Free vibration analysis of arbitrary thin shell structures by using spline finite element

Journal of Sound and Vibration (1995) 179(5), 763–776

FREE VIBRATION ANALYSIS OF ARBITRARY THIN SHELL STRUCTURES BY USING SPLINE FINITE ELEMENT S. C. F  M. H. L School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263 (Received 27 October 1992, and in final form 15 October 1993) This paper presents the free vibration analysis of arbitrary thin shell structures by using a newly developed spline finite element. The new element has three salient features in its formulation: (i) the use of B-spline shape functions for the interpolations of both in-plane and out-of-plane displacements of a general thin shell element; (ii) the use of ‘‘displacement constraints’’ and ‘‘parameters shifting’’ to construct a finite element model; (iii) that a proposed modified version of the Koiter’s thin shell theory is employed. The element is doubly curved, has nine primary nodes and eight auxiliary nodes, and has a total of 63 degrees of freedom. It is formulated through the conventional C1 displacement approach, and is capable of modelling sharp corners, arbitrary shapes and multiple junctions of thin shell structures. The numerical examples discussed include spherical panels, cylindrical panels and shells of revolution, as well as single- and double-cell boxes. These examples are typical shells of negative, zero and positive Gaussian curvatures. The effects of aspect ratios, distorted meshes and junctions of shells on the performance of the element are studied. It is shown that the new spline finite element is a reliable, versatile, accurate and efficient thin shell element suitable for the analysis of arbitrary thin shell structures.

1. INTRODUCTION

Thin shell structures have been widely used in many branches of engineering. Doubly curved shells such as hyperbolic paraboloids or spherical panels are frequently favoured for use as roofs of large column-free areas. In water-retaining structures, spherical or cylindrical tanks are most common. Other structures in which thin shells are applicable include box girder bridges, cooling towers, pressure vessels, aero structures, etc. This vast interest in the use of shell structures is mainly due to their aesthetic appeal and cost-effectiveness. For the analysis and design, an important item of consideration is the free vibration response and a considerable amount of research on the subject has been carried out by using various analytical and numerical methods (see, e.g., references [1–4]). New methods of analysis, especially numerical models which offer improvements in accuracy of solution or reduction in computational effort, also continue to be developed. Recent advances include the work of Heppler and Wahl [5], Ganesan and Sivadas [6], Yamada et al. [7], Narita and Leissa [8] and To and Wang [9]. It is also noticed that the application of spline functions in the analysis of shell structures has emerged to be an exciting topic of research in the literature [10–17]. Recently, the present authors developed a spline finite element for the analysis of arbitrary thin shell structures. The element is doubly curved, quadrilateral C1 element based on Kirchhoff–Love assumptions. These assumptions may be briefly stated as follows: (1) the shell is thin; (2) the deformation of the shell is small; (3) normal stresses 763 0022–460X/95/050763 + 14 $08.00/0

7 1995 Academic Press Limited

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. .   . . 

perpendicular to the middle surface can be neglected in comparison to the other stresses; (4) straight lines normal to the undeformed middle surface remain straight and normal to the deformed middle surface. Although no precise definition of thinness is available, the Kirchhoff–Love assumptions are expected to yield sufficiently accurate results when the ratio of the thickness to the minimum radius of curvature of the shell middle surface is less than 0·05 (i.e., t/R Q 1/20). In the present development, it is also assumed that the material is isotropic, homogeneous, and linearly elastic. Three salient features of the spline finite element formulation can be stated and these are, first, the use of bicubic spline functions as shape functions for the interpolations of both in-plane and out-of-plane displacement of the element; second, the use of ‘‘displacement constraints’’ and ‘‘parameters shifting’’ to construct the element nodal configuration and the corresponding B-spline shape functions; and, third, that a proposed modified version of the Koiter’s thin shell theory is employed. The new element is capable of modelling sharp corners, arbitrary shapes and multiple junctions of thin shell structures. The performance of the element for static analysis has been thoroughly investigated in another paper [18] and it was found that the proposed element is accurate, reliable and versatile. In this paper, the free vibration analysis of arbitrary thin shell structures by using the new spline element is reported. The examples of analysis include doubly curved shell panels, shells of revolution and cylindrical panels, as well as single- and double-cell boxes. In order to show the accuracy of the proposed element, both coarse- and fine-mesh solutions are presented in all cases. The results obtained are also compared with those reported by other investigators to validate the present method. 2. SPLINE FINITE ELEMENT MODEL

The formulation of the spline finite element employed in the present analysis has been presented in detail in another paper [18] and only an outline is given below. The element is quadrilateral, and has nine primary nodes and eight auxiliary nodes, as shown in Figure 1. Four co-ordinate systems are used in the formulation and these are defined as follows (see Figure 1). 1. Global Cartesian co-ordinate system (x, y, z). This system is used to define the geometry and global translational displacements of the element.

Figure 1. The nodal configuration and co-ordinate systems of a doubly curved quadrilateral thin shell element.

     

765

2. Natural curvilinear co-ordinate system (j, h, z). The shape functions for geometry as well as displacements are expressed in this system. The middle surface of the element contains the j- and h-ordinates, the values of which vary between −1 and 1 with the origin located at the element centroid (central node). The z-direction is normal to the element middle surface with j–h–z forming a right-handed system. 3. Orthogonal curvilinear co-ordinate system (a, b, z). This system is needed in the present formulation because the strain–displacement equations used are expressed in orthogonal curvilinear co-ordinates. The curvilinear axes a and b are orthogonal and are both tangential to the element middle surface. Note that a–b–z is also a right-handed system and that the direction of the a-axis is chosen to coincide with the direction of the j-axis. 4. Nodal orthogonal co-ordinate system (s, h, z). This system is defined at each nodal point along the edges of the element and is needed for expressing the degrees of freedom such that inter-element compatibility can be achieved. The s-axis is defined to coincide with the j- or h-axis along the edge of the element, the z-axis is in the direction of the shell’s normal, and the n-axis is perpendicular to both the s- and z-directions and is tangential to the shell’s middle surface. The present element has 63 degrees of freedom, with 21 each for the three components of displacement u, v and w (which are in the directions of a, b and z respectively). The shape functions for both in-plane (u, v) and out-of-plane (w) displacements are of the same order and are derived from B3 -spline functions with the techniques of ‘‘displacement constraints’’ and ‘‘parameter shifting’’ (see references [18, 19]). The displacement function may be written as {f} = [N]{d'},

(1)

where {f} = [u

v

w

]T,

{N} = [N1

N2

N3 ]T

(2, 3)

in which the Ni , for i = 1, 2, 3, are each a submatrix of shape functions for the individual T 1 Local and global degrees of freedom of shell element (see Figure 1) Nodal type and no. Corner (1, 2, 3, 4)

Local degrees of freedom {d'}

Global degrees of freedom {d}

u, v, w

u¯ , v¯ , w¯

Mid-side (5, 7)

u, v, w, 1w/1h

u¯ , v¯ , w¯ , fn

Mid-side (6, 8)

u, v, w, 1w/1j

u¯ , v¯ , w¯ , fn

Auxiliary (along edges 1–5–2 and 4–7–3)

u, v, 1w/1h

u', v', fn

Auxiliary (along edges 2–6–3 and 1–8–4)

u, v, 1w/1j

u', v', fn

u, v, w, 1u/1j, 1u/1h, 1 2u/1j 2, 1 2u/1h 2, 1v/1j, 1v/1h, 1 2v/1j 2, 1 2v/1h 2

u¯ , v¯ , w¯ , 1u/1j, 1u/1h, 1 2u/1j 2, 1 2u/1h 2, 1v/1j, 1v/1h, 1 2v/1j 2, 1 2v/1h 2

63

63

Central (9)

Total number of degrees of freedom

Note that auxiliary nodes are located at j (or h) = 2(0·5 + 0·5(1/z3)) = 20·78867513 . . . along the edges; fn is normal rotation about element edge; u¯ , v¯ and w¯ are global translations along the x-, y- and z-directions respectively; u' and v' are in-plane displacements along and perpendicular to the element edge respectively.

. .   . . 

766

Figure 2. A spherical panel of rectangular projection.

component of displacement u, v and w. The local displacement vector {d'} consists of a set of 63 local degrees of freedom. These degrees of freedom are given in detail in Table 1, together with the global degrees of freedom which satisfy inter-element compatibility. For geometric definition, a two-tier procedure is adopted in the present formulation. First, the Cartesian co-ordinates of the nine primary nodes are given, based on the actual geometry. Second, an additional 12 points are derived, with two points each along the six curvilinear lines j = −1, 0, 1 and h = −1, 0, 1, by assuming the curvilinear lines to be circular. Each of these circular curvilinear lines is uniquely defined by three primary nodes which are located along the line at j (or h) = −1, 0, 1; and the two additional points are obtained at j (or h) = 20·5. The co-ordinates of an arbitrary point on the element middle surface are then obtained by using the 21 nodal co-ordinates as 21

[x

y

z]T = s Li (j, h)[xi yi zi ]T,

(4)

i=1

where xi , yi and zi are the Cartesian co-ordinates of node i, and Li (j, h) are the corresponding shape functions. The geometric shape functions are derived from B3 -spline functions in a similar way as that of the displacement shape functions. T 2 Natural frequencies (radians per second) of a simply supported spherical panel of square projection (E = 1, n = 0·3, r = 1, t = 0·0191, R = 1·91, a = b = 1) Mode no. 1 2 3 4 5 6 7 8 9 10 11 12

Present method ZXXXXXXXCXXXXXXXV 1×1 2×2 4×4 8×8

Blevins [21]

Shen and Wan [10]

Geannakakes and Wang [14]

0·52570 0·61830 0·62139 0·63283 — — — — — — — —

0·53585(1, 1) 0·59621(1, 2) 0·59621(2, 1) 0·69454(2, 2) 0·77430(3, 1) 0·77430(1, 3) 0·90779(3, 2) 0·90779(2, 3) 1·10208(4, 1) 1·10208(1, 4) 1·15259(3, 3) 1·25531(4, 2)

0·52835 0·59151 0·59253 0·69040 0·77070 0·77307 0·90372 0·90745 1·10284 — 1·15263 1·25854

0·53146 0·59114 0·59641 0·68980 0·76283 0·77390 0·89397 0·89940 1·09537 1·13240 — —

0·54073 0·59503 0·59642 0·68043 0·76270 0·76727 0·86156 0·86260 — — — —

0·53386 0·59180 0·59221 0·68591 0·76116 0·76381 0·88983 0·89015 1·07915 1·07951 1·12383 1·21955

0·53263 0·59041 0·59080 0·68486 0·76020 0·76260 0·89010 0·89025 1·07837 1·07871 1·12711 1·22187

Note that the results of Blevins [21] are calculated from equation (16) with the values of (i, j) given in parentheses.

     

767

3. STRAIN–DISPLACEMENT RELATIONS

The strain–displacement equations employed in the present formulation are those of Koiter [20], but with a proposed modification by the authors. The modified strain–displacement equations are ea =

gab = xb =

1 1u w + Ka v + , A 1a R11

eb =

1 1v w + Kb u + , B 1b R22

1 1v 1 1u 2w + − K a u + Kb v + , A 1a B 1b R12

1 1f2 1 + Kb f1 − f, B 1b R12 3

2xab =

xa =

(5a, b)

1 1f1 1 f, + Ka f2 + A 1a R12 3

0

(5c, d)

1

1 1 1f2 1 1f1 1 + − Ka f1 − Kb f2− − f, B 1b R11 R22 3 A 1a (5e, f)

in which ea and eb are the membrane strains in the a- and b-directions respectively, gab is the membrane shear strain, xa and xb are the changes of curvature of the middle surface respectively in the directions of a and b, and xab is the middle surface change of torsion. The in-plane displacements u and v are measured in the directions of a and b respectively; and w is the out-of-plane displacement in the z-direction. The curvatures (1/R11 ) and (1/R22 ) are respectively measured along a and b, and (1/R12 ) is the initial torsion of the middle surface. The terms Ka and Kb are, respectively, the curvatures of the projected curvilinear lines a and b on the tangent plane to the shell surface at the point considered, and are given by

0

1

1 1r 1 2r , · A 2B 1b 1a 2

Ka = −

0

1

1 1r 1 2r , · AB 2 1a 1b 2

Kb = −

(6a, b)

T 3 Natural frequencies (radians per second) of a simply supported spherical panel of rectangular projection (E = 1, n = 0·3, r = 1, t = 0·0191, R = 1·91, a = 0·6, b = 1·2) Mode no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Present method ZXXXXXXXXXXXCXXXXXXXXXXXV 1×1 2×2 4×4 8×8 0·54912 0·56755 — — — — — — — — — — — — —

0·56671 0·60260 0·70280 0·84923 0·88307 0·94449 — — — — — — — — —

0·55784 0·60709 0·72162 0·84600 0·92198 0·93788 1·09706 1·22058 1·31841 1·54118 1·58400 1·61411 1·62001 1·64657 1·81564

0·55597 0·60595 0·72184 0·84450 0·92686 0·93732 1·10097 1·22653 1·34203 1·54017 1·58790 1·61716 1·64849 1·66322 1·83022

Blevins [21] 0·55978(1, 1) 0·61201(1, 2) 0·73439(1, 3) 0·85303(2, 1) 0·94966(2, 2) 0·94966(1, 4) 1·12025(2, 3) 1·26251(1, 5) 1·37154(2, 4) 1·55645(3, 1) 1·66885(3, 2) 1·66885(1, 6) 1·70651(2, 5) 1·85797(3, 3) 2·12547(2, 6)

Note that the results of Blevins [21] are calculated from equation (16) with the values of (i, j) given in parentheses.

. .   . . 

768

Figure 3. The geometry of a hyperboloidal shell of revolution. Equation of meridian; (r/a)2 − [(z − d)/b]2 = 1, b = ad/zrb2 − a 2.

in which r is the position vector. The physical components of rotation occurring in equations (5d)–(5f) are defined as 1 1w u v + + , A 1a R11 R12

f1 = −

2f3 =

1 1w v u + + , B 1b R22 R12

f2 = −

1 1v 1 1u − − Ka u + Kb v. A 1a B 1b

(7a–c)

Note that A and B are coefficients, evaluated as A=

$0 1 0 1 0 1 % 1x 1a

2

+

1y 1a

2

+

1z 1a

2

1/2

,

B=

$0 1 0 1 0 1 % 1x 1b

2

+

1y 1b

2

+

1z 1b

2

1/2

.

(8a, b)

T 4 Natural frequencies (in Hz) of a fixed–free hyperboloidal shell Mode sequence (j, n) 5, 1 6, 1 4, 1 7, 1 6, 2 3, 1 5, 2 4, 2 8, 1 7, 2 9, 1 2, 1 8, 2 10, 1 7, 3

Present method ZXXXXXXXXCXXXXXXXXV 2×3 4×6 8 × 12 1·0570 1·1555 1·2219 1·3231 1·3115 1·3570 1·3802 1·4008 1·4952 1·5279 — — — — —

1·0335 1·1482 1·1847 1·3032 1·3222 1·3726 1·4265 1·4386 1·4830 1·5069 1·6547 1·7638 1·8076 1·8472 1·9161

1·0332 1·1478 1·1817 1·3028 1·3245 1·3747 1·4281 1·4451 1·4783 1·5128 1·6501 1·7652 1·8040 1·8391 1·9217

Eight axisymmetric elements [12] 1·0331 1·1478 1·1809 1·3027 1·3250 1·3755 1·4290 1·4474 1·4778 1·5140 1·6494 1·7657 1·8050 1·8384 1·9221

Note that the larger number of elements along the meridian; j = circumferential wavenumber; n = meridional mode number.

     

769

Figure 4. The geometry and mesh pattern on one quadrant of a hemispherical shell. R/t = 100, n = 0·3.

By following standard finite element manipulation, equations (5a)–(5f) may be written as {e} = [B]{d'},

(9)

where {e} = {ea

eb

gab

xa

xb

2xab }T,

(10)

{d'} is the local displacement vector of the element, and [B] is the usual strain–displacement matrix.

4. ELEMENT STIFFNESS AND MASS MATRICES

The local stiffness and mass matrices of the element are obtained as [K'] =

g

g g +1

[B]T[D][B] dA =

A

−1

+1

[B]T[D][B]=J= dj dh,

(11)

−1

T 5 Frequency parameters l = (r/E)1/2 Rv of a fixed hemispherical dome (R/t = 100, n = 0·3) Mode sequence (j, n) 1, 1 0, 1 1, 2 2, 1 0, 2 3, 1 1, 3 2, 2 4, 1 0, 3 5, 1 3, 2

Present method ZXXXXXXXXXXCXXXXXXXXXXV Three elements 12 elements 48 elements 0·5799 0·7789 0·8973 0·9026 0·9113 0·9054 0·9255 0·9203 0·9409 0·9553 0·9447 0·9628

0·5698 0·7639 0·8944 0·9017 0·9386 0·9481 0·9657 0·9667 0·9696 0·9838 0·9851 0·9893

0·5680 0·7616 0·8936 0·9013 0·9382 0·9478 0·9657 0·9665 0·9694 0·9839 0·9851 0·9896

Note that j = circumferential wavenumber; n = meridional mode number.

Luah and Fan [12] 0·568 0·762 0·894 0·901 0·938 0·948 0·966 0·966 0·969 0·984 0·985 0·990

. .   . . 

770

Figure 5. A cylindrically curved panel of trapezoidal projection.

[M'] =

g

g g +1

[N]T[N]rt dA =

−1

A

+1

[N]T[N]rt=J= dj dh,

(12)

−1

where A is the mid-surface area of the element, =J= is the determinant of the Jacobian matrix, [D] is the rigidity matrix for a thin shell element, r is the mass density, and t is the thickness of the element. In order to achieve inter-element compatibility in an arbitrary thin shell structure, the element global degrees of freedom as given in Table 1 must be employed. The corresponding global stiffness and mass matrices are then obtained as [K] = [T]T[K'][T],

[M] = [T]T[M'][T],

(13, 14)

where [T] is the transformation matrix. Further details of the present formulation have been given in reference [18]. T 6 Natural frequencies (in Hz) of a cylindrically curved trapezoidal fan blade Mode no. 1 2 3 4 5 6 7 8 9 10 11 12

Present method ZXXXXXXXXXXXCXXXXXXXXXXXV 1×1 2×2 4×4 8×8 161·80 293·61 446·47 583·09 791·59 — — — — — — —

188·70 306·34 463·56 526·63 757·41 907·55 936·07 1160·67 1290·82 1431·33 1512·26 1628·83

191·57 307·41 472·58 546·20 774·15 930·33 951·98 1185·49 1290·11 1466·46 1561·57 1695·85

191·72 307·64 473·57 547·71 777·73 937·55 956·61 1187·61 1297·20 1482·23 1577·44 1701·71

Yang and Wu [22] 194·3 293·8 471·1 534·8 773·3 923·6 938·5 1180·0 — — — —

     

771

5. NUMERICAL EXAMPLES

In this section, five numerical examples on the free vibration analysis of various thin shell structures are presented. These examples include spherical panels, shells of revolution and cylindrical panels, as well as single- and double-cell boxes. In order to demonstrate the accuracy and efficiency of the spline finite element, both coarse- and fine-mesh solutions are given in all cases. The results obtained are also compared with those available in the literature to validate the present method. It is worth pointing out that in the numerical examples chosen, shells of negative, zero and positive Gaussian curvatures are included. The effects of aspects ratios, distorted meshes and junctions of segments on the accuracy of solutions are also studied. 5.1.        The first problem considered is the vibration of simply supported spherical panel of rectangular projection, as shown in Figure 2. The boundary conditions imposed are: w = 0,

u' = 0,

(15)

where u' is in-plane displacement along the edges of the element. Since the spherical panel is a shallow shell, most of the results reported by the other authors are based on shallow shell theory [10, 14]. An approximate formula to calculate the natural frequencies is also available and is [21] v=

$ 0

j2 p4 i2 2+ 2 12 a b

1

2

E Et 2 + r(1 − n 2) rR 2

%

1/2

,

(16)

where v is the natural frequency in radians per second, i and j are respectively the mode numbers along a and b dimensions, E and n are Young’s modulus and the Poisson ratio respectively, and R is the radius of curvature of the panel. In the present study, two spherical panels of different aspect ratios were considered. The first panel analyzed has a square projection and the results of natural frequencies for the first 12 modes obtained by using various meshes are presented in Table 2. It can be seen that the fundamental mode can be accurately predicted by using only one element for the whole panel. The results obtained by the 2 × 2, 4 × 4 and 8 × 8 meshes are also seen to be in close agreement. For purposes of comparison, the results computed from equation (16), as well as those reported by the other authors are quoted in Table 2. Fair agreement can be observed among the various methods. However, the present method is believed to be the most accurate as convergence of the results is observed, and it is based on elaborate deep shell theory. In order to examine the effect of aspect ratio on the performance of the element, a second panel of rectangular projection with an aspect ratio of 1:2 was analyzed. The results obtained by using various meshes are given in Table 3, together with those computed from equation (16). As equal number of elements are used in the two dimensions, the individual element has the same aspect ratio as that of the panel. It can be observed that despite of the 1:2 aspect ratio of the elements, the results are rapidly converging. It is also noticed that the natural frequencies computed from equation (16) are higher than those of the present method, and are unreliable for the higher modes of vibration. 5.2. –   The natural frequencies of hyperboloidal shells of revolution have been investigated by many researchers using various methods. The geometry of the shell is as shown in Figure 3, and the following geometric and material properties are assumed: a = 25·603 m,

772

. .   . . 

Figure 6. Mode shapes of vibration of a cylindrically curved panel of trapezoidal projection.

b = 63·906 m, d = 82·194 m, L = 100·787 m, t = 127 mm, E = 2·609 × 104 MN/m2, n = 0·15 and r = 2405 kg/m3. In the present analysis, only one quadrant of the shell was modelled. This necessitates separate analyses for the vibration modes with odd and even number of circumferential waves by the proper imposition of symmetric or antisymmetric boundary conditions along the lines of symmetry. The natural frequencies obtained for the lowest 15 modes of vibration are tabulated in Table 4. Comparisons are made in the table

     

773

Figure 7. A cylindrically curved panel of rectangular projection: (a) geometry; (b) distorted mesh pattern.

with the results obtained from an axisymmetric element also developed by the authors [12]. Excellent agreement of the results can be observed. It is also encouraging to see that the 4 × 6 mesh, which is relatively coarse, is able to produce accurate results. 5.3.    This example of fixed hemispherical shell was chosen for two reasons. First, this is the only deep shell problem considered and, second, the mesh employed in the analysis is an irregular one in which all the elements are skew, so that the effect of mesh distortion on the performance can be studied (to a certain extent). The geometry and mesh pattern on one quadrant of the hemisphere is depicted in Figure 4. Since only a quarter of the shell was modelled, separate analyses for the vibration modes with odd and even number of circumferential waves were required. The results of analyses are given in terms of a frequency parameter, defined as l = (r/E)1/2Rv,

(17)

in which R is the radius of the shell. The frequency parameters obtained for the lowest T 7 Natural frequencies (in Hz) of a cantilevered cylindrically curved panel of rectangular planform Mode no. 1 2 3 4 5 6 7 8 9 10 11 12

Uniform mesh ZXXXXXXXCXXXXXXXV 1×1 2×2 4×4 8×8 78·52 126·40 240·11 330·08 367·63 — — — — — — —

84·31 137·37 246·42 328·11 375·65 527·66 687·18 694·94 764·38 797·11 963·08 1144·79

85·76 138·35 246·84 340·56 384·68 528·98 719·77 723·62 769·59 801·77 999·62 1200·49

85·89 138·41 247·00 342·25 386·41 529·05 726·35 728·37 771·80 802·59 997·96 1206·49

Distorted mesh ZXXXXXCXXXXXV Four elements 16 elements 83·05 136·69 246·49 327·96 376·91 505·03 687·91 704·04 766·75 803·32 941·55 1100·04

84·74 137·77 245·95 341·69 384·84 524·57 719·33 722·57 768·10 799·72 994·91 1198·96

SemiLoof [23] 85·3 138·1 245·1 340·4 383·8 — — — — — — —

. .   . . 

774

Figure 8. The geometry of cantilevered shell boxes: (a) single-cell box; (b) double-cell box.

12 modes of vibration are given in Table 5, together with those of Luah and Fan [12] obtained by using an axisymmetric element. Excellent agreement of the results can be observed. Since, in this particular case, all the elements employed are skew, the accurate results obtained also prove that mesh distortion does not severely affect the accuracy of present element. 5.4.     In this example, two cylindrically curved cantilevered panels are presented. The first panel discussed is a fan blade with a trapezoidal planform, as shown in Figure 5. The trapezoidal planform tapers linearly from a circumferential width of 12 in at the base of the blade to a width of 6 in at the tip. The circumferential radius is 12 in and it is constant along the meridional direction. The blade has a uniform thickness of 0·125 in and the material properties are: E = 30 × 106 psi, r = 0·284 lb/in3 and n = 0·3. The natural frequencies for the lowest 12 modes of vibration are presented in Table 6. The results obtained by Yang and Wu [22] using a 48-degree-of-freedom skew quadrilateral thin shell finite element are also quoted for comparison. Good agreement among the results is observed. The mode shapes for the first 12 modes of vibration obtained by the present method with the 4 × 4 mesh are plotted in Figure 6. The second cylindrical panel discussed is of rectangular planform as shown in Figure 7(a). This structure has previously been studied by many investigators. The circumferential T 8 Frequency parameters l of a cantilevered single-cell box (B1 /L = 0·33, B2 /L = 0·25, t/L = 0·005, n = 0·32, l 2 = rL 2(1 − n 2)v 2/E) Present method ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV Symmetric Six elements 16 elements 64 elements mode no. (two axial elements) (two axial elements) (four axial elements) Irie et al. [24] 1 2 3 4 5 6 7 8 9 10 11 12

0·0432 0·0553 0·0722 0·0798 0·0997 0·1219 0·1277 0·1311 0·1674 0·2041 0·2074 0·2406

0·0434 0·0564 0·0727 0·0808 0·1053 0·1262 0·1290 0·1332 0·1742 0·2131 0·2158 0·2530

0·0434 0·0565 0·0728 0·0811 0·1058 0·1271 0·1292 0·1336 0·1754 0·2134 0·2161 0·2537

0·0437 0·0569 0·0736 0·0816 0·106 0·128 0·130 0·135 0·177 — — —

     

775

T 9 Natural

frequencies (radians per second) of a cantilevered (E = 30 × 106, n = 0·3, r = 7·35 × 10−4, t = 0·5)

Mode no. 1 2 3 4 5 6 7 8 9 10 11 12

double-cell

box

Present method ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV 14 elements 28 elements 56 elements (two axial elements) (two axial elements) (four axial elements) 126·76 157·07 176·16 189·38 196·18 212·61 215·16 228·87 246·19 251·09 251·18 264·95

126·51 156·58 175·61 190·70 195·72 214·05 215·07 230·55 248·31 253·39 253·49 267·81

127·42 159·81 180·88 192·42 203·20 217·76 225·73 235·61 254·74 274·42 274·53 275·65

width and the length of the panel are both 12 in and the thickness is 0·12 in. The circumferential radius remained at 24 in along the meridional direction. The material properties are E = 30 × 106 lb/in2, n = 0·3 and r = 7·35 × 10−4 lb s2/in4. Two mesh patterns, one with uniform meshes and the other with distorted meshes (Figure 7(b)) are considered. The natural frequencies obtained by using various meshes with the two mesh patterns as well as those reported by Martins and Owen [23], who used a semiLoof element, are presented in Table 7. The results clearly indicate the accuracy and efficiency of the present element. In particular, the good performance of the element with distorted meshes enhances greatly the versatility of the method. 5.5.    A single-cell and a double-cell cantilevered box are analyzed in this example. The geometries and dimensions of the structures are depicted in Figure 8. This example provides a check on the performance of the element in situations of junctions and corners of shells. The frequency parameters obtained for the first 12 symmetric modes of vibration (symmetrical about a vertical plane) of the single-cell box are given in Table 8 together with those reported by Irie et al. [24]. Since only the symmetric modes are considered, the analysis is carried out with half of the structure. Good agreement among the various schemes of solution can be observed. For the double-cell cantilevered box, three meshes with 14, 28 and 56 elements for the whole structure were used in the analysis. The natural frequencies for the lowest 12 modes of vibration are presented in Table 9. No published result is available for comparison. However, the good agreement of the coarse- and fine-mesh solutions of the present method indicates a fast rate of convergence of the spline finite element.

6. CONCLUSIONS

The free vibration analysis of arbitrary thin shell structures by using a newly developed spline finite element is presented. Five numerical examples, including spherical panels, cylindrical panels and shells of revolution, as well as single- and double-cell boxes, are

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analyzed. The effects of aspect ratios and distorted meshes on the performance of the element are studied. Shells of negative, zero and positive Gaussian curvatures are also included in the analysis. From the theoretical considerations and the numerical results obtained, it can be concluded that the spline finite element employed in this paper is a versatile, reliable, accurate and efficient thin shell element suitable for the analysis of arbitrary thin shell structures. REFERENCES 1. A. W. L 1973 Vibration of Shells NASA SP-288. Washington D.C.: U.S. Government Printing Office. 2. W. S 1981 Vibrations of Shells and Plates. New York: Marcel Dekker. 3. D. B 1984 Computers and Structures 18, 471–536. Computerized analysis of shells— governing equations. 4. D. G. A and A. S 1973 International Journal of Mechanical Sciences 4, 37–47. A new cylindrical shell finite element based on simple independent strain functions. 5. G. R. H and L. W 1992 Journal of Sound and Vibration 152, 263–283. Finite element analysis of free–free shells of revolution. 6. N. G and K. R. S 1990 Computers and Structures 34, 669–677. Free vibration of cantilever circular cylindrical shells with variable thickness. 7. G. Y, T. I and Y. T 1984 Journal of Sound and Vibration 95, 117–126. Free vibration of non-circular cylindrical shells with variable circumferential profile. 8. Y. N and A. L 1986 Journal of Applied Mechanics 53, 647–651. Vibrations of completely free shallow shells of curvilinear planform. 9. C. W. S. T and B. W 1991 Computers and Structures 40, 555–568. An axisymmetric thin shell finite element for vibration analysis. 10. P.-C. S and J.-G. W 1987 Computers and Structures 25, 1–10. Vibration analysis of flab shells by using B-spline functions. 11. W. Y. L, L. G. T, Y. K. C and S. C. F 1990 Journal of Sound and Vibration 140, 39–53. Free vibration analysis of doubly curved shells by spline finite strip method. 12. M. H. L and S. C. F 1989 Computers and Structures 33, 1153–1162. General free vibration analysis of shells of revolution using the spline finite element method. 13. T. M 1988 International Journal for Numerical Methods in Engineering 26, 663–676. Application of spline strip method to analyse vibration of open cylindrical shells. 14. G. N. G and P. C. W 1991 Computers and Structures 39, 489–492. Vibration analysis of arbitrarily-shaped shell panels using B3 -spline finite strips. 15. Y. K. C, W. Y. L and L. G. T 1989 Journal of Sound and Vibration 128, 411–422. Free vibration analysis of single curved shell by spline finite strip method. 16. S. C. F and M. H. L 1989 Journal of Sound and Vibration 132, 61–72. Spine finite element for axisymmetric free vibrations of shells of revolution. 17. G. J. H and S. C. W. L 1986 Thin-walled Structures 4, 269–294. Buckling of thin flat-walled structures by a spline finite strip method. 18. M. H. L and S. C. F (to appear) America Society of Civil Engineers, Journal of Engineering Mechanics. Spline finite element for curved thin shells of arbitrary geometry. 19. S. C. F and M. H. L 1992 American Society of Civil Engineers, Journal of Engineering Mechanics 118, 1065–1082. New spline finite element for plate bending. 20. W. T. K 1960 Proceedings of Symposium on Theory of Thin Elastic Shells (W. T. Koiter, editor), 12. Amsterdam, North-Holland. A consistent first approximation in the general theory of thin elastic shells. 21. R. D. B 1979 Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand Reinhold. 22. H. T. Y. Y and Y. C. W 1989 International Journal for Numerical Methods in Engineering 28, 2855–2875. A geometrically non-linear tensorial formulation of a skewed quadrilateral thin shell finite element. 23. R. A. F. M and D. R. J. O 1977 International Journal for Numerical Methods in Engineering 11, 481–498. Structural instability and natural vibration analysis of thin arbitrary shells by use of the semiLoof element. 24. T. I, G. Y and Y. K 1985 Journal of Sound and Vibration 102, 501–513. Free vibration of an oblique rectangular prismatic shell. (Errata in ibid., 107(2), p. 359.)