476.273

669.348

688.001

703.179

731.087

867.126

1014.350

1066.710

1070.690

1079.570

1079_!1570 w 0\

RATIO

S/r a/r

25o 15.92

53. 31.83

75. 47o75

100. 63.66

150. 95.49

200o 127.32

250. 159.16

300. 190.99

350. 222.82

400. 254.65

450. 286.48

500 .. 318.31

TABLE 8

SUMMARY OF SOLUTIONS FOR THEORY INCLUDING ROTATORY INERTIA AND SHEAR DEFORMATION

FIJCED-ENDED UNIFORM CIRCULAR ARCHES - eo = 900 , r = 0.1

FREQUENCY COEFFICIENT , Plmi+/EI SYMMETRIC MODES ANTI-SYMMETRIC MODES


35.194 61.293 109.735 157.800 164.622 32.595 77.442 92.096 136.035

63.854 100.801 188.509 294.006 342.891 46.715 131.832 184.176 243.426

79.877 124.383 225.069 365.132 522.210 154.155 252.768 300.256

90.483 152.454 251.061 419.599 606.271 52.758 168.132 313.757 360.358

99.279 209.229 283.663 480.062 723.425 54.405 180.709 362.755 513.631

102.505 248.972 322.226 510.315 783.315 55.028 185.815 381.742 622.592

104.017 265,.520 377.975 529.469 817.286 55.302 188.319 391.310 654.232

104.843 272o490 436.134 547.616 838.272 55.466 189.723 396.755 669.705

105.343 Z76.141 482.498 576.240 853.233 55.558 190.587 400.144 679.009

105.667 2781)341 5CJ7.654 624.089 865.053 55.629 191.156 402.371 685.144

105.890 279.784 519.152 682.774 876.068 55.663 191.548 403.928 689.494

106 .. 050 280.7F!f7 525.208 741.752 889.733 55.696 191.832 405.037 692.513

Fifth

186.274

351.(]72

442.269

515.749

605.812

706.124

861.113

981.210

1018.900

1032.610

1043.990

1052.160 CJ.:) -.._]

RATIO MODE

s/r a/r

12.5 7.96 First Second Third Fourth Fifth

25. 15.92 First Second Third Fourth Fifth

37.5 23.87 First Second Third Fourth Fifth

50o 31.83 First Second Third Fourth Fifth

TABLE 9

S~~y OF PERCENT ENERGY

FIXED-ENDED UNIFORM CIRCULAR ARCHES - 60 = 900

PERCENT ENERGY

CLASSICAL THEORY ROTATORY INERTIA INCLUDED ROTATORY INERTIA AND SHEAR

iflNT I-SYMMETRIC SYMMETRIC ~en- F1e- Ex:ten- F1e-sion xure sion xure 86 14 37 63 14 86 97 3 99 1 6 94

97 3 98 2

15 85 84 16 6 94 5 95 95 5

96 4 12 88 13 87 88 12

96 4 4 96

2 98 79 21 78 22 11 89 20 80 14 86 3 97 85 15

97 3 2 98

SHEAR DEFORMATION NEGLECTEr ANT I-8YMMETRIC SYMMETRIC Ex:ten- F1e- ~en- F1e-

sion xure sion xure

13 87 68 32 86 14 4 96 2 98 97 3

1 99 1 99 99 1

2 98 79 21 74 26 12 88 24 76 7 93 2 98 93 7

98 2 1 99

DEFORMATION ( ANT I-SYMMETRIC

Shear Exten- F1e-sion xure

62 3 35 44 40 16 28 61 11 83 2 15 81 10 9

r = 0.1 ) INCLUDED SYMMETRIC

Shear Exten- F1e-

13 62 80 46 50

sion xure

68 13 2

44 41

19 25 18 10 9

w 00


PERCENT ENERGY RATIO ROTATORY INERTIA INCLUDED

MODE SHEAR DEFORMATION NEGLECT~ S/~ a/:rr ~ I..SYAIIB.rRIC S:'fMMI:rRIC IA;NT I..SYMME:l'RIC SYMMETRIC

iEJcte:rr Fle- ~en- Fle- Exten- Fle- ~en- Fle-sion xure sion xure sion xure sion xure

53. 31.83 First Second Third Fourth Fifth

75. 47.75 First 1 99 56 44 55 45 Second 9 91 38 62 8 92 39 61 Thil!d 88 12 2 98 89 11 2 98 Fourth 2 98 95 5 2 98 92 8 Fifth 41 59 5 95 6 94 8 92

100. 63.66 First 0 100 Z1 73 0 100 26 74 Second 2 98 68 32 2 98 68 32 Third 88 12 3 ,97 87 13 3 97 Fourth 9 9ll. :44 996 10 90 3 97 Fifth 1 99 96 4 1 99 97 3

125. 79.58 First Second Third 34 66 Fourth 64 36 1 99 Fifth 94 6

ROI'ATORY INERTIA,AND SHEAR DEFORMATION ( r ~..ll INCLUDED

ANTI-SYMMETRIC ~ear Exten- Fle-

sion xure

28 1 71 44 9 47 7 85 8

59 5 36 70 1 29

32 3 65 8 79 13

39 16 45 56 1 43

10 0 90 22 1 77 23 32 45 12 64 24 43 2 55

SYMMETRIC Shear Exten- F1e-

13 25 54 64 4

15 10 39 51 56

13 4

26 39 42

sioa Xl.lr'e

59 32 2 4

94

38 55 3 2 9

20 72 6 1

13

28 43 44 32 2

47 35 58 47 35

67 24 68 60 45

w '>0

I


PERCENT ENERGY

RATIO CLASSICAL THEORY RC1l'ATORY INERTIA INCLUDED

SHEAR DEFORMATION NEGLECTEr S/r· a/r· ~NT I-SYMMETRIC SlMMETRIC ANT I-SYMMETRIC SYMME.l'RIC

ExteiP- Fla.- l!lx!teiP- Fla.- Ex:teiP- Fle- ~aD- Fle-sion xure sion xure sion xure sion xure

150. 95.49 First 0 100 s 92 0 100 s 92 Second 1 99 78 22. l 99 77 23 Third 7 93 13 frl 6 94 13 frl Fourth 90 10 :t 99 90 10 1 99 Fifth 2 98 6 94 2 98 5 95

200. 1Z7.32 First 0 100 4 96 0 100 4 96 Second 0 100 45 55 0 100 44 56 Third. l 99 49 51 1 99 49 51 Fourth 60 40 2 98 57 43 2 98 Fifth 38 62 1 99 40 60 11 99

250. 159.16 First 0 100 2 98 0 100 2 98 Second 0 100 15 85 0 100 15 85 Thll.rd 1 99 77 23 1 99 77 23 Fourth 5 95 5 95 5 95 5 95 Fifth 91 9 1 99 9 91 1 99

300. 190.99 First 0 100 1 99 0 :100 1 99 Second 0 100 7 93 0 100 7 93 Third 0 100 78 22 0 100 77 23 Fourth 2 98 l3 frl 1 99 l3 frl Fifth 59 41 1 99 66 34 1 99

"

ROTATORY INERTIA AND SHEAR DEFORMATION ( r = 0.1 ) INCLUDED

ANT I-SYMMETRIC Shear ExteiP- Fle-

sion xure

5 0 95 11 1 88 19 3 78 3 86 11

24 10 66

3 0 97 7 0 93

12 1 frl 15 18 67 4 78 18

2 0 98 4 0 96 8 1 91

12 3 85 2 frl 11

1 0 99 3 0 97 6 0 94 9 1 90 8 34 58

SYMMEI'RIC Shear ExteiP- Fle-

7 4

11 23 31

5 5 4

l4 20

3 5 1 9

l4 2 4 2 6

10

sion xure

7 70 20 1 1

3 ,36 57 3 1

2 13 77 6 1

1 6

73 17

1

86 26 69 76 68

92 59 39 83 79

95 82 22 85 85

97 90 25 77 89 ~ 0

.

I

!

!

I


PERCENT ENERGY RATIO CLASSICAL THEORY ROTATORY INERTIA INCLUDED

MODE SHEAR DEFORMATION NEGLECTED S/r a/r ANT I..SYMMEJrRIC SYMJ\IIETRIC ANT I-SYMMETRIC SYMMEI'RIC

Ex:ten- Fle- IExten- F1e- Ex:ten- Fle- Ex:ten- Fle'J sion xure sion xure sion xure sion xure

350o 2220 82 First 0 100 1 99 0 100 1 99 Second 0 100 4 96 0 100 4 96 Third 0 100 57 43 0 100 56 44 Fourth 1 99 36 64 1 99 37 63 Fifth 18 82 2 98 12 88 2 98

400. 254.65 First 0 100 1 99 0 100 1 99 Second 0 100 3 97 0 100 3 97 Third 0 100 26 74 0 100 26 74 Fourth 1 99 81 19 0 100 69 31 Fifth 4 96 3 97 4 96 3 97

450. 286.48 First 0 100 1 99 n 100 1 99 Second 0 100 2 98 0 100 2 98 Third 0 100 12 88 0 100 12 88 Fourth 0 100 79 21 0 100 79 21 Fifth 1 99 6 94 Q 100 6 94

500. 318.31 First 0 100 1 99 0 100 0 100 Second 0 100 1 99 0 100 1 99 Third 0 100 6 94 0 100 6 94 Fourth 0 100 78 22 0 100 77 23 Fifth 1 99 13 87 1 99 13 87

ROTATORY INERTIA AND SHEAR DEFORMATION ( r = Ojll ) INCLUDED

ANT I-SYMMETRIC Shear Ex:ten- F1e-

sion xure

1 0 99 2 0 98 4 0 96 7 0 93 9 4 87

1 0 99 2 0 98 3 0 97 5 1 94 7 1 92

1 0 99 1 0 99 3 0 o/7 4 0 96 6 2 92

0 0 100 1 0 99 2 0 98 4 0 96 5 0 95

SYMME:rRIC Shear Ex:ten- Fle-

2 3 3 3 8

1 3 3 1 6

1 2 3 1 5

1 2 3 1 3

sion xure

1 4

49 43

2

1 2

22 70

4

1 2

11 78 7

0 1 6

75 16

o/7 93 48 54 90

98 95 75 29 90

98 96 86 21 88

99 97 91 24 81 "'" I-'

!

'

i

I

r

RATIO

S/r a/n First

25. 31.83 Ex:tension 8 Flexure 92

50. 63.66 Extension 1 Flexure 99

75. 95.49 Extension Flexure

100. l'Zl.32 Extension 0 Flexure 100

150. 190.99 Extension Flesxure








TABLE 10

SUMMARY OF PERCENT ENERGY

FIXED-ENDED UNIFORM CmCULAR ARCHES- 9o = 45°

CLASSICAL THEORY

PERCENT ENERGY ANTI-sYMMETRIC MODES

Second Third Fourth Fifth First 2 35

98 65 94 5 1 99 67 6 95 99 1 33 6 94 64

94 6 36 1 97 2 0 79

99 3 98 100 21 4 95 0 58

96 5 100 42 1 94 5 28

99 6 95 72 4 95 l4

96 5 86 1 97 8

99 3 92 5

95 2

98

0 0 0 0 2 100 100 100 100 98

SYMMETRIC MODES Second Third Fourth

2 98 6 98 2 94 2 9 91

98 91 9

10 0 0 90 100 100 36 1 0 64 99 100 67 2 0 33 98 100 79 21 78 12 22 88 68 25 1 32 75 99 47 47 53 53

66 3 34 97

16 76 4 84 2.4 96

Fifth

94 6 1

99

99 1

97 3 0

100

,j::. 1:-.J

I I

' i

I

RATIO

S/r a/n First

12.5 3.98 Extension 58 Flexure 42


37.5 11.94 Extension Flexure









350. 111.49 Extension 0 . Flexure 100

---- -~- -~

TABLE 11


FIXED-ENDED UNIFORM CIRCULAR ARCHES - eo = 1800

CLASSICAL THEORY PERGil! ENERGY

ANTI-SYMMETRIC MODES Second Third Fourth Fifth First

43 96 44 59 57 4 56 41 81 13 92 46 76 19 87 8 54 24 70 25 84 30 75 16 34 61 8 94 24 66 39 92 6 76 8 78 13 20 8

92 22 87 80 92 3 '5I 57 3 4

97 63 43 97 96 ll 79 7 89 21 93

66 26 2 34 74 98

l 2 ll 79 1 99 98 89 21 99

0 1 1 40 1 100 99 99 60 99

0 l 2 11 0 100 99 98 89 100

0 0 1 2 0 100 100 99 98 100

SYMMEI'RIC MODES Second Third Fourth

88 20 93 12 80 7 23 91 17 77 9 83 48 23 52 77 71 16 86 2$ 84 14 74 18 68 26 82 32 39 55 7 61 45 93

7 74 17 93 26 83 3 23 69

97 77 31 1 6 75

99 94 25 l 3 34

99 97 66 1 2 10

99 98 90

Fifth

92 8

86 14 92 8 5

95 31 69 91 9

7 93 4

96 16 84 59 41 79 21 ~ w

TABLE 12

SUMMARY OF PERCEN'l' ENERGY

HINGED-ENDED UNIFORM CIRCULAR ARCHES - 9o = 45°

CLASSICAL THEORY

RATIO PERCENT ENERGY

S/r a/n First ANT I-SYMMETRIC MODES

Second Third Fourth Fifth First

5.89 7.5 E:rlension 98 2 100 Flexure 2 98 0

7.85 10. Extension 95 5 100 100 Flexure 5 95 0 0

ll.78 15. E:rlension 57 44 99 2 Flexure 43 56 1 98


31.42 40. Extension 1 98 1 99 1 84 Flexure 99 2 99 1 99 16

39.Zl 50. Extension 0 96 3 5 Flexure 100 4 97 95

47.12 60. Extension 68 32 0 Flexure 32 68 100

62.83 so. Exrliension 2 97 0 91 Flexure 98 3 ]00, 9

70.69 90. Extension 1 98 0 89 Flexure 99 2 100 ll


109.96 140. Extension 48 52 59 Flexure 52 48 41

125o66 1600 Extension 3 96 39 Flexure 97 4 61


99 3;, 98 1 97 2

2 98 45 98 2 55 1 99 1

99 1 99 2 36 64

98 64 36 2 1

98 99 5 0 99

95 100 1 7 0 97

93 100 3 21 0 79 100 39 0 61 100 59 41

Fifth

100 0

0 100

~ ~

I

RATIO

S/r a/r First

157..08 200o Extension Flexure

176.72 225. Extension Flexure


235.62 300. Ex:tenaion Flexure

'Zl4.89 350o Extension 0 Flexure 100


353.43 450o Extension Flexure

400.55 5llO. Ex:liension Flexure



549.78 700. Exitension Flexure



----------- - -- L ____


PERCENT ENERGY ANTL-SYMMETRIC MODES

Second Third Fourth Fifth Firat

1 99 ]6 99 1 84

96 4

43 57 1 4

99 96 0 0

100 100

1 99


82 18

89 11 79 17 21 83 55 42 45 58 'Z7 70 73 30 12 85 88 15 6 89

94 n 89 ll 82 18 65 35 35 65

Fifth

.t::. CJ1

!

RATIO

S/~Jt a/r First

7.85 2.65 Extension 42 Flexure 58

15.71 5. Extension 10 Flexure 90

23.56 7.5 Extension 14 Flexure l4







157o08 50. Extension Flexure

188o50 60 Extension Flexure

TABLE 13


HINGED-ENDED UNIFORM CIRCULAR ARCHES - Go = 1800

PERCENT ENERGY ANTI-SYMMETRIC MODES

Second Third Fourth Fifth First

58 97 ].()() 98 42 3 0 2 88 43 60 89 12 57 40 11 15 93 56 85 7 44

82 24 18 76

11 86 5 7 89 l4 95 93 4 86 11 3

96 l4 89 97 2 48 50 2

98 52 50 98 1 13 84 3 1

99 87 16 97 99 1 3 81 15 1

99 97 19 85 99 1 17

:·99 83 1 4

99 96


88 22 91 12 78 9 Z7 84 83 73 16 17 47 92 13 53 8 87 77 68 33 23 32 67 90 8 94 10 92 6 81 18 78 19 82 22 39 59 8 61 41 92 13 84 37 87 16 96 3 86 10

97 l4 90 2 37 60

98 63 40 1 8 88

99 92 12

1.--------- ---------------------

Fifth

3 97

'""" 0\


RATIO PERCENT ENERGY ANT I-SYMMETRIC MODES

S/r a/r First Second Third Fourth Fifth First

235o62 '75. Extension 0 1 Flexure 100 99

282.74 90. Extension 1 3 Flexure 99 97

314.16 100. Extension 0 2 Flexure 100 98

376.99 120. Ex:tension 1 Flexure 99


1 2 75 99 98 25

1 99 1 5

99 95 0 2

100 98

Fifth

'""" --...]

RATIO

S/r a/r

25. 15.92:

5o. 31.S3

75. 47.75

100. 63.66

150. 95.49

200. 1Z7.32

300. 190.99

TABLE 14

SUMMARY OF SOLUTIONS FOR THEORY INCLUDING ROTATORY INERTIA AND SHEAR DEFORlAATION

FIXED-ENDED UNIFORM CffiCULAR ARCHES - 6o = 90° , r = 0.3

FREQUENCY COEFFICIENT , P~~/EI SYMMETRIC MODES ANTI-sYMMEI'RIC MODES

First Second Third: Fourth Fifth First Second Third Fourth

36.Sl3 S2.49S l55o061 167.651 249.562 42.442 S4.'Z17 122.541

63.650 110.054 231.S07 321.267 3S7.349 51.3S3 150.473 1S3.473 307.611

S4.040 129.445 261.S62 448.022 4SS.529 53.719 173.645 257.745 355.979

94.659 154.559 Z76oS71 487.470 643o5S5 54o62S 1S1.?56 332.351 3?!7.lS3

101.861 211.592 295.892 521.162 Sll.011 55.284 1?!7.S66 3?!7.29S

326.356 536.2S? S42.762 55.531 190.093 397.555 651.60S

105o591 Z76.305 438.354 558.?15 869.439 55.694 -~9_1~7gl_ .. 404._449 689.932

Fifth

463.006

560.97S

613.046

719.098

1005.700

~ 00

I

RATIO

S/r a/r

25. 15.92

50. 31.83

75. 47.75

100. 63.66

150. 95o49

200. l27.32

300o 190.99

TABLE 15

SUMMAR! OF PERCENT ENERGY

FIXED-ENDED UNIFORM CIRCULAR ARCHES- 9o = 900

THEORY INCLUDING ROTATORY INERTIA AND SHEAR DEFORMATION- r = 0.3 PERCENT ENERGY

ANTI-SYMMETRIC MODES SYMMETRIC MODES First Second Third Fourth Fifth First Second Third Fourth

Shear 94 4 50 5 43 24 35 Extension 7 87 7 70 6 58 41 Flexure 59 9 43 25 51 18 24 Shear l3 16 10 37 33 4 15 31 2 Extension 1 38 59 1 28 73 17 3 95 Flexure 86 46 31 62 39 23 68 66 3 Shear 6 l3 1 22 30 5 5 18 23 Extension 1 6 89 4 1 49 45 2 14 Flexure 93 81 9 74 69 46 50 80 63 Shear 4 9 4 11 21 5 1 11 17 Extension 0 2 74 22 1 24 70 4 2 F1ezure 96 89 22 67 78 71 29 85 81 Shear 2 4 7 3 1 4 9 Extension 0 1 5 7 75 15 1 Flexure 98 95 88 90 24 81 90 Shear 1 2 4 4 3 1 5 Extension 0 1 1 40 57 52 2 Flexure 99 t:yr r ~5 56 40 47 93 Shear 0 1 2 3 2 1 1 1 2 Extension 0 0 0 1 57 1 7 76 14 Flexure 100 99 98 96 41 98 92 23 84

Fifth

67 1

32 41

3 56 4

85 11 1

96 3

l3 2

85 8 1

91 4 1

95 ~

"'

I

I

I

I

50

Fig; 1: COORDINATE AND DISPLACEMENT NOTATIONS

0

....... c <tl c

E .... <lJ

....... <lJ

0

1.5

0 6 0 : 90 , S/r : 100.

~ 1 ~ ~ ~ ~

~ I )y I I \:1 I I 11 ll I I I I I l'f Cn 01 I ~' I I A I I I I ' I fl I

Continuous 150

(a) A ntisymmetric Modes (b) Symmetric Modes

Fig. 2: TYPICAL PLOTS OF DETERJ\1INANT VS. FREQUENCY -

FIXED ENDED ARCHES C,/1 1-'

,..,.

c: u

. ..... c: a> ·-(.)

'+-'+-a> 0 u

>. (.)

c a> :::J 0" a> ....

1..1..

I I v

800

r-- 80: 900 I f-

I I 1000

~

600 r / v

~ / I 400 ))

~// v

200

0

~~ I I I I I I I i I I I I I I I I I

100 200 300 Slenderness Ratio, yr

(a) Antisymmetric Modes

-

400 500

1000

-(

I / 800

v / I I .,../

,--: --- --600

I I I / v

}) ----/ 400

,I( ~:..---

'-~)___-v 200

_!~ I I I I I I I I I I I I I I t I I I I I

0 100 200 300 400 500 Slenderness Ratio , S/r

(b) Symmetric Modes

Fig. 3: NATIJRAL FREQUENCIES FOR FIXED ENDED ARCHES- CLASSICAL THEORY CJ1 N

...

Third Moue , c3 = 409.7

Second Mode, c2 = 193.0

First Mode , c1 =55. 8


eo: 90° I Sjr : 50

~I ~ Third Mode, c3 : 532.6

~ zd 'C7. " I

Second Mode, c2 : 283.2

t1r

First Mode, c1 : 106.5

(b) Symmetric Modes

Fig. 4: PREDOMINANTLY FLEXURAL MODES OF VIDRATION- FIXED ENDED ARCHES C/l w

r•

c u

-c Cl)

(.)

-Cl)

0 u

>(.)

c Cl)

:::J 0" Cl) ,_

u..

1000

0

eo= 900

~ EE:. .38 FE=.62

~~~----=-~ EE::.. 59 FE::.. 41

EE: . 91 FE= .09

,___~

EE = .60 FE: .40

~ EE =. 78 FE=.22

EE = .90 FE=.10

~

~ EF =. 34 FE=. 66

;:;:;:~ EE =. 78 . FE=.22

~ FE=.16

100 200 300 400 500 Slenderness Ratio , S/r

(a) Anti symmetric Modes

0

EE=.49 EE=.26 FE=.51 ~FE=.74

~ EE:.-.57 EE = 78 FE = . 43 . \ FE=.22 VI ~ -~ ~ ·\~ =-r '

Slenderness Ratio , S/r

(b) Symmetric Modes

EE =. 45 FE=.55

400

Fig. 5: FREQUENCIES ASSOCIATED WITH PREDOMINANTLY EXTENSIONAL MODES OF VIBRATION

FIXED ENDED ARCHES

500

C/1

'""'"

"1

55

Second Mode for S/r = 25~

Second Mode for S/r = 50.

Third Mode for S/r = 100~

v

-......_. /~I s 7 -


Fourth Mode for S/r = 125.

Fig~ 6a: PREDOMINANTLY EXTh~SIONAL ANTISY}mETRICAL MODES OF VIBRATION

- FIXED ENDED ARCHES -

56


I

<::-----..... ~ -........... I v ~~_z ____ S:....::::==--~:Z:::-...Ji

I

I Fourth Mode for S/r = 200.

~---:?

Fifth Mode for S/r = 200~

~~----::::::::, v ~ :s;::

~.

Fifth Mode for S/r = 250. I

::/'.. ~ ~ ~~"'C/~s / . Fifth Mode for S/r = )oo; <;;:;r

Fig. 6b: PREDOMINANTLY EXTENSIONAL ANTISYMMETRICAL MODES OF VIBRATION


57

First Mode for S/r = 12.5

'W ~------v

First Mode for S/r


~._.__2? ___ :::;=>J Second Mode for S/r = 150~

I I

'W ~--- ~

~2v ?::s~ 7i Second Mode for S/r = 200~

Fig~ 7a: PREDOMINANTLY EXTENSIONAL SYMMETRICAL MODES OF VIBRATION


Third Mode for

> 2v

S/r = 200~


____ :s;~--~ 4v ~----~-




58

Fig~ 7b: PREDOMINP~LY EXTENSIONAL SYMMETRICAL MODES OF VIBRATION


•

c u

..... c Q)

u --Q)

0 u

>u c Q) :::::s 0' Q) ..... u.

1000~ ~

0

eo= 900 100+

Co' EE = .0

~ FE:.1.0 ~ EE:. .0 '~ --:7'--~

EE:. .20 ~--- ----·--- ~~

EE = .0 '\\_\

100 200 300 400 500 0 100 200 300 400 500 Slenderness Ratio , S/r Slenderness Ratio , S/r

(a) Antisymmetric Modes (b) Symmetric Modes

Fig. 8: MODES CORRESPONDING TO FREQUENCIES OF NEARLY EQUAL VALUES -FIXED ENDED ARCHES

C/1 \0

60

v

Third Mode for S/r = 50.

v

Second Mode for S/r = 500;

I

~~~ 5v -><:...___ i

Fourth Mode for S/r = 75~

Fig~ 9a: ANTISYMMETRICAL MODES CORRESPONDING TO FREQUENCIES OF NEARLY

EQUAL VALUES - FIXED ENDED ARCHES -

61

5v s ~ <::::::::::--~--Third Mode for S/r = 500;

I ~~I ~'<::7 ~ 2v 1

I Fifth Mode for S/r = 150;

Fourth Modo for S/r = 500~

Fig; 9b: ANTISYMMETRICAL MODES CORRESPONDING TO FREQUENCIES OF NEARLY


•

62


v

First Mode for S/r = 500.

2v


2v


Fig~ lOa: SYMMETRICAL MODES CORRESPONDING TO FREQUENCIES OF NEARLY


~ I

-><----Fifth Mode for S/r = 50~

lOv

Fourth Mode for S/r = 200~

~ c--......... I ~· ·~<::::7 ~z::::J


Fig~ lOb: SYMMETRICAL MODES CORRESPONDING TO FREQUENCIES OF NEARLY

EQUAL VALUES- FIXED ENDED ARCHES -

63

c u -

-c (!) -u ·-..,_ ..,_ <ll 0 u

I >-u --t c I (!)

::I I

0" I (!) I '-

u... I 200

0

e-0 = 450

I I I I

100 200 300 Slenderness Ratio I S/r


[

400 500 200 300 400 Slenderness Ratio I S/r

(b) Symmetric Modes

Fig. 11: COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS -

FIXED ENDED ARCHES WITH eo= 4 5°

500

0\ ..,..

•

c: u

-c: Q)

u ..... ..,_

Q)

0 u >. u c: Q)

::l cr Q) ,_ u.

----------------- ,--:..::=- I

9o = 90o

I I.

" " lOOOl /

I

, , , , ~

I ---------- --- -,.L-------

~ , ,

II'. ,., , , , ,

------7L--- - --- I

r I I I J ~~~~~~~~_L~~~_L~~~~

100 200 300 400 500 Slenderness Ratio , S/r


II'. ~

, ,

100 200 300 400 Slenderness Ratio , S/r

(b) Symmetric Modes

Fig. 12: COMPARISON OF APPROXIMATE AND EXACT SOLUTIONSFIXED ENDED ARCHES WITI-I eo = 9 0°

500

0\ CJl

c: u

-c: Q)

u "+

"+-Q)

0 u >. u c: Q)

::::l cr Q) '

LJ..

0

8.0 = 180°

-'--L---'--'- 1 . j _____1_1 __j_ L L _l___l_ I I I l__l_l - I I I I

100 200 300 400 Slenderness Ratio , S/r


r- ---,---10001 I ------- --

r 1 I 1 r I ,'

- I I ---- I

I-+.--- I

I --1 l - - - -,L - - - -I I -----

1 I

I.

. I -~~~I I I I I I I I I I I I I I I I I I I I I

0 100 200 300 400 Slenderness Ratio , S;t

(b) Symmetric Modes

Fig. 13: COMPARISON AND APPROXIMATE AND EXACT SOLUTIONS -

FIXED ENDED ARCHES WITH 9o = 18 0° Q\ Q\

•

900 Classical Theory

800 ----- Solution Included Rotatory Inertia

700

s::l 0 600

.A .. ~ (J)

"8500 •rl

t: Q) 0

0 400 fi s::l

~ ar 3oo t:

200

100

0 100 200 300

Slenderness Ratio , S/r Q\ '-J

Figo 14: EFFECTS OF SHEAR DEFORMATION AND ROI'ATORY INERTIA FOR ANTISYMMETRIC MODES - FIXED ENDED ARCHES -

900

800

700

OS:: 600 .. ~ ~ 500 C)

"1"1

t: 8

0 400

~ ~ g 300 l::

200

100

0

1.:.

A

----------8------A-----------

----------------

100 200 300


Fig. 15: EFFECTS OF SHEAR DEFORMATION AND ROTATORY INERTIA FOR SYMMETRIC MODES - FIXED ENDED ARCHES -

0\ 00

500

400

300

c:f .. 200

~ Q)

ori 0

ori lH

100 lH Q) 0

0

~ ~ 0

~

200

100

0

----- Classical Theory 69

-·-With Rotatory Inertia, No Shear Deformation

-··-With Rotatory Inertia And Shear Deformation { r = 0.1)

....

r-

r·' .., .. /

/.· . .

~ --··-·· --··-··--v .. ~ .. / ~·

.~

/ L/ ./

100

Third Mode

--··-·· ··-··-. Secon Mode

First Mode


200

Fig. 16a: EFFECTS OF SHEAR DEFORMATION AND ROTATORY INERTIA

FOR ANT !SYMMETRIC MODES - FIXED ENDED ARCHES -

1000

900

800

700

.. § 600 ..-1 0

..-1

~ Q) 0

0 >. 500 g

~ t:

300

~------~---------.---------.--------~--------,70

/ .·

-·-Fo h Mode

100

. . / ./

--·· ..----· .

Fifth Mode

200


Fig. 16b: EFF~TS OF SHEAR DEFORMATION AND ROI'ATORY INERTIA

FOR ANT ISYMrmRIC MODES - FIXED ENDED ARCHES -

~ <I)

•r-i C)

----- Classical Theory 71

-·-With Rotatory Inertia , No Shear Deformation

-··-With Rotatory Inertia And Shear Deformation ( f = 0.1)

600 ~------~--------~--------.---------.--------.

500 ~-------+--------~------~,--------r------__,

400 t----------t--

Third Mo

8 200 <I) 0

0

~ C)

§ 100 L-.-----'

& <I)

rt

0 100 200


Fig. 17a: EFFECTS OF SHEAR DEFORPAATION AND ROTATORY INERTIA

FOR SYMMEI'RIC MODES - FIXED ENDED ARCHES -

~ 0 500 ~-----+-#--~----~--------~------~---------~ ..

. . I

.. --··--

100 ~~~~~~-L~~~~._~~~~~~~~~~~ 0 100 200


Fig. 17b: EFFECTS OF SHEAR DEFORMATION AND ROTATORY INERTIA

FOR SYMMETRIC MODES - FIXED ENDED ARCHES -

72

c u

-c (l)

u ...... ......

(l)

0

u >u c (l)

::::1 o(l) .....

u...

0

____/ eo= 900

600 ~

E~ FE = :4

~ EE =.44 FE=.56

200 Sir


401 '-

0

~

~ ~ EE =.52 FE= .48

~ FE =,.34

100 200 Slenderness Ratio , s/r (b) Symmetric Modes

FE=.49

Fig. 18: FREQUENCIES ASSOCIATED WITH PREDO:MINANTLY EXTENSIONAL MODES -

HINGED ENDED ARCHES

300

-.....) w

Second Mod0 for S/r = 31~42

v ::::;:~::=.....cc::::...:::::::===:.;;>'~l· Second Mode for S/r = 43~98

Third Mode for S/r = 43~98

Third Mode for S/r = 70.69

Fig~ 19a: PREDOMINANTLY EXTENSIONAL ANTISYMMETRICAL MODES

OF VIBRATION - HINGED ENDED ARCHES -

74

•

~ I

1 7s

Third Mode for S/r = 102~1

~---:_...;7'

Fourth Mode for S/r = 102.1

Fourth Mode for S/r = 141~4

I """" I s ~

v ~ ..... -7 <=:?. Fourth Mode for S/r = 188~5

Fig~ 19b: PREDOMINANTLY EXTENSIONAL ANTIS'YMMETRICAL MODES

OF VIBRATION - HINGED ENDED ARCHES -

I

76

v

First Mode for S/r = 50;27

5v

Second Mode for S/r = 50~27

w~ ~ ~ ---....a ~



Fig; 20a: PREDOMINANTLY EXTENSIONAL S'IMMETRICAL MODES OF VIBRATION

- HINGED ENDED ARCHES -

~ I

77



I

~cz::s<J I

Third :Mode for S/r = 314~2 I

w I ~~~~

10v 1

Fourth Mode for S/r = 314~2 I

Fig~ 20b: PREDOMJ:NANTLY EXTENSIONAL SYMMETRICAL MODES OF VIBRATION

- IflNGED ENDED ARCHES -

c u

c Q)

u -...... Q)

0 u >. u c Q)

:::1 oQ) .....

1..1...

/ /

------------------~~ I

eo= 450

I

I I

I

100 200 Slenderness Ratio , S/r

(a) Anti symmetric Modes

I I

I I

I


(b) Symmetric Modes

Fig. 21: COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS -

HINGED ENDED ARCHES WITH 9o = 45°

300

--.) cc

c u

... c C'l)

u --Q)

0 u

oQ) ....

·u..

0

7 I

-------- ,' eo= 1800 ----7---;;...:::oo--I.

IJ IJ

100 200 Slenderness Ratio , ~r

--_j


I I

I I

20

I I

I

100 200 Slenderness Ratio , S/r

(b) Symmetric Modes

Fig. 22: COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS

HINGED ENDED ARCHES WITH eo = t 8 Oo

300

-...]

""

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