Vibration Frequencies of Buried Pipelinesunderground piping system, this paper developes and provides the basic fundamentals of dynamics of buried pipelines. The dynamic fundamentals

Seismic Vulnerability, Behavior and Design

of Underground Piping Systems

Vibration Frequencies of Buried Pipelines

by

Sponsored by National Science Foundation

Research Applied to National Needs (~~)

Grant No. ENV76-l4884

Technical Report (SVBDUPS Project) No. 2R

Januar.y 1978

Department of Civil EngineeringRensselaer Polytechnic Institute

Troy, New York 12181

Any opinions, findings, conclusionsor recommendations expressed in thispublication are those of the author(s)and do not necessarily reflect the viewsof the Nltional Science Foundation.

,t;}

50272 '101

_.._----- -----~-~----

(C)

8. Performing Organization Rept. No.

-- --- --- ._--- ----------11. Contract(C) or Grant(G) No.

2R-- .~---------------___'l10. Project/Task/Work Unit No.

Rensselaer Polytechnic InstituteDepartment of Ci vil EngineeringTroy, New York 12181

7. Author(s)

REPORT ~~MENTATION Jc-~~~iTR;~'780339 .____ 1

2

• __ ~3"J~+cffljnt?~O1214. Title and Subtitle .-- -.. -- I 5. Report Date

Vibration Frequencies of Buried Pipelines (Seismic Vulnerabil"l'1 January 1978ity, Behavior and Design of Underground Piping Systems, Techni-~-('(l-' ..... Nn ?R) _ .___ _

t--:-="L~. .:..:R"-..-,W.:.=a:.:.;nc..;;1qL- . . . _9. Performing Organization Name and Address

(G) ENV7614889- -- ---- ----- ----------- --_._-_._---------

13. Type of Report & Period Covered

---------_._._------- ----{

12. Sponsoring Organization Name and Address

Applied Science and Research Applications (ASRA)National Science Foundation1800 GStreet, N.W.Washinqton, DC 20550

~-=..:..=-=-:..:...:....:.'-.oL.:::....::....:..:'-L-----'=-"-'=-=-.c'--------- _

15. Supplementary Notes

Techni cal-- --~- --_._---

14.---

- --- ----_._----_._--16. Abstract (Limit: 200 words)

Overall aims of this research are to develop a systematic way of assessing theadequacy and vulnerability of water/sewer distribution systems subjected to seismicloads and also to develop future design methodologies for water/sewer systems. Thispaper develops and provides the basic fundamentals of dynamics of buried pipelines.The dynamic fundamentals reported include the determination of fundamental frequenciesof continuously elastic-supported straight pipelines subjected to axial, torsional,and flexural motions. Various boundary conditions, which can represent the actualconstruction, have been considered. Using a finite element and consistent massapproach, the matrix formulation of buried pipeline is developed.

17. Document Analysis a. Descrip,tors

Desi gnEa rthq uakesEarthquake resistant structuresDynamic structural 'analysis

Water pipesWater supplySewer pipes

HazardsPiping systemsSubsurface structures

b, Identifien;/Open·Ended Terms

Matrix formulation of buried pipelineEarthquake engineering

REPRODUCED BYNATIONAL TECHNICALINFORMATION SERVICE

U S DEPARTMENT OF COMMERCE• •SPRINGFIELD, VA. 22161

c. COSATI Field/Group

18. Availability Statement

NTIS.

119. Security Class (This Report)

120. Security Class (This Page)

I

I~i7c:Y/!Fi~(See ANSI-Z39.18) See InstructIons on Reverse OPTIONAL FORM 272 (4-77)

(Formerly NTlS-35)Department of Commerce

TABLE OF CONTENTS

ACKNOWLEDGEMENT .

KEY WORDS.

ABSTRACT

INTRODUCTION

BACKGROUND

AXIAL VIBRATION FREQUENCY OF A STRAIGHT BURIED PIPE •

TORSIONAL VIBRATION OF A STRAIGHT BURIED PIPE .

FLEXURAL VIBRATION OF A STRAIGHT BURIED PIPE

MATRIX FORMULATION OF BURIED PIPING SYSTEM

CONCLUSION

REFERENCES

TABLE.

FIGURES

PAGE

ii

iii

iii

1

1

2

6

9

15

22

24

27

29

NOTATION .,

ib

41

ACKNOWLEDGEMENT

This is the revised version of Technical Report No. 2 titled 'Dynamics

of Buried Pipelines' produced under the general title of 'Seismic Vulner­

ability, Behavior and Design of Underground Piping Systems' (SVBDUPS)., for

the publication in ASCE Journal of Technical Councils.

The research has been sponsored by the Earthquake Engineering Program of

NSF-RANN under grant no. ENV76-l4884 and Dr. S.C. Liu is the Program Manager

of this Project in which Dr. Leon Ru-Liang Wang is the Principal Investigator.

The overall aims of this research are to develop a systematic way .of assessing

the adequacy and vulnerability of water/sewer distribution systems subjected

to seismic loads and to develop future design methodologies.

The author wishes to express his appreciation for the inputs and discus­

sions from Dr. Michael O'Rourke, Senior Investigator, and Mr. Kwong M. Cheng,

Research Assistant of the Project and from several reviewers of the original

paper.

Appreciation also goes to th~ Advisory Panel which consists of Mr. Holly A.

Cornell, President of CHZM Hill, Inc., Corvallis, Oregon; Mr. Warren T. Lavery,

Superintendent of Latham Water District, Latham, N.Y.; Dr. Richard Parmelee,

Professor of Civil Engineerng, Northwestern University, and Drs. Jose Roesset

and Robert V. Whitman, Professors of Civil Engineering, M.I.T. for their

constructive comments and suggestions.

The typing and proofreading of this report by Mrs. Jo Ann Grega is also

appreciated.

Please note that although the project is sponsored by the National Science

Foundation, any opinions, findings and conclusions or recommendations expressed

in this publication are those of the author and do not necessarily reflect the

view of NSF.

ii

Key Words:

Buried Pipelines; Dynamics; Lifeline ~arthquake Engineering;

Soil-Structure Interactions; Vibrations; Beams on .Elastic Foundation;

Natural Frequency

ABSTRACT

To aid the research on seismic vulnerability, behavior and design of

underground piping system, this paper developes and provides the basic

fundamentals of dynamics of buried pipelines.

The dynamic fundamentals reported include the determination of funda-

mental frequencies of continuously elastic-supported straight pipelines

subjected to axial, torsional and flexural motions. Various boundary con-

ditions, which can represent the actual construction, have been considered.

Using a finite element and consistent mass approach, the matrix formu-

lation of buried pipeline is developed.

iii

INTRODUCTION

An earlier study(27) has shown that buried water/sewer pipelines have

been damaged heavily by earthquakes. Other than the catastrophic failures

caused by landslides or liquefaction of soil, substantial failures of buried

pipelines reported were resulted from seismic shaking/vibration. Many papers

on the dynamic analysis and design of above ground buildings can be found(3,25),

but very little has been done for underground pipelines~ Only until recently,

(5 27) (11 21)several state of the art' and behavioral study , papers have peen

published.

To aid the research on seismic vulnerability, behavior and design of

underground piping system, several aspects of the basic fundamentals of dy­

namics of buried pipelines have been studied(26). Based on the.report(26),

this paper presents the fundamental frequencies of continuously elastic-

supported pipelines subjected to axial, torsional and flexural motions.

Using a finite element and consistent mass approach(4), the matrix formula-

tion of a buried pipeline system is developed.

BACKGROUND

Continuously supporteJ structures on ground or below ground may be an­

alyzed by using the anology of beams on an elastic foundation(9). Dynami-

cally, soil resistant springs have been used to handle pile-soil foundation

(18) (8 22)problems and other underground structures '

In a recent paper, Parmelee and Ludtke(17) formulated the dynamic equ-

ation of motion for buried pipelines which were treated as a plane strain

problem. The spring constant was obtained analytically using elastic half

space theory originally developed by Mindlin and Cheng(15). It was found

that the value of the spring constant is a function of the Young's modulus

at the site, diame.ter of pipe and buried depth. The static soil reaction

1

modulus has been evaluated and shown in another' paper (16) •

In another paper, Sakurai and Takahashi (20) have studied the dynamic

longitudinal stresses of underground pipelines during earthquakes theoreti-

cally as well as experimentally. In this study, the resistance to the motion

was assumed to be the friction force between the pipe and the soil. They

further assumed that the friction force was linearly proportional to the re-

lative displacement. Their discussion was extended briefly to include lateral

motion, but not vertical or torsional vibrations.

AXIAL VIBRATION FREQUENCY OF A STRAIGHT. BURIED PIPE

A buried pipe restrained by friction forces surrounding the pipe and an

elastic spring at the right end, subjected to axial motion is shown in Fig. 1.

Using the notations shown in Fig. 1, the dynamic equilibrium equation of an

undamped beam is

m dx y + f 21TR dx = Adao

(1)

where m, A, R are mass per unit length along the pipe, cross sectional areao

and outer radius of pipe; f, a are frictional force per unit area and axial

stress; y, absolute displacement of pipe. Note that the mass along the pipe

may include the mass of water in pipe, and a portion of the soil mass that

. . (17)might move with the p1pe as descr1bed by Parmelee in addition to pipe mass

itself. This mass may be expressed as

m = A mp

(2)

where A is a constant; m is mass per unit length of pipe itself.p

Assuming that the frictional force is proportional to the relative dis-

placement, u = y - Ys' the equation of motion in terms of u becomes

2

•• 21TR A ....u + __0_ k u acr

A m a = Am ax - y sp p

(3)

where ka is the frictional spring constant and Ys is the absolute displace­

ment of soil medium.

aSubstituting the stress-strain (cr = EE) and strain-displacement (E = a~)

relationships, aIl~ e~.fn:~~a,~~::g~~he excitation function ys for the undamped

frequency study, Eqn. (3) becomes:

21TRo EA a2uu+--k u=----

A m a Am.... 2. p p aX

(4)

Letting u(x,t) = ~(x) f(t), Eqn. (4) can be transformed into two dif-

ferential equations after separation of variables.

in which

2ep"(x) + a ep(x) = 0 (5)

(6)

2a

2a Am

=--p",-EA

anq 2w = (7)

Note that Eqn. (5) determines the modal shape function and Eqn. (6) deter-

mines the frequency of the system. Both equations are related by a constant

2a shown in Eqn.

Eqn. (5) to find

(7) .

2a •.

For frequency calculation, one needs only to solve

Eqn. (7) will yield the frequency of the system.

The solution to Eqn. (5) is

3

~(x) = Cl

sinax + C2

cosax

which depends on boundary conditions.

Axial Frequency of Buried Free-Free Pipe

(8)

First, let us examine the vibration of a buried free-free pipe with the"

following boundary conditions:

x = 0 and x = L au€ = ax. = $' (x) = 0 (9)

The eigen value and the eigen function will be

aL = nTI and $(x) = cos(nTIX/L) n=0,1,2, •...

Substituting Eqn. (10) into Eqn. (7) and then Eqn. (8) the axial frequency

of the system is found to be:

wn

=2TIR k

a aA m

p+ n = 0,1,2, .... (11)

where n determines the modal frequencies and shapes. When n

mode frequency is

0, the zero

w = w =o

2'!fR ko a

A mp

(12)

which is the rigid motion of pipe.

Normalizing the frequencies by wo ' Eqn. (11) becomes

wn-=

wo

EA21TR. k

o an = 0,1,2, ...• (13)

Using AWWA Standards(1,2), the periods for the rigid body modes for concr-

ete and cast-iron pipes are shown in Fig. 2. One can see from Fig. 2 that

the period decreases with increasing soil friction spring constant but in-

creases with added masses. One will further note that the period increases

4

with increasing pipe sizes. This is because the increase of pipe size and

the added masses from soil and water for the rigid mode means an increase

of mass of the system which in turn increases the period.

in Fig. 3 for n = 1 to 5 modes.

Axial Frequency of Buried Free-Fixed Pipe

Next, we examine the vibration of a buried free-fixed pipe. The boun-

dary conditions are:

x _.

x = L

aue: = - = 4>' (0)ax

u(L) = <j>{L) = 0

o

(14)

Using the solution shown in Eqn. (8) we find the eigen value and eigen

function as follows:

aL = n~ and ~(x) = cos (n~/2L)2

n 1,2, .... l (15)

Therefore the frequency for the free-fixed pipe is

wn

2~R__0_ kA m a

pn = 1,2, .... (16)

and the frequency ratio as compared to the rigid body motion is

wn-=

wo

1 + n = 1,2, .... (17)

which is presented in Fig. 4 for n = 1 to 5 modes.

5

Axial Frequency of Buried Free-SEring Restrained Pipe

For general application of free-spring restrained pipe, we refer back

to Fig. 1. The boundary conditions are:

0au

ep'(O) = 0x = E = -=ax

R(18)

x = L u(L) = -K

for which R is the resistant force from the spring and K is the end spring

constant. For equilibrium, the resistence of the spring can be determined

from the end force of the member, i.e.

auR = - E E(L)A = - EA ax = - EA ~'(L) (19)

Substituting Eqns. (18) and (19) into Eqn. (8), the characteristic equation

for the system becomes

EAcot aL = KL • aL (20)

which ~ay be solved graphically or numerically for a given ~~ value. Once

aL is solved, the frequencies of the system are obtained by Eqn. (7). Figs.

Sand 6 indicate the frequency ratios of first two modes for various ~

values.

TORSIONAL VIBRATION OF A STRAIGHT BURIED PIPE

Due to rocky action of earthquakes, buried pipeline may subject to

torsional vibration.. Referring to Fig. 7, the dynamic equilibrium for

the torque of the system can be written:

p I ~ dx + f 2nR2 dx = dr

p 0

+ f 2nR2 dr

or p I ~ . =p 0 dx (21)

6

where G, I , P are shear modulus, polar moment and mass density of pipe;p

r is the applied torque, f is the interface friction force per unit

area.

Relating the friction resistance to Ro with a spring constant, k t ,

Eqn. (21) becomes:

2 'I1R3

kG

~ + o t~ = ~"I . pP

(22)

Following the same procedures of separation of variables as given for

the axial vibration, Eqn. (22) is transformed into two differential equations

2<p"(x) + a. (x) = 0

.. 2f(t) + w f(t) = 0

Both equations are related by

22

22nR3k

2a pand 0 t +a. =-- w = a

G p IP

The solutL, •. of Eqn. (23) is

Torsional Frequency of Buried Free-Free Pipe

For free-free pipe, the boundary conditions are

(23)

(24)

(25)

(26)

x = 0 and x L; d<j> = 0dx

(27)

It is found that the eigen values and eigen functions are

a. =~ and <j>(x) = cos nnx!LL

.7

n = 0,1,2, .••• (28)

The torsional frequency of the free-free buried pipe is

wn

For n = °=

2 2+ n 1T G

L2 pn = 0,1, .... (29)

w =o(30)

which is the frequency of the rigid body mode.

The ratios of frequency of various modes to the rigid body mode are

wn-=

wo

n = 1,2, .... (31)

Note that if the frictional resistance on the surface of the pipe is the same

for axial and torsional motions, the torsional frequency ratio for the free-

free pipe will be the same as the axial frequency except for the conversion1

Torsional Frequency of Buried Free-Fixed Pipe

Without further discussion, the torsional frequency for the buried free-

fixed pipe will be obtained from the axial vibration solution as

wn

=wo

(32)

The graphical presentations for torsional frequencies will be the same

as for axial frequencies except for the conversion of E to G, ka to kt

and

AIR to I IR3 and thus will not be repeated.

o p 0

8

FLEXURAL VIBRATION OF A STRAIGHT BURIED PIPE

Flexural vibration of a buried pipe may result from earth motion in

vertical or lateral directions by earthquakes. The spring resistance from

all sides of a buried pipe are assumed to be the same. Then, the problem

can be considered as a common beam on an elastic foundation.

Referring to Fig. 8, the equation of flexural motion without forcing

(6,23)function of a buried pipe is formulated as follows:

4m y + kfY + EI l-Y = 0

dX4 (33)

where EI is the flexural stiffness of the pipe and m, kf

are mass and soil

flexural sl,;:ing constant per unit length. Note that the mass described here

may include the mass of water and soil that are vibrating with the pipe, in

addition to the mass of the pipe itself as defined by Eqn. (2) .

Again by separation of variables, Eqn. (33) reduces to two ordinary

differEntial equations

IV 44> (x) - a 4>(x)

.• 2f(t) + w f(t) = 0

o (34)

(35)

The two equations are related by

w = (36)

in which a must be obtained after solving Eqn. (34). Once a is found, the

frequency of the system is~etermined by Eqn. (36). The general solution of

Eqn. (34) is

~(x) = A sinax + B cosax + C sinhax + D coshax

which is governed by the boundary conditions:

9

(37)

.F1exura1 Frequency of Buried Free-Free Pipe

The boundary conditions for the free-free pipe are:

x = 0; M(O) = <1>"(0) = 0

V(O) = <jJ'" (0) = 0(38)

x = L; M(L) = <jJ"(L) = 0

VeL) = <jJ"'(L) = 0

The characteristics equation is obtained by substituting the above boundary

conditions into Eqn. (37) as

coshaL = secaL (39)

The solution of the abo':e characteristic equation yields the eigen values

and the eigen function:

- n 1TaL =-

2

<jJ(x) sin~x + sinhax

n = 0,3,5 (odd numbers)

When n

+ sinaL - sinhaL (cosax + coshax)coshaL - casaL

°which is the rigid body mode, the frequency is

(40)

(41)

The shape function for the rigid body motion is obtained from Eqn. (34)

without the presence of a as

(42)

By applying the boundary conditions as shown in Eqn. (38) the rigid body

mode has two arbitrary constants

10

(43)

In other words, there will be two possible rigid body modes with the same

frequency (Eqn. 41), one changes its position without changing its slope

and the other changes its slope without changing its position. These two

mode shapes are shown in Fig. 9.

Using spring constants for fine grain soils observed experimentally by

Howard(10~ the rigid body mode periods for AWWAconcrete and cast iron pipes

are shown in Fig. 10 and Fig .11 respectively. It is noted that these

rigid mode periods for lateral or vertical motion are very similar to

those for axial motion reported earlier.

The frequency for other modes are normalized by the rigid mode fre-

quency to yield the frequency ratio:

or

wnw

o=

4 41 +~ EI

24 k L4f

n = 3,5 (44)

wo1 + 1f4 (;.

4+ 1:.) EI

2 k L4f

n = 1,2, .... (45)

where ii = n - 2

The frequency ratios for n = 1 to 5 which represent 3rd to 7th modes

are shown in Fig. 2.

Flexural Frequency of Free-Hinged Pipe

The boundary conditions for free-hinged pipes are

x = 0; MeO) = ~"(O) = 0

V(O) 4>'" (0) = 0

(46)x = L; y(L) = ~(L) = 0

M(L) = f'(L) = 0

11

The characteristics equation is found to be

tanaL = tanhaL

The eigen value and eigen function are

- . 3aL = 0 and aL = (n - -)

4

<f>(x) = sinax + sinhax

+ sinaL - sinhaLcoshaL _ cosaL (cosax + coshax)

n = 2,3, ....

(47)

(48)

There is one rotational rigid mode rotating about the hinge and the fre-

quency ratios for :-he higher modes are

l.Iln-=

l.Ilo

34

4- -) 'lr

4n = 2,3, .... (49)

or l.Il­n-=

l.Ilo 0+ 4 - 1 4'lr (n + 4) n=1,2, .•.. (50)

where n = n - 1

The frequency ratios for n = 1 to 5 which represent 2nd to 6th modes

are shown in Fig. 13.

Flexural Frequency of Free-Fixed Pipe

The boundary conditions are:

x = 0; M(O) ~" (0) ='0 (51)

x = L;

V(O) = <P" 1 (0) = 0

y(L) = <j>(L) = 0

y'(L) = ~'(L) = 0

The characteristics equation is found to be:

cosaL coshaL + 1 = 0

12

(52)

and the eigen value and eigen function are:

ilL = 1.875, 1(n - -)71'

2 n = 2,3, ..•.

~(x) = sinax + sinhax

+ cosaL + coshaLsinaL - sinhaL (cosax + coshax)

The frequency ratios are:

:01 = J1 + 0.12688 EI71'4k L4

f

and

CI.l 4 1 4 EIn-= 1 + 71' (n - -)CI.l 2 k L4

0 f

or CI.l- 4 - 1 4 E2n-= 1 + 71' (n + 2)CI.l k L4o .

f

where n = n - 1

(53)

(54)"

n=2,3, .... (55)

n=1,2, ..•. (56)

Except CI.ll

which is very close to the rigid mode frequency CI.lo ' the fre­

quency ratios for ~ = 1 to 5 which represent 2nd to 6th modes are a1so··-shown

j.n Fig. 12.

Flexural Frequency of Hinged-Hinged Pipe

Boudary Conditions:

x = 0; ;/..0) = ~(O) = 0

M(O) = f'(O) = 0(57)

x = L; y(L) = HL) = 0

M(L) = <jl"(L) = 0

13

Characteristic Equation:

sina.x = 0

Eigen value and Eigen function:

aL = n~; and ~(x) = sinax

Frequency Ratio:

n = 1,2, ••.•

(58)

(59)

wn-=

wo

4 4n ~

EI

k L4f

n=1,2, ..•. (60)

The frequency and mode shape for n = 1 to 5 are shown in Fig. 14.

Flexural Frequency of Hinged-Fixed Pipe

Boundary Conditions:

x = 0; yeO) = <p (0) = 0

M(O) = <p"(0) = 0

x = L; y(L) = <p(L) = 0

y'(L) = <jl'(L) = 0

Characteristic Equation:

tanaL = tanhaL

Eigen value and eigen function

(61)

- (62)

1aL = (n + 4)~;

sina.L~(x) = sinax - sinhax'i' sinhaL

Frequency ratio

n=1,2, •••.

(63)

wn-=

wo

n = 1,2, •••. (64)

14

Fig. 13 shows the frequencies for the hinged-fixed pipe.

Flexural Frequency of Fixed-Fixed Pipe

Boundary Conditions:

x = 0; yeO) = $'(0) = 0

y'(O) = ~'(O) = 0

x = L; y(L) = $(L) = 0

y'(L) = $'(L) = 0

Characteristic Equation:

cosaL coshaL = 1

Eigen value and Eigen function:

(65)

(66)

- 1aL = (n + 2)

$(x) = sinax - sinhax

n=1,2, ...• (67)

+ CosaL - coshaLsinaL + sinhaL (cosax - coshax)

Frequency Ratio

(68)

tiln-=

tilo

41 41 + iT (n + 2) EI

k L4f

n=1,2, ....

Fig. 12 shows the frequencies for the fixed end pipe.

MATRIX FORMULATION OF BURIED PIPING SYSTEM

For the application of dynamic analysis to an actual water/sewer dis-

tribution system which may consist of several mains and many branches of

different sizes, a simplified but accurate method that can be handled with

15

".

reasonable amount of effort must be developed. Using the analytical continuous

solution to the differential equation for a buried pipe described in previous

sections, it will be very difficult, if not impossible, to get any reasonable

solution.for the large degrees of freedom system. This leads to the adoption

"f h 11 k ' f" 1 h(19,24)o t e we nown matr~x ~n~te e ement approac •

Since seismic excitation may come from any direction to·the piping sys-

tem, it is anticipated that some pipes in the system will be dominated by

the axial motion or torsional motion, while other pipes may be dominated by

flexural motions either in vertical or· lateral direction. Therefore for

generality, there will be i2 degrees of freedom for each member with six de-

grees of freedom at each end.,

Note that flexural vibration, axial vibration and torsional vibration are

all uncoupled, The developments of element mass matrix or stiffness matrix

can be worked out separately for each of the above mentioned motion.

For simplicity, the spring constants are assumed to be the same

and uniform along the pipe in all directions,

Element Flexural Stiffness Matrix

A buried pipe with a distributed flexural resistant soil spring kf is

shown in Fig. 15. At each end, there is a linear displacement coordinate and

a rotational displacement coordinate, denoted as Yl

to Y4

which are a function

of time. The distributed displacement function of the pipe can be represented

by the discrete modal displacement coordinates:

which may be written in a matrix form

16

,Yl

(t)

Y2(t)

Y3

(t)

Y4

(t)

(70)

Note that the strain energy of the system consists of the energies from

pipe and soil spring.

u=u. +u .p~pe spr~ng

The strain en'~::gy from pipe is

1 I 2u. = - EI(ylf) dxp].pe 2

in which EI;~ flexural stiffness

tyl!(x,t) = [<PI!(x)] {y}

o.f pipe and

(71)

(73)

Substituting Eqn. (71) into Eqn: (70); the strain energy of the pipe

becomes

u . 1 [y] t ( lEI {<jllf} [f'] t dx)= -p1pe 2

= 1- [Y] t [K ] {y}2 p f

{y} t

(74)

where [K ] is the stiffness matrix of a pipe element with its 'expanded formas p f

L

<pI! <P" <Pi <jl" <p U <jl" <jll! <1>"[K ] 1 1 2 1 3 1 4-

P fEI

<jll! <pI! <jll! <pI! <pit <jllt <pit <pitdx (75)2 1 2 2 2 3 2 4

<p" <p u <jl" <pI! <pI! <jllt <jl" <p"3 1 3 2 3 3 3 4

0 L<P4$" ¢If ¢I! <p" <p" <pI! <pI!1 4 2 4 3 4 4

17

conditions:

~l(x) = 2x3

/L3

- 3x3 - 3x2

/L2 + 1

~2(x) = L(x3

/L3

- 2x2

/L2 + x/L)

~3(x) = 3x2

/L2

- 2x3/L3

~4(x) = L(x3

/L3

- x2/L2)

(76)

and working out the integrations, one will find the stiffness of the pipe element

as the common flexural stiffness of a beam element.

12EI

7 -symmetric

6EI 4EI

L2 L

[K ] (77) "p f -12EI -6EI 12EI-2 L

2 2

6EI 2EI -6EI 4EI

L2 L 7 L

The strain energy from the soil spring is

1fk f

2U = - y dx

spring 2

1 ['1] t ( Jkf{~ } [4>] t dx) {y} t= -2

1 ['11 t [K 1 {y} ,(78)= -2 s f

where [K] is the stiffness matrix from the soil spring for flexural motion.S f

18

Using the Hermitian polynomials, Luk(13) worked out the details and reported

as

13 k L35 f Symmetric

on 2 k L3f

210 k f L 105

9 13 2 13

[K 170 kfL 420 kfL 35 kfL

= (79)s f

k L3

-13 2-k 13

-11 2f f420 kfL 140 210 kfL 105

The flexural stiffness matrix of a buried pipe element will be

[~ Jp f

= [K Jp f

+ [K Js f (80)

Element Flexural Mass Matrix

Using the consistent mass approach(4) the mass matrix of a buried pipe

element is developed.

Note that the kinetic energy of the system is

1 LT = 2" J

o

·2Am y dx

p(81)

Using the Hermitian polynomials of Eqns. (76) and Eqn. (81) can be

transformed to

=

o

~ [~lt [MbpJ {y}f

19

° (82)

Note that Eqn. (82) has the same form as the soil spring stiffness. Thus,

by interchanging k f to Amp' in Eqn. (79) the flexural mass matrix of a

buried pipe element is obtained.

Element Axial Stiffness Matrix

The Hermitian polynomials for the axial displacement are

xL

(~3)

x= -L

The axial strain energy of pipe is

L

Upa1= - EA2 f

o

(84)

which, in turn, the axial stiffness of pipe is lfound,...,.

EA -EA[K ] = L L

P a

-EA EAL L

The strain energy of soil spring in axial direction

u =! f 2nR k y2 dxsa 2 0 a

20

(85)

(86)

The stiffness matrix from the soil spring is found by integrating the

deflection function

kL kLa a

[K ] 21TR3 6 (87)=s 0 --k L k La a- a

6 3~

Element Axial Consistent Mass Matrix

Without further explanation, the element axial consistent mass matrix is

mL m L.-:L -.L

[~ ] A 3 6

p amL m L-.L -.L

6 3

Element Torsional Stiffness and Mass Matrixes

G I G I--.E. _ ---.E.

[K ] L L=P t

G I G I- --.E. ---.E.

L L

k L k L

21TR3 t t

[K ] = 3 6s 0t

k L k Lt t6 3

I L I L.-:L -.L

[~p]3 6

= APt

I L I L.-:L -.L

6 3

21

(88)

(89)

(90)

(91)

General Buried Pipe Element Stiffness and Mass Matrices

By combining the contributions from flexural, axial and torsional mo­

tions, the· generalized buried pipe element stiffness and consistent mass

matrices are summarized in Table 1.

Formulation of Buried Piping System

Upon the determination of these buried pipe element stiffness and con-

sistent mass matrices, they can be input into available generalized computer(12) (14) .

programs such as ICES-STRUDL or NASTRAN . for solution. With the de-

velopments shown, it is not difficult at all to write a computer program to do

the frequency analysis for the discrete system.

(7)Using the above formulations, Davis has been able to obtain frequency

values for one and two pipe systems with AWWA Sections. For a straight pipe,

the discrete frequencies and mode shape obtained by Davis are comparable to

the solutions obtained by solving the differential equations given in this

paper•.

CONCLUSIONS

. The vibration characteristics (axial, flexural and torsional) of buried

pipelines are greatly influenced by the properties of both the pipe and the

surrounding soil medium. For engineering practice, the problem can be

successfully handled by the analogy of beams on an elastic foundation.

22

With the assumptions that both the pipe and the soil medium have uni­

form-continuous.properties, analytical solutions of natural frequencies

(natural periods) of buried straight pipes can be obtained. However, for

buried piping systems, finite element approach, which converts a continuous

system to a system of discrete coordinates, must be employed. Since both

the stiffness and mass matrices for the buried piping system have been

worked out in the paper, it is only a matter of computer program to obtain

the numerical solutions.

ACKNOWLEDGE!:'1ENT

The author wishes to thank the RANN Program of National Science Founda­

tion for the finar~ial support (Grant No. ENV76-14884) under the general

title 'Seismic Vulnerability, Behavior and Design of Underground Piping

Systems' from which this paper is developed.

23

APPENDI~ I - REFERENCES

1. American Water Works AssociationAWWA Standard for Thickness Design of Cast-Iron Pipe,AWWA ClOO, 1972

2. American Water Works AssociationAWWA Standard for Reinforced Concrete Pressure PipeAWWA C300, 1974

3. Applied Technology CouncilAn Evaluation of A Response Spectrum Approach to Seismic

Design of BuildingsReport to Center for Building Technology,National Bureau of Standards, Sept. 1974

4. Archer, J.S.Consistent Matrix Formulations for Structural Analysis Using

Finite-Element TechniquesAIAA Journal, Vol. 3, No. 10, Oct. 1965

5. Ariman, T.A Review of the Earthquake Response and Aseismic Design of

Underground Piping SystemsProceedings of the First Specialty ASCE Conference on the

Current State of Knowledge of Lifeline Earthquake EngineeringASCE 282-292, August 1977

6. Biggs, J.M.Introduction to Structural DynamicsMcGraw-Hill Book Company, 1964

7 Davis, G.D •.Parametric Study of the Natural Frequencies and Mode Shapes for

an Underground Piping SystemMaster of Engineering Project Report, Department of Civil Engineering,

Rensselaer Polytechnic Institute, Troy, New York, May 1977

8. Hemant, S. and Chu, S.L.Seismic Analysis of Underground Structural ElementsJournal of Power Div., ASCE,Ju1y 1974, pp. 53-62

9. Hetenyi, M.Beams of Elastic FoundationThe University of Michigan PressAnn Arbor, Michigan, 1946

10. Howard, A.Modulus of Soil Reaction Values For Buried Flexible PipesJournal of Geotechnical Division, ASCE, Jan. 1977

24

11. Isenberg J., Weidlinger, P., Wright, J.P., and Baron, M.L.Underground Pipelines in a Seismic EnvironmentProc. of ASCE First Specialty Conference on Lifeline Earthquake

Engineering, UCLA, Aug. 1977

12. Logcher, R.D., Flachshart, et a1ICES-STRUDL II - Engineering User's ManualCivil Engineering Department, M.I.T., Nov. 1968

o

13. Luk, N.N.Dynamic Response of a Simple Piping System on An Elastic Foundation

FoundationMaster of Engineering Project Report, Department of Civil Engineering,

Rensselaer Polytechnic Institute, Troy, N.Y., Sept. 1976

14. McCormick, C.W.The NASTRAN User's ManualScientific and Technical Information Division,NASA, Washington, D.C., 1970

15. Mindlin, R.D. and Cheng, D.H.Nuclei of Strain in·the Semi-Infinite SolidJournal of Applied Physics, Vol. 21, No.9, Sept. 1950

16. Parmelee, R.A. & Corotis, R.B.Analytical and Experimental Evaluation of Modulus of Soil ReactionTransportation Research Record 518Transportation Research Board, Washington, DC, 1974

17. Parmelee, R.A. and Ludtke, C.A.Seismic Soil-Structure Interaction of Buried PipelinesProc. of U.S. National Conference on Earthquake EngineeringAnn Arbor, Michigan EERI, 1975, pp. 406-415

18. Penzien, J •.Soi1~Pile Foundation InteractionChapt~r 14 of Earthquake Engineeringed. by Robert L. Wiegel, Prentice-Hall, Inc. 1970

19. Przemieniecki, J. S•Theory of Matrix Structural AnalysisMcGraw-Hill Book Company, 1968

-20. Sakurai, A. and Takahashi, T.DYnamic Stresses of Underground Pipelines During EarthquakesProc. of 4th World Congress of Earthquake EngineeringSantiago, Chile, 1969, pp. 81-95

25

21. Shinozuka M. and KamakamiUnderground Pipe Damages and Ground CharacteristicsProc. of ASCE First Specialty Conference on Lifeline

Earthquake Engineering, UCLA, Aug. 1977

·22. Tajimi, R.Dynamic Analysis of a Structure Embedded in an Elastic StratumProc. of 4th World Conference of Earthquake EngineeringSantiago, Chile, 1969, pp. 53-56.

·23. Timoshenko, S. and Young, D.H.Vibration Problems in EngineeringD. Van Nostrand, Co., 1955

2"4. Wang, L. R. L.Parametric Matrices of Some Structural MembersJournal of Structural Division, STS, ASCEAugust, 1970

25. Wang, L.R.L.Some Aspects of Vibration Resistant Structural DesignProceedings of the First National Conference on Earthquake EngineeringTaiwan, Republic of China, June 1973

2"6. Wang, L.R.L.Dynamics of Buried PipelinesTechnical Report (SVBDUPS Project) No. 2Department of Civil Engineering, Rensselaer Polytechnic Institute,

Troy, N.Y., April 1977

27. Wang, Leon R.L. and O'Rourke, M.State of the Art of Buried Lifeline Earthquake EngineeringTechnical Report (SVBDUPS Project) No.1Department of Civil Engineering, Rensselaer ~olytechnic I~stitute,

Troy, New York, January 1977; Also Pr~ceed:ng of ASCE F1rstConference on Lifeline Earthquake Eng1neer1ng, UCLA, Aug. 1977

26

Table 1

Non-zero Terms of Stiffness and Mass Matricesof A Buried Pipe

Non-zero Term

(1,1)

(1,7)=(7,1)

(2,2)

(2,6)-(6,2)

(2,8)=(8,2)

(2,12)=(12,2)

(3,3)

(3,5)=(5,3)

(3,9)=(9,3)

(3,11) = (11,3)

(4,4)

(4,10)=(10,4)

(5,5)

(5,9)=(9,5)

(5,11)= (11,5)

(6,6)

(6,8)=(8,6)

(6,12)=(12,6)

[K ]P

EA/L

-EA/L

12EI/L3

6EI/L2

-12EI/L3

6EI/L2

12EI/L3

-6EI/L2

-12EI/L3

-6EI/L2

GI /LP

-GI /Lp

4EI/L

6EI/L2

2EI/L

4EI/L

-6EI/L2

2EI/L

[K ]s

21TR '. 1:. k Lo 3 a

21TR . 16

kLo a

1335 kfL

11 2210 kfL

970 kfL

·13 2-. 420 kfL

1335 kfL

11 2- 210 kfL

970 kfL

13 2420 kfL

21fR~ • ; ktL

21fR3 '. 1:. k Lo 6 t

1 3105 kfL

13 2- 420 kfL

1 3- 140 kfL

1 3105 kfL

13 2420 kfL

1 3- 140 kfL

27

1:. Am L3 p

1:. Am L6 P

1335 Am L. P

11 2210 AmpL

970 AmpL

-13--· ._-._.....- - Am L-----··420 p ~

1335 AmpL

11 2- 210 AmpL

970 AmpL

13 2420 AmpL

1- p I L3 p

1- p I L6 p

1 3105 AmpL

13 2- 420 AmpL

1 3- 140 AmpL

1 3105 AmpL

13 2420 AmpL

1 3- 140 AmpL

Non-zero Term [K ]P

Table 1

[K ]s

(Continued)

EA/L

___ .___ - 4EI/L

12EI/L3

12EI/L3

(7,7)

(9,9)

(10,10)

(11,11)

(12,12) __

12'ITRo • "3 kaL

1335 kfL

1335 kfL

GI /L 2'ITR3 l k L_________P ~ Q--3..----.t -

1 34EI/L 105 kfL

1 3105 kfL _

28

1'3" AmpL

1335 Am'pL

13 .35 AmpL

1- p I L3 p

1 3105 AmpL

1 3105 AmpL

77777777777777777777777777

X--t:> Ys

y~----

L-f

c:z-

0"~G -----c> 0" + dO"

co--

~Fig~ I Axial Vibration of a Buried Pipe

29

91.44 em (36 11) pipe

0.06

0.05Q)

~0>. 0.040

" "....0 .0

0.03Q,)(J)

0.02

0.01

~~- 60.96 em (24") pipe

·30.48 em. (12") pipe

234 5 6 7

--N/em2/em

8 9 10--"'t---...,.....-'__-_.,_--.Jl._,_-&.---ro-'----r-.l-----,r-I--r-&---t;>

. 0 4 . . 8 12 16 20 24 28 32 36 2ka Ibs./in

: . Fig 20 Rigid Mode Natural Period of Buried AWWAConcrete Pipe - Axial Motion

60.96 em (24") pipe

7

30.48 em (12 II) pipe

4562 3'---l.r--"-r-----r--J,--r---.J..---r---.--,.-!.o---.~-.-&--r'--->

o 4 8 12 16 20 24 28 32 36ko Ibs. / in2

/ in.

Fig. 2b - Rigid Mode Natural Period of Buried AV/VJACast. Iron Pipe - 'Axial Motion

0.06

0.05Q)

~0~

0.04·0

" "~

.0Q) 0.03en

0.02 15.24 em (6") pipe

0.01

10080

n =nodal number

I ]

f:- L e:1

20 40 60. 2

(EA/21TRo L k a ) ~/

Fig. 3 - Axial Frequency Ratio of Buried Free-Free Pipe

n =0; rig id body modeI~-----------------ao

10 -

50

20

70

30

60

80

o:3

'c 40:3 ..'.

n =modal number

7 77777/7"7/"';-

c::::=======~1---- L I

1008020 40 60(EA/21TRoL2 ka )

I0­o

10

30

20-

70

60

.50

o 403

"'-c

3

. Fig.4 - Axial Frequency Ratio of Buried Free-Fixed Pipe j~

10 - \ -, ,-j'~

1000 10~

cI3 5

o,I

20 40 602EA/27TR L k

. 0 a

80 100

....•.•__ ...__ ..----Fig.5 - First Mode Axial Frequency Ratio of Buried

"'. __ .. _ Free -~pring Restrai~,~,9~_~P~_.__10010 .

1KL/EA

,..----------,.VI/M~l-.-L -DI K

-'/1 III /11 717 firYl/ VI

5

15

10

25

20

Io 20 40 60 80 100

EA/27TRo L2 kaFig.6 - Second Mode Axial Frequency Ratio of Buried

Free - Spring Restrained Pipe J3

o3·

"'­ca

Fig. 7- Torsion in a Pipe Segment

. --- - .. - ..----.... ~ ·.....------------··----.- ....T --- .--..-#---.--~~.--------------- ... ~ - ----------------

I

---------M (trmlp M+dM

v~ V+dV~dX-~

Fig.8\- Flexural Vibration of Buried Pipe

lr~rtrr~rr fu' Neutural-- --. Position

~?/~;> ?),.)/ ~ )7 ~ j 727~ 5;5,.

(0) Deflection Rigid Mode

Neutural

Position

(b) Rotation Rigid Mode

Fig .9-" - Rig id Modes in Flexural Vibration. ,".. ,....

35

. 10

CD0.08~o

"0.06CDen

~.04.....

o~ .02

pipe

60.96em (24") pipe

30.48 em (12 11) pipe .

J300

1o

o---q---- ------ -----t----r----] ----r--·--r------J-----T :>

400 800 1200 1600 2000I I - I I I I f N / e{:,;em

900 1500 2100 2700k f Ibs/in/in

Fig.la-Rigid Mode Natural Period of Buried A'W'vVA Concrete Pipe- Lateral Motion

. I 0

60.96 em (24") pipe

16001200

30.48 em (12") pipe

15.24 em(6") pipe

800400o

Q)

"0 .08~o.....o .06Q)(J)

_~ .04

o......02

I2100

I900

I

o 300I

1500k f

Fig.ln- Rigid Mode Natural Period of Buried" - Lateral Motion

36

-....'-.

.. • ·.t.

-: ::: :::;;

n=n

n=n-In ==n-2

Fixed - Fixed Pipe

Free - Fixed Pipe

Free -Free. Pipe.-

- n=I,2,3,·-~·

......---...----I..... ..J J~___J..l__'__~ I20 30 . 40 50 6C

kf L4

7T4 EI

2

14

18

16

12.

10

8

22

20

I .

·24

~j . ~I

~

"w~

~

c;;t--It\.!+Ie...........-+

_----------.-- - •__..... ...-.__-- ~.. _:r....._..:.-.. ..._-__._. -----....

'._.Fig. 12 - flexural Frequency Ratio For Buried Fixed- )

--'...,. _.. ,. 'Yiie-ci 'p-iPe~ -Fr~e~fiee--I'{~e:-;~d-i-r~~~ii;:;dl'ip~'-

37

Hinged - Fixed Pipe n =n

Free-Hinged Pipe n=n-I

n = 1,2,3'"

24

22

~-I20

.....18~

"'-w 16~

l::: 14.go.........

--IN !2I ',+Ie

10..........+

8II

3° 6"'-c

43

2

00 10 20 30k L4

f

40 50 60

7T4 EI

Fig. "f3t:-: Flexual Frequency Ratio for Buried Hinged-Fixed~

Pipe and Free - Hing ed Pipe

38

n=I,2,3'"

::

24

22

~ 20:..J

4-18x

..........

w 16'¢

~14v

c+ 12

10

8II

3° 6"-c:3 4

2

402010"--__-"'-- ...l.oo..-~-=-_J", ..J.______'L_.pp_.........-..!_,_

50 60

.--Fig.14- Frequency

7T4 EI

Ratio for Buried Simply Supported Pipe

39

;r'-T7777777'

Fig. 15, - A Buried Pipe with Flexural

Nodal Coordinates

--. -oJ - _.- .... - ... - ..---""' -.. .. - _ ..... _ .. -...-----------_~ ... ._-- --- ..-,-.__--__-_._--,.._-_._- -_ ... _ . .... _ -.._.~'-_._- __'_

40

NOTATION

A

E

f

G

I

IP

ka

[K ]P

[K ]s

L

m

mp

M

[M ]P

[M ]s

n

R

Ro

T

u

Cross Sectional Area of Pipe

Young's Modulus of Pipe

Frictional Force Per Unit Area Between Soil and Pipe--

Shear Modulus of Pipe

Flexural Moment of Inertia of Pipe

Polar Moment of Inertia of Pipe

Distributed Axial Soil Spring Constant (Friction)

Distributed Flexural Soil Spring Constant (Compression)

Distributed Torsional Soil Spring Constant (Friction)

Concentrated Axial Soil Spring Constant at End of Pipe

Stiffness Matrix of Pipe Element

Stiffness Matrix of Soil Medium

Stiffness Matrix of Buried Pipe System

Length of Pipe

Mass Per Unit Length of Buried Pipe SystemI

Mass Per Unit Length of Pipe Itself

Moment in Pipe

Mass Matrix of Pipe

Mass Matrix of Soil Medium

Mass Matrix of Buried Pipe System

Modal Index

Axial Resistant Force From Concentrated Axial Soil Spring K

Outer Radius of Pipe

Kinetic Energy

Relative Displacement Coordinate Between Pipe and Soil

Medium (v-y ). - s

41

v

x

y,y,y

{Y}

r

e:

A

p

0

<P

<P (x)

oj

w0

Shear Force in Pipe

Linear Space Coordinate

Continuous Displacement, Velocity and Acceleration

Coordinates of Pipe

Continuous Displacement, Velocity and Acceleration

Coordinates of Soil Medium.

Discrete Displacement Vector of Pipe

Torque in Pipe

Axial Strain in Pipe

Mass Ratio (m/m ). p

Mass Density of Pipe

Axial Stress in Pipe

Angle of Twist of Pipe

Modal Function

Circular Frequency of Buried Pipe System

Rigid Mode Circular Frequency of Buried Pipe System

42