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Natural frequencies and mode shapes of curved pipes
Computers & Stm-rum
Vol. 63, No. 3, pp. 46-73, 1997 0 1997 Elwvier Science Ltd Printed in Great Britain. All rights reserved 004%7949/97 s17.00 + 0.00
Pergamou PII: soo45-7949(%)00365-3
NATURAL
FREQUENCIES AND MODE SHAPES OF CURVED PIPES Dewu Huang, David Redekop and Bo Xu
Special Equipment Research Institute, Shenyang Institute of Technology, Shenyang, Liaoning 110015, People’s Republic of China (Received 19 Ocrober 1995)
Abstract-The
linear elastic Sanders shell theory is developed in toroidal coordinates to solve the problem of the vibration of a curved pipe with rigid diaphragm end supports. In-plane and rotary inertias are neglected. The theory, which is valid for any value of the pipe curvature, has been developed to give natural
frequencies for symmetric mode vibration. Sample results are calculated and compared with previously published results found using the Mushtari-Vlasov-Donnell theory. A numerical solution based on the finite element method is also given. Natural frequencies and mode shapes are determined. Frequency results from the two methods showed good agreement. Finally fundamental frequencies are given of 90” bends for a wide range of curved pipe geometries. 0 1997 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
GEOMETRY
The problem of the vibration of thin cylindrical shells (straight pipes) ha.s attracted considerable attention over the years. Some of the more significant results are summarized in the monographs by Gonthevich [I 1, Leissa [2], and Soedel[3]. Toroidal shells (curved pipes) have attracted less attention. Application of such work is to the design of piping systems and special purpose storage tanks. Analytical solutions for vibration problems in toroidal coordinates have been presented using several different shell theories [4-111. The work relates mainly to complete toroidal shells, and curved pipes have received little attention. Recent work includes that of Redekop [12], who used the Mushtari-Vlasov-Donnell (MVD) theory, specialized for pipes with moderate curvature, to determine natural frequencie:s of a pipe bend. In this paper the linear elastic Sanders shell theory is developed in toroidal coordinates to cover the problem of the vibration of curved pipes with rigid diaphragm end supports (Fig. 1). In-plane and rotary inertias are neglected. The theory has been developed only for symmetric mode vibration, but is valid for any value of the pipe curvature. Sample results are calculated and compared with those of the MVD theory from Ref. [12]. A numerical soluxion based on the finite element method (FEM) is also given. Natural frequencies and mode shapes are evaluated. Frequency results from the two methods are compared. Finally using the shell theory solution results are given for fundamental frequencies of 90” bends for a wide range of
The curved pipe has a toroidal radius R, a cross-sectional radius r, a thickness h, and an angular length $ (Figs 1 and 2). The linear length L of the pipe center-line is given by R$. A general point P on the pipe mid-surface is defined by the circumferential
curved
pipe geometries.
and longitudinal angular coordinates radius vector T to P is given by
B and 4. The
r = (R + r cos @sin c#A + (R + r cos 19)cos +j + r sin Bk (1) where i, j, k are unit vectors forming a base in the Cartesian x, y, z coordinate system. Both 4 and r] = &5 are used in the following as the longitudinal coordinates, with p = R/r. The Lame parameters A,
and AZ, and the principal radii of curvatures RI and R2 are given by A,=r;
Az=rS;
5 = 1 + y cos
RI = r;
8;
y =
RZ = rl’/(y
r/R = l/p.
cos 0) (2)
The dependence of Al and Rz on 0 cause the complication in the analysis of shells in toroidal coordinates. Natural frequencies are to be determined for pipes having end conditions corresponding to support by diaphragms rigid in their own planes. In Ref. [12] it was shown that the vibration in symmetric and anti-symmetric modes, with respect to the horizontal plane (z = 0), takes place at nearly the same
465
Dewu Huang et al.
466
frequencies. Thus in the present paper the theory is developed only for the symmetric modes. SANDERS
TOROIDAL
SHELL
THEORY
The shell theory of Sanders [13] is considered one of the most accurate first-order theories. Mainly due to its relative complexity it has been used sparingly in toroidal coordinates. The governing equations are written in terms of the three displacement components u, u and w (Fig. 3). Membrane stress resultants NB, N,,, No,, and bending stress resultants MO, M,,, iWe,,are defined in the theory, as well as normal shear resultants Qo, Q, (Fig. 3). The theory was developed in Ref. [14] for a static loading problem. In the present paper extensions to the theory are made to cover the vibration problem. In the Sanders theory the governing equations for the displacement components may be expressed as L,,u + L12v + L,3w = 0 L*,ll + L**v + L23w = 0 L31u + L32v + L33w = - (r4/D)h4q,
(3)
where the linear operators L,, depend on 0 and on the material parameters v (Poisson ratio) and E (Young’s modulus). These operators are defined in full in Ref. [14]. In eqn (3) D = Eh’/[l2(1 - v*)] and q, = q,(e, q) is the normal load. This load is included initially to permit accounting of the inertial effects later on, using the D’Alembert Principle. Inertial effects in the circumferential and longitudinal directions are taken as zero, as are rotary inertial effects.
To determine the natural frequencies for the symmetric mode vibration, Fourier series expansions are written for the displacement components and the normal load as u = Z;Cu, sin ntl sin prntj sin cot v = CCv,, cos n0 cos pLmtjsin ot w = CCw,, cos n0 sin pmfj sin wt
qr = =qrm” cos n0 sin prnrjsin cot
(4)
pm = mxYl*
(5)
where
and w is the natural frequency, while J/ equals the shell angular length (Fig. 1). In eqn (4), and in the following summations involving sin no are understood to extend for n = 1, 2, . , N, while sumextend over mations involving cos n0 n=0,1,2 ,...) N. The summations involving sin p,,,q and cos p,,,q extend over m = 1,3, . , M. Quantities N and M (unsubscripted) are the series truncation integers. These are selected large enough to ensure convergence of the solution. The selection of the series expansions in eqns (4) implies support at the two ends of the shell by diaphragms which are rigid in their own planes. Thus the following boundary conditions are satisfied at the shell ends 4 = 0” and 4 = $
L
Fig. 1. Geometry.
N,, = u = w = M,, = 0.
(6)
Mode shapes of curved pipes
461
Fig. 2. Toroidal coordinate system.
The Fourier series expansions of eqn (4) are next substituted into the governing eqn (3), and the coefficients of like products of trigonometric terms are collected. All equations for a given m form a set, containing only unknown displacement coefficients pertaining to that m. The three generic equations for a typical value of m then become a19m,n - 4th
- 4 +
a21h,n - 4th.”
- 4 +
a39,,”
+ a3hnwmn + +
a24m,n+Iv,c+I
+
a13~+
+
a33mr + 2wm.” +
+ 2hn
al8ms
- 3%!
- 3 +
+
a38m.n - 3wm.n - 3 +
+
a27m,n - 2vmVn - 2 +
+
ath
-
turn,- I +
+
ahfl
-
Iw,,,,- I +
a28m.n - 3&c
- 2wm,” - 2
a26m.n - IV,,~
alSmnumn
+
+
+ 2 +
2+
I
a34nr,n+Iwm,n+I
a23m,n + 2fhn
+ 2
+ 3hn
+
3
a31mfl + 3w,,,
+
3
ahn
(7)
+ 3 +
- 3
a17m.n - 2Um.n - 2
a3hn
+ lumc +
- 4~m,n - 4
+ a2h,”+3hn +
ah
-
I
aZ5dmn
+ al hC+4th +4 + +
bww +
a3h,n+4~m.n+4
4unt.n - 4 +
hn,n
=
u2h..
+ 4th
0
b29m.n - 4vm.n - 4 +
-sun,,
- 3 +
Fig. 3. Displacement components and stress resultants.
+ 4
bmn,m -4wm.n
bmn - a,,n -
3
- 4
Dewu Huang et al. + bwn,n+ ~um,n + , + bwnn+ , w,, + ,
+ h3m.n+ 2an.” +
b33m.n + zwm,.
+
b**m,”
+
bl ,m,n + d4n.n
+
b31m.n+4Wm,n+4
CI9m.n - 4&,.n
Fig. 4. FEM mesh for shell with $ = 0.4 rad.
+ 2 +
+ 2 +
+ 3U.r.n + , +
- 4 +
+ 4 +
=
b23m.n + 2Vrn.” + 2
h2m.n
+ 3Um.n + 3
632m.”
+ 3Wrn.”
+ 3
bfh,”
+ 4um,n + 4
0
C29m,n - 4fh.n
- 4 +
c39,,,
- 4w,n,.
+
CI8m.n - 3thn.n - 3 +
C28m.n - 3um.n - 3
+
c38m.n - 3wm,,
- 3 +
Cl7m.n
+
C27m,n - 2Um.n
2 +
C3lm,.
+
CMrn,”
I +
C26rn.”
+
C36m,n -
+
U3SmnWmn
+
C34nw + IWrn,n
+
c23m,
+
C12m,n + 3%n,n + 3 +
-
I Urn,” -
- 2um.n
- 2
- 2wm,.
IV,.”
- 4
- 2
-
I
+ b38m.n - 3~ln,?l- 3 + bllrn,”- 2Um.n - 2
+ b27m,n - 2Gfl,”
+
bl6m.n
- IUm,n
+
hl.n-
+
~35m.Wmn
,wm,n
+
- 2 +
- 1 +
I +
b14m.n
b3h,”
b&!,”
- 2Wm.n
-
IUrn,.
blSmnUmn
+ l%v7
+ I
+
1wm.n
-
I +
ClsmnUmn
+
C2SmnVmn
- 2 +
Cl4m.n
+ 1Um.n + I +
C24m.n + lum,n
- I
b2SmnUmn
Fig. 5. FEM mesh for 90” bend.
+ 2&n,”
+ I +
+ 2 +
C13mnt,n+ 2um.n
+ 2
C33m.n i 2Wm.n + 2
C22m.n + 3&n,”
+ 3
+ I
Mode Table
M
n
1 1
0 1 2 3 4
1 1 1
1
5
1 1 1 1
6 7 8 9
+
Cmn,n+3~m.+3
+
C2lrn,”
Straight MVD
+4&n,,,
+4
+
of curved
1. Natural
frequencies
469
pipes ( x IO-‘)
Curved pipe
pipe Sand.
5.1621 3.6762 1.9815 1.1683 0.8999 0.9757 I .2467 1.6269 2.0853 2.6120
+
shapes
5.1621 3.6757 1.9783 1.1580 0.8783 0.9464 1.2149 1.5946 2.053 1 2.5799
MVD Symm.
MVD Asymm.
5.1629 3.6057 2.0243 1.3102 0.7246 1.0617 1.3102 1.7481 2.0243 2.6995
3.6648 2.0247 1.3101 0.7246 1.0617 1.3101 1.7481 2.0247 2.6995
Sand.
Symm.
5.1670 3.6361 1.9914 1.2820 0.7056 1.0420 1.3878 1.7191 2.1589 2.6701
Invoking the D’Alembert Principle the terms on the right hand member of eqns (9) are transformed as
Clln,n+4Um,n+4
C3lm.n + 4Wm.n
(9)
+ 4
qm,+ - pho2w,. = dsq,,?l -4 +
dsqm,.
+
dlqm,“+4
+
d,qm,
- 3 +
+
dqmt,+
I +
d7qm
- 2 +
d3qm,,+*
+
d5qw
-
d2qm.n+
Transferring these terms to the appropriate position on the left-hand sides leads to a homogeneous set of equations. For a non-trivial solution the determinant must be zero, i.e.
3
where the coefficients aijmn,b,,, and cijmnand d, are lengthy, but known, functions of n, p,,,, v, h and E [14]. These generic expressions involve negative-indexed terms for n := 0, 1, 2,3 which are not present in the original expansions. It is necessary to modify the equations for these values of n, setting the appropriate terms to zero and employing the identities cos( -nf3) = cm nf3; sin( -nO) = -sin
n0
IA] = 0
Case 1 2 3 4 5 6 7 8 9 10
frequencies
(12)
where A represents the matrix of the coefficients of the transformed equation set (7H9). Equation (12) is a non-linear equation in the frequency parameter B = r4phu2/D. The roots of this equation may be found using the real form of the Muller method [ 151. Where frequencies are closely spaced, separation is obtained using the bisection method. The natural frequencies for corresponding straight pipes may be used as starting values. Soedel[3] gives the MVD predictions for the frequency parameters /Imn for straight pipes as
(10)
for the cases of negative arguments of the cos and sin functions. The qua.ntities q,,,”are related to the inertial effect and are discussed further below. By equating coelhcients of like trigonometric terms on the two sides of the equations, one obtains M’ = (M - 1)/2 + 1 sets of simultaneous equations. Each set consists of 3N + 2 equations, and there are in each set also 3,Y + 2 unknowns.
Table 2. Fundamental
(11)
I
pmn =
(n’+ cl:)’+ b4
(13)
(n’ + dJ*
where k = 12( 1 - v2)(r/h)2, and the p,,,have the values defined for curved pipes.
for 90” bends ( x lo-“) (distances
r
R
1
L
r
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.6 0.6
0.00025 0.00033 0.00048 0.00114 0.00180 0.00450 0.00620 O.OlGQO 0.01060 0.01800
0.471 0.471 0.471 0.471 0.471 0.471 0.628 0.628 0.943 0.943
133.3 101.0 69.4 29.2 18.5 7.4 4.0 2.5 1.6 0.9
Straight MVD 0.1342 0.1592 0.1862 0.2954 0.3599 0.6033 0.5114 0.7300 0.5480 0.6033
pipe Sand. 0.1313 0.1549 0.1812 0.2853 0.3388 0.5678 0.4337 0.5844 0.5165 0.5170
are in m) Curved MVD 0.0911 0.1056 0.1295 0.2083 0.2703 0.4682 0.4251 0.5867 0.4099 0.5539
pipe Sand. 0.0968 0.1051 0.1223 0.1919 0.2477 0.4202 0.3496 0.4607 0.2696 0.3326
Dewu Huang ef al. VALIDATION
AND RESULTS
The Sanders theory developed in the preceding was incorporated in the FORTRAN computer code OMSAND. To partially validate the solution numerical results were calculated for the following values of the problem parameters, corresponding to a short steel curved pipe p = 7770 kg m-’ L/r = 2.0;
v = 0.3;
h/r = 0.02;
E = 207,000 MPa
r/R = 0.2;
II, = 0.4 rad. (15)
Fig. 6. Global mode shape for shell with $ = 0.4 rad.
FINITE ELEMENT
METHOD
The equations formed for the vibration analysis contain the system stiffness and mass matrices [16]. These equations may be represented in matrix form as [Ml@) + [K](u) = 0
(14)
where [K] represents the system stiffness matrix, [M] the system mass matrix, (ti) the vector of nodal accelerations, and (u) the vector of nodal displacements. The commercial FEM program ANSYS was used to determine both the natural frequencies and mode shapes of curved pipes. The curved thin-shell element available in this program is an eight-noded isoparametric element with five degrees of freedom per node. This element was used for the free vibration analysis. Solutions were developed using the FEM for two basic shell geometries. The curved pipe shown in Fig. 4 served as the first example. The mesh selected consisted of 400 elements, with a total of 1288 nodes. The 90” bend shown in Fig. 5 served as the second example. The mesh consisted of 480 elements, with a total of 1488 nodes. Using this second mesh several problems, involving different choices for some of the geometric parameters, were solved.
Table 3. Comparison of natural frequencies for 90” bends (x 10-d) Case
MVD
Sand.
FEM
Mode shape
2 6 8 10
0.1056 0.4682 0.5867 0.5539
0.1051 0.4202 0.4607 0.3326
0.1015 0.3963 0.4465 0.2937
Fig. Fig. Fig. Fig.
7 8 9 10
Natural frequencies found using the MVD theory are available in Ref. [12] for this geometry, which is specifically within the range of validity of that theory. Results for the natural frequencies from the Sanders theory for this case are given in Table 1. The results are for the first harmonic in the longitudinal direction, m = 1, for which typically the lowest frequencies are obtained. Straight pipe and MVD curved pipe values from Ref. [ 121are also given in the table. The value of n in the table indicates the harmonic of the straight pipe solution. For the MVD theory results for both the symmetric and anti-symmetric toroidal modes were available. Table 1 contains corrections to Ref. [12] involving multiple roots. The two straight pipe values found using the MVD and Sanders theory agree well for this geometry, as do the curved pipe values. The lowest frequency occurs for m = 1, n = 4. For this curved pipe the MVD values for symmetric and anti-symmetric modes, and the Sanders values for symmetric modes, show close agreement. The fundamental frequency is some 20% lower than the lowest value for the straight pipe. Examination of eqns (7)-(9) and (12) leads to the relation
(rw,#=
(16)
Thus for a curved pipe the product of frequency and cross-section radius depends on the material parameters, and on ratios involving the pipe length, thickness and center-line radius. A pipe bend of some practical significance is the 90” bend (Fig. 5). Static loading results for some specific practical geometric cases were given in Ref. [14]. To characterize the geometry the factor F = r2(Rf)-’ is often used. In Table 2 natural frequencies are given for 90” bends for a wide range of the r factor. These bends correspond to those considered in Ref. [14]. The frequency values given correspond to the lowest result for the m = 1 case for symmetric mode vibration. Results from both the MVD and Sanders theory are given for straight and
471
Mode shapes of curved pipes
W
(4
(4
Fig. 7. Case 2 mode shape: (a) global; (b) plane z = 0; (c) plane 4 = x/4.
curved pipes. The fstraight pipe and MVD curved pipe results were found using the program OMTUBE [ 121.
Examination of results shows that substantial differences arise in the prediction of values from the two theories for some geometries. The results from the FEM will now be discussed. Corresponding to the pipe bend described by the eqn (15) a solution was calculated using ANSYS with the mesh of Fig. 4. The fundamental frequency obtained was 5759 Hz. The corresponding value found using the Sanders theory (Table 1) is 7056 Hz. Figure 6 shows the mode shape found using the FEM for this case. Results for some of the 90” bend cases were also calculated using the FEM method with the ANSYS
program. Table 3 lists the natural frequencies for cases 2, 6, 8 and 10, together with the shell theory results. The last column in the table indicates the figure number of the paper (Figs 7-10) in which the mode shape is depicted. Figure 7 shows the mode shape for case 2 of Table 3. Part (a) of the figure represents the global mode, part (b) represents the section in the plane z = 0, while part (c) the section in the plane 4 = n/4. Figures 8, 9, 10 give similar representations for cases 6, 8 and 10 of Table 3. It can be seen from Figs 7-10 that the FEM mode shapes are geometrically complex. It is difficult to identify symmetric or antisymmetric modes, and in fact mixed modes are indicated. Furthermore
(b) Fig. 8. Case 6 mode shape: (a) global; (b) plane z = 0; (c) plane 4 = */4.
472
Dewu Huang et al.
(4
(b) Fig. 9. Case 8 mode shape: (a) global; (b) plane z = 0; (c) plane $J= n/4.
(c) Fig. 10. Case 10 mode shape: (a) global; (b) plane z = 0; (c) plane I$ = n/4. the mode shapes on the intrados and extrados are of different complexity. In the case of the extrados there is typically one wave, while for the intrados there are several. The convergence of the FEM results was verified. With calculations for meshes of increasing refinement the fundamental frequency was found to gradually converge. The meshes depicted in Figs 4 and 5 represent a balance between computational accuracy and expense. longitudinally
CONCLUSION A Sanders theory solution for the linear elastic vibration of curved pipes with rigid diaphragm supports has been presented. Sample results were given for the natural frequencies of a shell recognized as being within the range of validity of the MVD theory. These results compared well with previously
published results found using the MVD shell theory. An FEM solution for the natural frequencies and mode shapes was also given. Results for the funda mental frequencies compared closely with those of the Sanders theory. It was shown that the funda mental mode involves a single harmonic longitudinally and two harmonics circumferentially. For some cases of 90” bends frequency results from the Sanders and MVD theories showed substantial differences. Acknowledgements-The
authors wish to thank Engineers
Jianhui Sui, Bashai Qie of the China North Industries Group for their support of this research. REFERENCES 1. V. S. Gonthevich, Natural Vibrations of Plates and Shells-A Handbook. Akad. Nauk., Kiev, Ukraine (1964) (in Russian).
Mode shapes of curved pipes 2. A. W. Leissa, Vibration of Shells. NASA, sp-288 (1973). 3. W. Soedel, Vibration of Shells and Plates, 2nd edn. Marcel1 Dekker, New York (1992). 4. T. Balderes and A. E. Armenakas, Free vibrations of ring-stiffened toroidal shells. AIAA Journal, 11 16371644 (1973). 5. S. P. Gavelya and N. I. Kononenko, Calculation studies of the dynamic characteristics of spherical and toroidal shells. Sou. Appt. Mech. 2, 129-132 (1973). 6. A. V. Bulygin, Calculation of lowest eigenfrequencies and vibration modes of toroidal shells. Sou. Aeron. J. 18, 39-43 (1975). 7. S. P. Gavelya and N. I. Kononenko,
oscillations and waves on a toroidal Mech. 11 31-35 (1975). 8. T. Kosawada, K. Suzuki and Free vibrations of toroidal shells. 2041-2047 (1985). 9. T. Kosawada, K. Suzuki and S. vibrations of thick toroidal shells. 30363042 (1986).
Characteristic shell. Sou. Appl.
S. Takahashi, Bull. JSME 28, Takahashi, Bull.
JSME
Free 29,
473
10. Y. Kobayashi, G. Yamada and T. Nagai, Free vibration of a toroidal shell with interior partition. Nippon Kihai Gahhai Ronbunshu C Hen 55, 2306231 I (1989) (in Japanese). 11. A. Y. T. Leung and N. T. C. Kwok, Free vibration analysis of a toroidal shell. Thin-walled Struct. 18, 317-332 (1994). 12. D. Redekop, Natural frequencies of a short curved pipe. Trans. CSME 18, 3545 (1994). 13. J. L. Sanders Jr, An improved first approximation
theory for thin shells, Langley Research Center, TR-24 (1959). 14. F. Zhang and D. Redekop, Surface loading of a thin-walled toroidal shell. Cornput. Struct. 43, 10191028 (1992). 15. D. E. Muller, A method for solving algebraic equations using an automatic commuter. Math. Tables Aidr Comput. 10, 208-215 (1956). 16. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ, pp.
499-566 (I 982).
Vol. 63, No. 3, pp. 46-73, 1997 0 1997 Elwvier Science Ltd Printed in Great Britain. All rights reserved 004%7949/97 s17.00 + 0.00
Pergamou PII: soo45-7949(%)00365-3
NATURAL
FREQUENCIES AND MODE SHAPES OF CURVED PIPES Dewu Huang, David Redekop and Bo Xu
Special Equipment Research Institute, Shenyang Institute of Technology, Shenyang, Liaoning 110015, People’s Republic of China (Received 19 Ocrober 1995)
Abstract-The
linear elastic Sanders shell theory is developed in toroidal coordinates to solve the problem of the vibration of a curved pipe with rigid diaphragm end supports. In-plane and rotary inertias are neglected. The theory, which is valid for any value of the pipe curvature, has been developed to give natural
frequencies for symmetric mode vibration. Sample results are calculated and compared with previously published results found using the Mushtari-Vlasov-Donnell theory. A numerical solution based on the finite element method is also given. Natural frequencies and mode shapes are determined. Frequency results from the two methods showed good agreement. Finally fundamental frequencies are given of 90” bends for a wide range of curved pipe geometries. 0 1997 Elsevier Science Ltd. All rights reserved.
INTRODUCTION
GEOMETRY
The problem of the vibration of thin cylindrical shells (straight pipes) ha.s attracted considerable attention over the years. Some of the more significant results are summarized in the monographs by Gonthevich [I 1, Leissa [2], and Soedel[3]. Toroidal shells (curved pipes) have attracted less attention. Application of such work is to the design of piping systems and special purpose storage tanks. Analytical solutions for vibration problems in toroidal coordinates have been presented using several different shell theories [4-111. The work relates mainly to complete toroidal shells, and curved pipes have received little attention. Recent work includes that of Redekop [12], who used the Mushtari-Vlasov-Donnell (MVD) theory, specialized for pipes with moderate curvature, to determine natural frequencie:s of a pipe bend. In this paper the linear elastic Sanders shell theory is developed in toroidal coordinates to cover the problem of the vibration of curved pipes with rigid diaphragm end supports (Fig. 1). In-plane and rotary inertias are neglected. The theory has been developed only for symmetric mode vibration, but is valid for any value of the pipe curvature. Sample results are calculated and compared with those of the MVD theory from Ref. [12]. A numerical soluxion based on the finite element method (FEM) is also given. Natural frequencies and mode shapes are evaluated. Frequency results from the two methods are compared. Finally using the shell theory solution results are given for fundamental frequencies of 90” bends for a wide range of
The curved pipe has a toroidal radius R, a cross-sectional radius r, a thickness h, and an angular length $ (Figs 1 and 2). The linear length L of the pipe center-line is given by R$. A general point P on the pipe mid-surface is defined by the circumferential
curved
pipe geometries.
and longitudinal angular coordinates radius vector T to P is given by
B and 4. The
r = (R + r cos @sin c#A + (R + r cos 19)cos +j + r sin Bk (1) where i, j, k are unit vectors forming a base in the Cartesian x, y, z coordinate system. Both 4 and r] = &5 are used in the following as the longitudinal coordinates, with p = R/r. The Lame parameters A,
and AZ, and the principal radii of curvatures RI and R2 are given by A,=r;
Az=rS;
5 = 1 + y cos
RI = r;
8;
y =
RZ = rl’/(y
r/R = l/p.
cos 0) (2)
The dependence of Al and Rz on 0 cause the complication in the analysis of shells in toroidal coordinates. Natural frequencies are to be determined for pipes having end conditions corresponding to support by diaphragms rigid in their own planes. In Ref. [12] it was shown that the vibration in symmetric and anti-symmetric modes, with respect to the horizontal plane (z = 0), takes place at nearly the same
465
Dewu Huang et al.
466
frequencies. Thus in the present paper the theory is developed only for the symmetric modes. SANDERS
TOROIDAL
SHELL
THEORY
The shell theory of Sanders [13] is considered one of the most accurate first-order theories. Mainly due to its relative complexity it has been used sparingly in toroidal coordinates. The governing equations are written in terms of the three displacement components u, u and w (Fig. 3). Membrane stress resultants NB, N,,, No,, and bending stress resultants MO, M,,, iWe,,are defined in the theory, as well as normal shear resultants Qo, Q, (Fig. 3). The theory was developed in Ref. [14] for a static loading problem. In the present paper extensions to the theory are made to cover the vibration problem. In the Sanders theory the governing equations for the displacement components may be expressed as L,,u + L12v + L,3w = 0 L*,ll + L**v + L23w = 0 L31u + L32v + L33w = - (r4/D)h4q,
(3)
where the linear operators L,, depend on 0 and on the material parameters v (Poisson ratio) and E (Young’s modulus). These operators are defined in full in Ref. [14]. In eqn (3) D = Eh’/[l2(1 - v*)] and q, = q,(e, q) is the normal load. This load is included initially to permit accounting of the inertial effects later on, using the D’Alembert Principle. Inertial effects in the circumferential and longitudinal directions are taken as zero, as are rotary inertial effects.
To determine the natural frequencies for the symmetric mode vibration, Fourier series expansions are written for the displacement components and the normal load as u = Z;Cu, sin ntl sin prntj sin cot v = CCv,, cos n0 cos pLmtjsin ot w = CCw,, cos n0 sin pmfj sin wt
qr = =qrm” cos n0 sin prnrjsin cot
(4)
pm = mxYl*
(5)
where
and w is the natural frequency, while J/ equals the shell angular length (Fig. 1). In eqn (4), and in the following summations involving sin no are understood to extend for n = 1, 2, . , N, while sumextend over mations involving cos n0 n=0,1,2 ,...) N. The summations involving sin p,,,q and cos p,,,q extend over m = 1,3, . , M. Quantities N and M (unsubscripted) are the series truncation integers. These are selected large enough to ensure convergence of the solution. The selection of the series expansions in eqns (4) implies support at the two ends of the shell by diaphragms which are rigid in their own planes. Thus the following boundary conditions are satisfied at the shell ends 4 = 0” and 4 = $
L
Fig. 1. Geometry.
N,, = u = w = M,, = 0.
(6)
Mode shapes of curved pipes
461
Fig. 2. Toroidal coordinate system.
The Fourier series expansions of eqn (4) are next substituted into the governing eqn (3), and the coefficients of like products of trigonometric terms are collected. All equations for a given m form a set, containing only unknown displacement coefficients pertaining to that m. The three generic equations for a typical value of m then become a19m,n - 4th
- 4 +
a21h,n - 4th.”
- 4 +
a39,,”
+ a3hnwmn + +
a24m,n+Iv,c+I
+
a13~+
+
a33mr + 2wm.” +
+ 2hn
al8ms
- 3%!
- 3 +
+
a38m.n - 3wm.n - 3 +
+
a27m,n - 2vmVn - 2 +
+
ath
-
turn,- I +
+
ahfl
-
Iw,,,,- I +
a28m.n - 3&c
- 2wm,” - 2
a26m.n - IV,,~
alSmnumn
+
+
+ 2 +
2+
I
a34nr,n+Iwm,n+I
a23m,n + 2fhn
+ 2
+ 3hn
+
3
a31mfl + 3w,,,
+
3
ahn
(7)
+ 3 +
- 3
a17m.n - 2Um.n - 2
a3hn
+ lumc +
- 4~m,n - 4
+ a2h,”+3hn +
ah
-
I
aZ5dmn
+ al hC+4th +4 + +
bww +
a3h,n+4~m.n+4
4unt.n - 4 +
hn,n
=
u2h..
+ 4th
0
b29m.n - 4vm.n - 4 +
-sun,,
- 3 +
Fig. 3. Displacement components and stress resultants.
+ 4
bmn,m -4wm.n
bmn - a,,n -
3
- 4
Dewu Huang et al. + bwn,n+ ~um,n + , + bwnn+ , w,, + ,
+ h3m.n+ 2an.” +
b33m.n + zwm,.
+
b**m,”
+
bl ,m,n + d4n.n
+
b31m.n+4Wm,n+4
CI9m.n - 4&,.n
Fig. 4. FEM mesh for shell with $ = 0.4 rad.
+ 2 +
+ 2 +
+ 3U.r.n + , +
- 4 +
+ 4 +
=
b23m.n + 2Vrn.” + 2
h2m.n
+ 3Um.n + 3
632m.”
+ 3Wrn.”
+ 3
bfh,”
+ 4um,n + 4
0
C29m,n - 4fh.n
- 4 +
c39,,,
- 4w,n,.
+
CI8m.n - 3thn.n - 3 +
C28m.n - 3um.n - 3
+
c38m.n - 3wm,,
- 3 +
Cl7m.n
+
C27m,n - 2Um.n
2 +
C3lm,.
+
CMrn,”
I +
C26rn.”
+
C36m,n -
+
U3SmnWmn
+
C34nw + IWrn,n
+
c23m,
+
C12m,n + 3%n,n + 3 +
-
I Urn,” -
- 2um.n
- 2
- 2wm,.
IV,.”
- 4
- 2
-
I
+ b38m.n - 3~ln,?l- 3 + bllrn,”- 2Um.n - 2
+ b27m,n - 2Gfl,”
+
bl6m.n
- IUm,n
+
hl.n-
+
~35m.Wmn
,wm,n
+
- 2 +
- 1 +
I +
b14m.n
b3h,”
b&!,”
- 2Wm.n
-
IUrn,.
blSmnUmn
+ l%v7
+ I
+
1wm.n
-
I +
ClsmnUmn
+
C2SmnVmn
- 2 +
Cl4m.n
+ 1Um.n + I +
C24m.n + lum,n
- I
b2SmnUmn
Fig. 5. FEM mesh for 90” bend.
+ 2&n,”
+ I +
+ 2 +
C13mnt,n+ 2um.n
+ 2
C33m.n i 2Wm.n + 2
C22m.n + 3&n,”
+ 3
+ I
Mode Table
M
n
1 1
0 1 2 3 4
1 1 1
1
5
1 1 1 1
6 7 8 9
+
Cmn,n+3~m.+3
+
C2lrn,”
Straight MVD
+4&n,,,
+4
+
of curved
1. Natural
frequencies
469
pipes ( x IO-‘)
Curved pipe
pipe Sand.
5.1621 3.6762 1.9815 1.1683 0.8999 0.9757 I .2467 1.6269 2.0853 2.6120
+
shapes
5.1621 3.6757 1.9783 1.1580 0.8783 0.9464 1.2149 1.5946 2.053 1 2.5799
MVD Symm.
MVD Asymm.
5.1629 3.6057 2.0243 1.3102 0.7246 1.0617 1.3102 1.7481 2.0243 2.6995
3.6648 2.0247 1.3101 0.7246 1.0617 1.3101 1.7481 2.0247 2.6995
Sand.
Symm.
5.1670 3.6361 1.9914 1.2820 0.7056 1.0420 1.3878 1.7191 2.1589 2.6701
Invoking the D’Alembert Principle the terms on the right hand member of eqns (9) are transformed as
Clln,n+4Um,n+4
C3lm.n + 4Wm.n
(9)
+ 4
qm,+ - pho2w,. = dsq,,?l -4 +
dsqm,.
+
dlqm,“+4
+
d,qm,
- 3 +
+
dqmt,+
I +
d7qm
- 2 +
d3qm,,+*
+
d5qw
-
d2qm.n+
Transferring these terms to the appropriate position on the left-hand sides leads to a homogeneous set of equations. For a non-trivial solution the determinant must be zero, i.e.
3
where the coefficients aijmn,b,,, and cijmnand d, are lengthy, but known, functions of n, p,,,, v, h and E [14]. These generic expressions involve negative-indexed terms for n := 0, 1, 2,3 which are not present in the original expansions. It is necessary to modify the equations for these values of n, setting the appropriate terms to zero and employing the identities cos( -nf3) = cm nf3; sin( -nO) = -sin
n0
IA] = 0
Case 1 2 3 4 5 6 7 8 9 10
frequencies
(12)
where A represents the matrix of the coefficients of the transformed equation set (7H9). Equation (12) is a non-linear equation in the frequency parameter B = r4phu2/D. The roots of this equation may be found using the real form of the Muller method [ 151. Where frequencies are closely spaced, separation is obtained using the bisection method. The natural frequencies for corresponding straight pipes may be used as starting values. Soedel[3] gives the MVD predictions for the frequency parameters /Imn for straight pipes as
(10)
for the cases of negative arguments of the cos and sin functions. The qua.ntities q,,,”are related to the inertial effect and are discussed further below. By equating coelhcients of like trigonometric terms on the two sides of the equations, one obtains M’ = (M - 1)/2 + 1 sets of simultaneous equations. Each set consists of 3N + 2 equations, and there are in each set also 3,Y + 2 unknowns.
Table 2. Fundamental
(11)
I
pmn =
(n’+ cl:)’+ b4
(13)
(n’ + dJ*
where k = 12( 1 - v2)(r/h)2, and the p,,,have the values defined for curved pipes.
for 90” bends ( x lo-“) (distances
r
R
1
L
r
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.6 0.6
0.00025 0.00033 0.00048 0.00114 0.00180 0.00450 0.00620 O.OlGQO 0.01060 0.01800
0.471 0.471 0.471 0.471 0.471 0.471 0.628 0.628 0.943 0.943
133.3 101.0 69.4 29.2 18.5 7.4 4.0 2.5 1.6 0.9
Straight MVD 0.1342 0.1592 0.1862 0.2954 0.3599 0.6033 0.5114 0.7300 0.5480 0.6033
pipe Sand. 0.1313 0.1549 0.1812 0.2853 0.3388 0.5678 0.4337 0.5844 0.5165 0.5170
are in m) Curved MVD 0.0911 0.1056 0.1295 0.2083 0.2703 0.4682 0.4251 0.5867 0.4099 0.5539
pipe Sand. 0.0968 0.1051 0.1223 0.1919 0.2477 0.4202 0.3496 0.4607 0.2696 0.3326
Dewu Huang ef al. VALIDATION
AND RESULTS
The Sanders theory developed in the preceding was incorporated in the FORTRAN computer code OMSAND. To partially validate the solution numerical results were calculated for the following values of the problem parameters, corresponding to a short steel curved pipe p = 7770 kg m-’ L/r = 2.0;
v = 0.3;
h/r = 0.02;
E = 207,000 MPa
r/R = 0.2;
II, = 0.4 rad. (15)
Fig. 6. Global mode shape for shell with $ = 0.4 rad.
FINITE ELEMENT
METHOD
The equations formed for the vibration analysis contain the system stiffness and mass matrices [16]. These equations may be represented in matrix form as [Ml@) + [K](u) = 0
(14)
where [K] represents the system stiffness matrix, [M] the system mass matrix, (ti) the vector of nodal accelerations, and (u) the vector of nodal displacements. The commercial FEM program ANSYS was used to determine both the natural frequencies and mode shapes of curved pipes. The curved thin-shell element available in this program is an eight-noded isoparametric element with five degrees of freedom per node. This element was used for the free vibration analysis. Solutions were developed using the FEM for two basic shell geometries. The curved pipe shown in Fig. 4 served as the first example. The mesh selected consisted of 400 elements, with a total of 1288 nodes. The 90” bend shown in Fig. 5 served as the second example. The mesh consisted of 480 elements, with a total of 1488 nodes. Using this second mesh several problems, involving different choices for some of the geometric parameters, were solved.
Table 3. Comparison of natural frequencies for 90” bends (x 10-d) Case
MVD
Sand.
FEM
Mode shape
2 6 8 10
0.1056 0.4682 0.5867 0.5539
0.1051 0.4202 0.4607 0.3326
0.1015 0.3963 0.4465 0.2937
Fig. Fig. Fig. Fig.
7 8 9 10
Natural frequencies found using the MVD theory are available in Ref. [12] for this geometry, which is specifically within the range of validity of that theory. Results for the natural frequencies from the Sanders theory for this case are given in Table 1. The results are for the first harmonic in the longitudinal direction, m = 1, for which typically the lowest frequencies are obtained. Straight pipe and MVD curved pipe values from Ref. [ 121are also given in the table. The value of n in the table indicates the harmonic of the straight pipe solution. For the MVD theory results for both the symmetric and anti-symmetric toroidal modes were available. Table 1 contains corrections to Ref. [12] involving multiple roots. The two straight pipe values found using the MVD and Sanders theory agree well for this geometry, as do the curved pipe values. The lowest frequency occurs for m = 1, n = 4. For this curved pipe the MVD values for symmetric and anti-symmetric modes, and the Sanders values for symmetric modes, show close agreement. The fundamental frequency is some 20% lower than the lowest value for the straight pipe. Examination of eqns (7)-(9) and (12) leads to the relation
(rw,#=
(16)
Thus for a curved pipe the product of frequency and cross-section radius depends on the material parameters, and on ratios involving the pipe length, thickness and center-line radius. A pipe bend of some practical significance is the 90” bend (Fig. 5). Static loading results for some specific practical geometric cases were given in Ref. [14]. To characterize the geometry the factor F = r2(Rf)-’ is often used. In Table 2 natural frequencies are given for 90” bends for a wide range of the r factor. These bends correspond to those considered in Ref. [14]. The frequency values given correspond to the lowest result for the m = 1 case for symmetric mode vibration. Results from both the MVD and Sanders theory are given for straight and
471
Mode shapes of curved pipes
W
(4
(4
Fig. 7. Case 2 mode shape: (a) global; (b) plane z = 0; (c) plane 4 = x/4.
curved pipes. The fstraight pipe and MVD curved pipe results were found using the program OMTUBE [ 121.
Examination of results shows that substantial differences arise in the prediction of values from the two theories for some geometries. The results from the FEM will now be discussed. Corresponding to the pipe bend described by the eqn (15) a solution was calculated using ANSYS with the mesh of Fig. 4. The fundamental frequency obtained was 5759 Hz. The corresponding value found using the Sanders theory (Table 1) is 7056 Hz. Figure 6 shows the mode shape found using the FEM for this case. Results for some of the 90” bend cases were also calculated using the FEM method with the ANSYS
program. Table 3 lists the natural frequencies for cases 2, 6, 8 and 10, together with the shell theory results. The last column in the table indicates the figure number of the paper (Figs 7-10) in which the mode shape is depicted. Figure 7 shows the mode shape for case 2 of Table 3. Part (a) of the figure represents the global mode, part (b) represents the section in the plane z = 0, while part (c) the section in the plane 4 = n/4. Figures 8, 9, 10 give similar representations for cases 6, 8 and 10 of Table 3. It can be seen from Figs 7-10 that the FEM mode shapes are geometrically complex. It is difficult to identify symmetric or antisymmetric modes, and in fact mixed modes are indicated. Furthermore
(b) Fig. 8. Case 6 mode shape: (a) global; (b) plane z = 0; (c) plane 4 = */4.
472
Dewu Huang et al.
(4
(b) Fig. 9. Case 8 mode shape: (a) global; (b) plane z = 0; (c) plane $J= n/4.
(c) Fig. 10. Case 10 mode shape: (a) global; (b) plane z = 0; (c) plane I$ = n/4. the mode shapes on the intrados and extrados are of different complexity. In the case of the extrados there is typically one wave, while for the intrados there are several. The convergence of the FEM results was verified. With calculations for meshes of increasing refinement the fundamental frequency was found to gradually converge. The meshes depicted in Figs 4 and 5 represent a balance between computational accuracy and expense. longitudinally
CONCLUSION A Sanders theory solution for the linear elastic vibration of curved pipes with rigid diaphragm supports has been presented. Sample results were given for the natural frequencies of a shell recognized as being within the range of validity of the MVD theory. These results compared well with previously
published results found using the MVD shell theory. An FEM solution for the natural frequencies and mode shapes was also given. Results for the funda mental frequencies compared closely with those of the Sanders theory. It was shown that the funda mental mode involves a single harmonic longitudinally and two harmonics circumferentially. For some cases of 90” bends frequency results from the Sanders and MVD theories showed substantial differences. Acknowledgements-The
authors wish to thank Engineers
Jianhui Sui, Bashai Qie of the China North Industries Group for their support of this research. REFERENCES 1. V. S. Gonthevich, Natural Vibrations of Plates and Shells-A Handbook. Akad. Nauk., Kiev, Ukraine (1964) (in Russian).
Mode shapes of curved pipes 2. A. W. Leissa, Vibration of Shells. NASA, sp-288 (1973). 3. W. Soedel, Vibration of Shells and Plates, 2nd edn. Marcel1 Dekker, New York (1992). 4. T. Balderes and A. E. Armenakas, Free vibrations of ring-stiffened toroidal shells. AIAA Journal, 11 16371644 (1973). 5. S. P. Gavelya and N. I. Kononenko, Calculation studies of the dynamic characteristics of spherical and toroidal shells. Sou. Appt. Mech. 2, 129-132 (1973). 6. A. V. Bulygin, Calculation of lowest eigenfrequencies and vibration modes of toroidal shells. Sou. Aeron. J. 18, 39-43 (1975). 7. S. P. Gavelya and N. I. Kononenko,
oscillations and waves on a toroidal Mech. 11 31-35 (1975). 8. T. Kosawada, K. Suzuki and Free vibrations of toroidal shells. 2041-2047 (1985). 9. T. Kosawada, K. Suzuki and S. vibrations of thick toroidal shells. 30363042 (1986).
Characteristic shell. Sou. Appl.
S. Takahashi, Bull. JSME 28, Takahashi, Bull.
JSME
Free 29,
473
10. Y. Kobayashi, G. Yamada and T. Nagai, Free vibration of a toroidal shell with interior partition. Nippon Kihai Gahhai Ronbunshu C Hen 55, 2306231 I (1989) (in Japanese). 11. A. Y. T. Leung and N. T. C. Kwok, Free vibration analysis of a toroidal shell. Thin-walled Struct. 18, 317-332 (1994). 12. D. Redekop, Natural frequencies of a short curved pipe. Trans. CSME 18, 3545 (1994). 13. J. L. Sanders Jr, An improved first approximation
theory for thin shells, Langley Research Center, TR-24 (1959). 14. F. Zhang and D. Redekop, Surface loading of a thin-walled toroidal shell. Cornput. Struct. 43, 10191028 (1992). 15. D. E. Muller, A method for solving algebraic equations using an automatic commuter. Math. Tables Aidr Comput. 10, 208-215 (1956). 16. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ, pp.
499-566 (I 982).