Comprers & .Sfmcrwes Vol. 37, No. I. pp. 7140, 1990 Printed in Great Britain.

004s7949/90 s3.00 + 0.00 Pergamon Press plc

DYNAMIC RESPONSE OF BURIED ORTHOTROPIC CYLINDRICAL SHELLS TO AN INCLINED

AXISYMMETRIC MOVING LOAD

J. SINGH and V. P. SINGH Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University,

Varanasi 221005, India

(Received 11 September 1989)

Abstract-The aim is to study the effect of orthotropy on the steady-state dynamic response of a buried pipeline subjected to a radial and/or tangential load moving along the axis of the pipe. For this purpose only axisymmetric response has been studied. In the formulation of the problem the pipeline has been modelled as an infinitely long orthotropic circular cylindrical shell buried in an infinite medium. A perfect bond between the shell and the surrounding medium has been assumed and thin shell theory formulation has been employed. Results have been obtained for hard (rocky), medium hard and very soft soil conditions surrounding the shell. Effects of shell orthotropy have been determined by varying the non-dimensional orthotropy parameters over a wide range. It is observed that the introduction of orthotropy reduces the amplitude of axial and radial displacements.

c

E = c/c,

u,

W’= woGx,/P’

wo

r=(x -cr)/R

NOTATION

speed of the moving line load non-dimensional speed of the moving line load longitudinal wave speed in the surround- ing medium (soil) shear wave speed in the surrounding medium (soil) displacement at any point in the medium displacement due to scattered field orthotropic elastic moduli of the shell shear modulus of shell material thickness of the shell intensity of applied load (force per unit length) mean radius of shell radial coordinate time

non-dimensional axial displacement of the shell

axial displacement of the shell at the middle surface

non-dimensional radial displacement of the shell

radial displacement of the shell at middle surface

non-dimensional moving axial coordinate

coordinate axis coincident with the shell axis non-dimensional wavelength parameter non-dimensional orthotropy parameters, defined by eqn (22) tangential coordinate Lame’s constant of surrounding medium Lame’s constant (modulus of rigidity) of surrounding medium (soil) circular wave number

Plll PO R 0

density of the medium (soil) density of the shell material non-dimensional frequency circular frequency

INTRODUCTION

Buried pipelines perform a very important role in conducting and distributing water, energy, communi- cation and transportation. Any damage to buried pipelines even at one place causes sufferings to mankind over a vast area. Due to this fact the dynamic response of buried pipelines has become an active area of research. In recent years the response of buried pipelines due to seismic excitation has been extensively investigated. Another source of dynamic excitation to buried pipelines are loads moving along the axis of the pipeline. These moving loads may arise due to some pressure pulse in the fluid which is flowing in the pipeline or they may result from the viscous forces of the fluid on the pipeline.

During the last 10-12 years many papers have appeared which have been concerned with the dy- namic response of buried pipelines subjected to a line load moving along the axis of the pipe. Mindlin and Bleich [l] investigated the transient response of a long, circular, cylindrical shell subjected to a plane acoustic wave travelling normal to the shell axis, when the cylindrical shell is surrounded by or contains an acoustic medium. Parnes [2] studied the response of an infinite elastic medium subjected to travelling ring loads in a cylindrical bore. Mann- Nachbar [3] studied the response of an infinite cylin- drical shell surrounded by an acoustic medium and symmetrically loaded by an axially propagating pressure wave. The problem was also discussed by Forrestal and Herrmann [4], who included the effects

71

12 J. SINGH and V. P. SINGH

of shear deformation and rotary inertia. Mangrum and Burns [5] analysed the dynamic response of infi- nitely long orthotropic cylindrical shells subjected to

[$(&+$)+L&,+p’$]w

an axisymmetric pressure load moving in the axial direction with a constant velocity. All the papers cited above, however, concern situations where the shell is either filled with some acoustic medium or there is an and acoustic medium surrounding the shell. None of these works deals with a buried shell structure. Chonan 161 and Datta and Chakraborty [7l studied the dynamic response of underground pipelines made of isotropic materials under axisymmetric ring pressure moving in the axial direction along the interior of the shell. Ariman and Muleski [8] emphasized the need to analyse pipes made of orthotropic materials and pointed out that new materials were becoming quite popular in pipeline systems, and were slowly replac- ing cast iron and steel. Moreover, reinforced plastic mortar (RPM) has already found acceptance as a substitute for cast iron and steel in a number of applications.

The main object of the paper is, therefore, to look into the relative influence of orthotropic parameters on the shell response when the shell is under a tangential moving load and the combined effect of radial and tangential moving loads when they act in different ratios. Only the axisymmetric response of the shell has been studied. The buried pipe has been treated here as a shell using Flugge’s shell theory. It is observed that orthotropy has an influence on shell response as it is placed in different ground conditions.

Ev a 3

---$$ w+ R ax 1 [

a2 a2 Exax2-p’s u+p:=O,

1 (3)

where t is the time, and u and w are the displacement components of the middle surface in the axial and radial directions, respectively.

The constants of eqns (2) and (3) are given as

E.x = h-K,

Dx = ; Ex, ,

E, = hEe,

D,=;E,,

E, = hE,,

p’=b,, (4)

where p. is the mass density of the shell material. The

FORMULATION AND GOVERNING EQUATIONS p,? are the traction components per unit area exerted on the shell by the surrounding medium and are due

Consider an infinitely long thin orthotropic cylin- to the actual moving load (radial or tangential), and drical shell of mean radius R and thickness h lying in also to the scattered field created in the surrounding an infinite, linearly elastic, homogeneous and medium. The traction components are given as isotropic medium. The shell is excited by a radial line load moving along the axis of the shell. The load distribution considered is axisymmetric and makes an angle 4’ with the radial direction in the r-x

p:=(l+~)~rri=~+~*,*,

plane. Let a cylindrical polar coordinate system (r, 0, x) p: = (1 +ii).ll/,_ir+(h,2)~ w

be defined such that x coincides with the axis of the shell. The stress-strain relations of the shell material where 0, and o,~ are, respectively, the normal and are assumed to be of the form shear stresses generated at the outer surface of the

shell (i.e. at r = R + (h/2)) due to the motion of the

c,, = E,, 6, + &coo surrounding medium. The displacement d(r, 0, x, t) at any point in the

QOO = Eu, cx.r + Eo, coo

(1)

surrounding medium satisfies the equation of motion

c;v(v.d)-c;vxvxd=$, (6)

where gii and cii are, respectively, the components of where c, = ((1 + 2p)/p,,,}“’ and c2 = (p/p,,,)“’ are, the stress and strain, and E,, , E,, , E,, and G,, are the respectively, the longitudinal and the shear wave four independent moduli. speeds, depending upon the values of the Lame’s

The equations governing the axisymmetric motion constants 1 and p and the density, pm, of the medium of such a shell can be written as (surrounding soil).

Steady-state dynamic response of buried pipelines 73

Because of a scatterer (shell) lying in the medium, the displacement d at any point in the medium is the scattered displacement field d(@ whose radial and axial components are written as

,=[la~,(~,~)+j*~~,(~*r)]ei~(~-cl)

and

d~)=~i~K~(~,~)+B~K~(4r)]ei~(~-Cf), (7)

where K, (n = 0, 1) are the modified Bessel functions of the second kind, and

5, = SJ(1 - (c/c*)*)

c2 = BJU - (c/c2)2)

where A ( = 2x/<) is the wavelength and c ( = o/t;) is the speed of the moving line load. Constants A and B in eqns (7) are determined by applying the condition of continuity of displacements at the outer surface of the shell, i.e.

(W)r=R+(h,Z) = &’ r=R+(h/Z)

@),=R+(h,Z) = &’ (9 r-Rt(h/2)

Now, assuming shell displacement in the form

W = WOe5(x-c1,

U = UOeit(x-cO (10)

and using two boundary conditions (9), the con- stants A and B in eqn (8) are determined in terms of a, and w,,, the displacements at the middle surface of the shell. The scattered displacement field results as

d!“’ = ; [{

(F, E,)K,,(a,)K, f E, ( )

- B*Ko(a,)K, f G ( >>

wo

+ 1(K,(a,)K,(it2)

- K, (a2)K, (f E, )}(E, /l)uo]eit(x-cf)

and

d$,=l D

(E,E,)K,(a,)& IE ( ) R2

-82K,(a2)% it, ( )I

u.

(11)

where

a2 = E2 1 +? ( 3

(12)

Now, d(*) and dt) being known, a!$ and c$! can be r calculated from

and

a!;‘= [P{’ +y}],=R+(*,2), which result in

(8) - I” t2 vzx2&(dKob2) 2 =w - R D

eiC(x-cO

-it!! v2x2&@l)Kl@2) R D

W-4 (13)

and

0% = & [{(f, v2x2)Kl (aI K (a2>>uo

+i((2ElE2B)~(a2)Kl(al)

-/?(8* + E~)~(a,)K,(a2)}wo]ei~~“-“‘, (14)

where

y 2/j Cl

14 J. SINGH and V. P. SINGH

and and

c: 2(1-m) -- x2=2- 1-h ’ (19)

m being the Poisson’s ratio of the surrounding medium (ground).

Until now, the main aim has been to determine the

For calculation of pt. the tractions at the shell components of traction per unit area exerted on the

surface, i.e. the radial and tangential stresses at the shell by the surrounding medium. When these values

outer surface, are written as of p: [from eqns (17) and (18)] and the displacements u and w from eqns (10) are substituted into the shell equations (2) and (3) we obtain, after some lengthy

u~l,=~+~~,2~=~~)l,=~+~*,2~+~~(~ -ct) (15) a1gebra

2 m21 -/ml - Q2[Nl

and

where 6( ) is the Dirac delta function defined as

where w’ ( = wOGXl /P’) and U’ ( = uOGX, /P’) rep- resent the non-dimensional displacements of the shell,

6(x - ct) = 1 at x = ct o for all other values of x,

P’ is the intensity of applied load (force per unit length) and 4’ is the angle which it makes with the radial direction.

Substituting the values of stresses a$ and as from eqns (13) and (14) into eqns (15) and (16), the values of pt are calculated, which may be expressed as

p 7 =A $ [P, , w. + P12 uO]eit(x-cr)

-6(x.- ct) (17)

and

is the non-dimensional frequency, where p = pn/po and fi = p/G,, are, respectively, the density of the surrounding soil (medium) and modulus of rigidity, and E = c/c, is the apparent wave velocity.

Matrix [P] is as given by eqn (19) and the com- ponents of the other matrices [Q] and [N] are

and

p: = $ [P2, w, + P22u0]eiC(x-cr)

-6(x - ct), (18) are the orthotropy parameters.

A new moving coordinate is now defined as

where y-(x -ct) --

R

and the Fourier transforms given by

P,2= -Pr,= -i v2X2Rr(a2)l(o(al)

D + 2 1 ’ f*(B)= f (f)e’fiR dx

--x (23)

Steady-state dynamic response of buried pipelines 75

and

are used in eqn (20). With these transformations the matrix equation [eqn (20)] becomes

For different values of radial and tangential loads, the matrix equation (25) can be solved to obtain w’* and U’*. To get back the values W’ and U’ from W’* and U’*, the inverse transformation given in eqn (24) is used.

In this paper results have been presented for differ- ent values of angle 4’. From the equations it can be observed that for 4’ = 0, the shell is subjected to radial moving load only, and the moving load is strictly a tangential one when 4’ = 90”. Obviously, when angle 4’ # 0 or 90”, the shell is subjected to a combination of radial and tangential moving loads.

RESULTS AND DJSCUSSION

Results have been presented mainly from the point of view of determining the effect of varying the orthotropy parameters q, , tfz and t/3 on the radial and axial displacements of the shell for different values of soil parameters and at different moving load con- ditions. The load speed 5 (= c/cl) has been kept constant throughout. Ranges of orthotropy par- ameters have been decided on the basis of orthotropy constants given for different kinds of composite materials. For Q, Q and r,tS three values have been taken (q, = 0.05, 0.50 and 1.0; t/* = 0.005, 0.050 and

1500

F

\T4s=o.20 @‘=OO

fi=o.o5, e=o.ao \ \

m=0.25, c=O.Ol

i

\ \ --_-- \ q2=0.05, qs=o.o2

\- ‘1, =0.50, q3=0.02

1000 \ ----- \

~,=0.50,1)2=0*05

Fig. 1. w’ versus x at E = 0.10 with q,, q2 and ‘1, as Fig. 2. w’ versus K at E = 0.10 with ql, ~2 and ~3 as parameters. parameters.

0.100; ns = 0.01,0.02 and 0.20) which are expected to cover a wide range of composite materials. While studying the effect of variation of one orthotropy parameter, the other two have been kept constant. For different kinds of combinations of ground and pipe, the rigidity ratio has been taken as 0.01 and 10.0. The thickness to radius ratio of the shell, x, ratio of the shell and soil densities, I, and the Poisson’s ratio of the soil, m, have been kept constant through- out; 7i = 0.05, i = 0.30 and m = 0.25.

The radial response of the shell, at an angle 4’ = 0” i.e. for radial moving load, is shown in Figs 1 and 2 which give displacement IV’ ( = w,G,,/P’) versus 8 ( = (x - ct)/R) plots with Q , q2 and Q as parameters. In the figures the displacement is maximum under the load (i.e. X= 0) and decreases as we move away from the load. Figure 1 predicts that variations of v, and qS have strong intIuence on radial displacement when the surrounding soil is very soft and sandy (J = 0.01). With increase in r,t, the radial displacement decreases, whereas with increase in qI it increases. As compared to ‘1,) variations in Q bring more marked changes in the maximum radial displacement, whereas change in t/* does not influence the radial displacement. In fact, it is observed that, excepting for q1 = 0.05 and qS = 0.2, the rate of decay of displacement amplitude is very, very slow in all the orthotropic cases. There- fore, in very soft and sandy soil conditions, the radial displacement under the load can be reduced to a remarkable level by selecting a proper combination of orthotropy parameters. Figure 2 indicates that when the shell is buried under very hard and rocky ground (J = lo), the maximum radial displacement under the load (x= 0) is not much effected, even with appreciable changes in ‘1,) qz and vs. Thus, from these two figures it is interpreted that orthotropy parameters ‘I, and TV, strongly influence the maximum

l$‘=Oo

ii=o.o5,;=0.30 m=0.25,(1=10.0

-----q2=0.05,tj3=0.02 -q,=0.50,Ij3=0.02

~,=0.5o,Tj2=0.05

1100

J. SINGH and V. P. SINGH

nz=o.o5, I)a=o.o2 - 71=0.50,n3=0.02

Th =0.50, n*=o.o5 \

‘1. \. ‘\

\\ ‘\

\;\ \ ‘\

ic Fig. 3. W’ versus K at c‘ = 0.10 with ‘I,, q2 and qj as

parameters.

radial displacement only when the surrounding soil is very soft compared to the shell.

Figures 3 and 4 show the radial response when shell is subjected to a moving load which makes an angle of 45” with radial direction, i.e. 4’ = 45”. This is the case when the shell is under the influence of a combination of radial and tangential moving loads. From Fig. 3 it is predicted that when the shell is buried under a very soft surrounding medium (ji = 0.01) the magnitude of radial displacement de- creases as compared to when the shell is subjected to a radial moving load only, although the nature of the changes in W’ due to alterations in ‘I,, q2 and Q were found to be similar to those observed at 4’ = 0”. As

m=0.25,ji=lO.O

---- q2=o.05,q3=o.02 - q,=O.50, q3=0.02 ---_- ‘1, =0.50, q2=0.05

q~=0.005,0.050,0.100

ii=O.O5,Q=O.30 m=0.25,p=lO.O

2 ---.- q2=0.05,q3=0.02 - q,=O.50,1j3=0.02

q,=o.5o,q,=o.o5

B "0 '1

0- 0 0.25 0.50

Fig. 4. W’ versus x at f = 0.10 with q,, p2 and ql as Fig. 6. W’ versus R at E = 0.10 with gl, Q and ~3 as parameters. parameters.

6. p=o.20

t

@'=90" ‘\ l=l=0.05,~=0.30

\ \ l_

m=0.25,~=0.01

----- q2=0.05,q3=0.02 - ~),=0.5O,q3=0.02 ----_ qr=0.50,q~=0.05

0 0.25 0.50

ic Fig. 5. W’ versus x at E = 0.10 with q,, p2 and ‘1, as

parameters.

in Fig. 1, the introduction of orthotropy through tf, , qz and q3 decreases the maximum deflection under the load and the rate of decay is much faster when Q 7 0.2. Figure 4 indicates that when the shell is buried under very hard and rocky surrounding such that the rigidity of the ground is more (ji = 10) than the shell material, the radial displacement under the load is slightly affected for a very narrow range of r. From Figs 3 and 4 it is visualized that with the combination of radial and tangential moving loads (4’ = 45”) the effect of changing r~~, qz and Q remains qualitatively the same as discussed for a radial moving load. Although it is not shown, a similar behaviour is observed when 4’ = 30 or 60”. As observed, the magnitude of radial displacement decreases with increasing angle 4’.

The radial displacement when the shell is subjected to a tangential moving load only (i.e. d’= 90’) is shown in Figs 5 and 6. As can be seen from Fig. 5,

Steady-state dynamic response of buried pipelines

the changes in orthotropy parameters q, and q3 strongly influence the maximum radial displacement of the shell in a similar manner as for other values of 4’. However, it is observed that when shell is buried under very soft soil and being excited by a tangential moving load, a change in tf2 affects the radial response which increases with increasing values of t/r. There- fore, unlike in the case of a radial moving load, changes in r,rr can be helpful in reducing the radial displacement of the shell buried in very soft and sandy soil conditions. When the shell is buried under a hard and rocky surrounding medium, as shown in Fig. 6, then variation in q3 has some effect on W’, but for only a very narrow range of X. Variation in q, and qz does not affect the radial displacement at all.

Figures 7 and 8 represent the displacement W versus K plots for different values of angle 4’ and fixed value of orthotropy parameters q, , t/z and q3 . It appears from Fig. 7 that radial displacement is much higher when the shell is buried under a soft surround- ing medium. The amplitude is maximum when angle 4’ = 0” and decreases with increasing values of 4’, and attains a minimum when 4’ = 90”, i.e. the shell is subjected to a tangential moving load. It is ob- served from Figs 7 and 8 that the rate of decay of the displacement amplitude is much faster when the shell is buried in a soft surrounding medium (p = 0.01). Figure 7 predicts that there is no variation in radial displacement when the shell is surrounded by soft soil and excited by a tangential moving load. At the same time displacement is constant when R= 0.45, irre- spective of values of 9’. From Fig. 8 it is concluded that the nature of variation of w’ remains almost unchanged for any value of 4’.

The effects of changing r,+r, q2 and q3 on the axial component of the displacement, U’ ( = (u,J G,, /P’), for different values of 4’ are shown in Figs 9-16.

200 $‘=OO

k/

ii=o.o5, c,=o.30 m=0.25, F=O.Ol

30” ~,=0.50,~*=0.05,qs=0.02

150

Fig. 7. W’ versus r at E = 0.10 with 4’ as parameter.

8 I$‘=0

h

~=0.05,~=0.30 m=0.25,~=10.0

30 qr=0.50,q2= 0.05,q3=0.02

0’ I I 3 ) ’ a ’ ’ ’ ’ 0 0.25 0.50

2

Fig. 8. w’ versus X at E = 0.10 with 4 as parameter.

Figure 9 shows the variation of axial displacement along the shell under a very soft and clay-like sur- rounding medium (ji = 0.01). Variations in q, and q3 are observed to have a strong effect on axial displace- ment as compared to those in qr. However, the nature of their effects is different. Whereas an increase in q, decreases the magnitude of axial displacement, with increase in q2 or q3 it increases. Figure 10 shows the variation of axial displacement U’ when the shell is surrounded by a very hard and rocky medium (ji = 10.0). Compared to variations in q, and r,r2, the alteration in ‘1, produces significant changes, especially of the maximum axial displacement under the load. Therefore it may be said that the insertion of orthotropy will have a significant effect on the reduction of maximum axial displacement of the shell under all soil conditions. However, when the medium is hard this effect is, of course, very small with respect to changes in q, and q2, but variations in q, are very

t

@‘=O

60 -,3<=0.20 ti=0.05,~=0.30

‘\ m=0.25, F=O.Ol

‘. 50 -

40

-1

?.I “0 30

Fig. 9. U’ versus X at E = 0.10 with q,, q2 and q, as parameters.

78 J. SINGH and V. P. SINGH

@‘SO”

r , fi=0.05,~=0.30 _\ \ m=0.25,~=10.0

- \p=o.20 - \ _ \ -_-

21 :

q~=o.o5,Tj3=0.02 - y1,=0.50,~\~=0.02

_ I ---- ,~,=0.50,~*=0.05 \ - \

0- 0 0.25 0.50

ic

Fig. 10. U’ versus R at E = 0.10 with q, , q2 and qj as parameters.

effective in decreasing the maximum displacement under the load.

Figure 11 shows the variations of axial displace- ment when the shell is subjected to a combination of radial and tangential moving loads (4’ = 45”). From Fig. 11 it appears that as the shell is surrounded by soft soil, axial displacement is strongly affected as qs is varied and increases with increasing qs . Changes in values of q, or q2 have negligible effects on the axial displacement of the shell. When the shell is buried in hard and rocky surrounding soil (Fig. 12), the variations in q,, again, are very effective in reducing the maximum displacement under the load. However, alterations in q, and qz do not produce any significant change at all. Therefore, it may be inferred that as the shell is subjected to combined radial and tangential moving loads, a change in orthotropy parameter q3 strongly affects the axial displacement, U’, of the shell

,500

f

_ 113=OJO ---me_ -----__ ---__ ----_

$I’=450

1300 fi=o.o5, ij=o.30

m=0.25,ik=0.01

----- q2=0.05,q3=0.02 3 - 1lj=0.50,q~=O.O2 “0 1100 -----'I,=o.5o,q*=o.o5

-_-- o~i--_--__--_---'- 700 2 I ' a a s '

0 0.25 0.50

ic

Fig. 11. U’ versus X at E = 0.10 with q,, q2 and ‘1, as Fig. 13. U’ versus I’ at E = 0.10 with ‘I,. qz and q3 as parameters. parameters.

8 \Ij3=0.20 E \ \ I$'=45

fi=O.O5,~=0.30 m=0.25,~=10.0 -*--- IJ~=0.05,q3=0.02

- q,=O.50,113=0.02 ---- ~,=0.50.~*=0.05

x

Fig. 12. U' versus X at E = 0.10 with rt,. q2 and ‘13 as parameters.

regardless of the surrounding soil condition, whereas U’ is not influenced due to variations in q, or Q.

Figures 13 and 14 show the results when the shell is subjected to a tangential moving load, i.e. r$ = 90”. The amplitude for axial displacement increases but the nature of variation for different values of orthotropy parameters qr, q2 and qS remains almost the same as discussed for 4’ = 45”, although

t

---_ '1>=_0.20 --A_ ---_

---__ 2000

t

I$,=90 fi=0.05,~=0.30 m=0.25,k=0.01

1 -------q~=o.o5,q~=o.o2 -q,=0.50,q3=0.02 -----q,=o.5o,q2=0.05

3 "o 1500 c t

Steady-state dynamic response of buried pipelines 19

12- l+‘=90”

L li=0.05,~=0.30 \ -\

m=0.25,ii=lO.O

\ \

- ‘?3=o.20 lo \ ---.- q2=0.05, q3=0.02

\ - q1 =0.50, q3=0.02 \ \

-----‘11=0.50,r)~=0.05

I

6

0 0.25 0.50

x

Fig. 14. U’ versus X at E = 0.10 with tt,, q2 and ‘I, as parameters.

displacement seems to be hardly affected as we move away from the load. Although not shown, similar behaviour is obtained for angle 4’ = 30 or 60”.

Figures 15 and 16 show the variation of U’ with _? for different values of 4’ and one set of ‘I,, q2 and vs. As appears from Fig. 15, when the shell is surrounded by very soft and sandy soil (ji = 0.01) axial displace- ment remains unchanged for any value of 2 but varies largely as 4’ is changed. Amplitude is maxi- mum when the shell is disturbed by tangentially moving load, and decreases with decreasing values of 4’. Figure 15 shows that as the value of 4’ is decreased at regular intervals, the axial amplitude also decreases accordingly. However, when the

1200 Q=90°

*ooy

400

t

ri=o.o5, Q=0.30 m=0.25, G=O.Ol

q, =0.50, q2=0.05, q3=0.02

01 ,o: ” ” 4

0 0.25 0.50

x Fig. 15. U’ versus X at E = 0.10 with qb’ as parameter.

Fl = 0.05, ij=o.30

8 m=0.25, G=lO.O

q,=O.50, q2=0.05, q3=0.02

Fig. 16. w’ versus 1 at E = 0.10 with 4’ as parameter.

surrounding soil is hard and rocky (Fig. 16), the axial displacement decreases as we move away from the load. The nature of the variation of U’ is the same for any value of 9’.

CONCLUSIONS

Based on the results discussed in the preceding section, the following conclusions can be drawn.

(1) For a radial moving load, the maximum deflec- tion of the shell can be reduced by adjusting the orthotropy parameters vi, q2 and Q . In very hard and rocky soil the orthotropy parameter variations do not have any effect on the shell response. In most of the cases the rate of decay of displacement amplitude is very slow.

(2) As the shell is subjected to a tangential moving load the rate of decay of radial amplitude is much slower. The axial amplitude is very high, under tangential moving load, but can be reduced by adjust- ing the orthotropy parameters vi, tar and vs.

(3) As the shell is subjected to an inclined moving load, the axial displacement varies only when the orthotropy parameter qr changes. Variation in vi and qr does not affect displacement U’.

(4) The ratio of soil to pipe rigidity (p) has a very significant effect on displacement, irrespective of angle 4’.

In a manner similar to that for the effect on radial displacement, introduction of orthotropy through q,, q2 and ns is found to lower the axial displacement of the shell quite significantly. This observation is of special interest because most of the failures of buried pipes have been reported as being due to excessive axial displacement.

REFERENCES

1. R. D. Mindlin and H. H. Bleich, Response of an elastic cylindrical shell to a transverse, step shock wave. ASME, J. appl. Mech. 20, 189-195 (1953).

CAS 3711-F

80 J. SINGH and V. P. SINGH

2. R. Pames, Response of an infinite elastic medium to travelling loads in a cylindrical bore. Trans. Am. Sot. me& Engrs 36, 51-58 (1969).

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3. P. Mann-Nachbar, On the role of bending in the dynamic response of thin shells to moving dis- continuous loads. J. Aerospace Sci. 29, 648-657 (1962).

4. M. J. Forrestal and G. Herrmann, Response of a sub- merged cylindrical shell to an axially propagating step wave. Trans. Am. Sot. mech. Engrs 32,788-792 (1965).

5. E. Mangrum and J. J. Bums, Orthotropic cylindrical shells under dynamic loadings. ASME, J. Mech. Des. 101, 322-329 (1979).

7.

8.

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