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A displacement variational method for free vibration analysis of thin walled members
Journal of Sound and Vibration (1995) 181(3), 503–513
A DISPLACEMENT VARIATIONAL METHOD FOR FREE VIBRATION ANALYSIS OF THIN WALLED MEMBERS W. Y. L W. K. H Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hong Kong (Received 24 June 1993, and in final form 17 January 1994) By introducing an important modification into Vlasov theory [1], a general displacement variational method is proposed for free vibration analysis of thin walled members to take into account the shear effect. The differential equations obtained are solved by the Galerkin method. The method can be applied to arbitrary cross sections including open and closed section thin walled structures and tube-in-tube structures. The results compare well with those of other authors and of a general finite element program COSMOS [2] in which the rigid section assumption and shear effect have also been considered. The method requires less computer time than the standard finite element method and even the finite strip method.
1. INTRODUCTION
Thin walled members have been widely applied in engineering structures: e.g., box girder bridges, and shear wall and core wall structures in tall buildings. These structures have to resist dynamic loads such as wind, traffic and earthquake loadings, so that the understanding of the dynamic behavior of the structures becomes increasingly important. Efforts have been made by many authors to include the shear effect and the non-linear distribution of warping displacement in the analysis of thin walled structures, since they become more and more important nowadays. Timoshenko [3] first provided a beam model accounting for the effect of shear strain. Timoshenko [3], Gere [4] and Garland [5] analyzed the coupled vibration of an open section beam by considering the transverse motion of the centroid and rotation about the shear center. Vlasov [1] considered the warping effect and introduced rotational and warping inertia terms in flexural torsional vibration analysis. However, none of the models mentioned, due to their somewhat inaccurate assumptions, can take account of the shear lag phenomenon, which has been recently noted as an important factor in thin walled structure analysis. The Vlasov theory is a systematic theory for thin walled member analysis. In many cases, when the shear lag effect is small, the theory gives reasonable results. His theory has two basic assumptions: i.e., (i) the rigid section assumption and (ii) the assumption that the shear deformation along the central line of the cross-section is zero (in an open cross-section) or constant (in a closed cross-section). Many previous works have shown that Vlasov’s second assumption, which makes the theory incapable of accounting for the shear lag effect, is inaccurate. Abandoning this second assumption, Koo and Cheung [6] proposed a mixed variational principle, and Li and Wu [7] proposed a general displacement variational method for the analysis of static problems. In these two methods, the shear lag effect has successfully accounted for. In references [8] and [9] a model for vibration analysis of thin walled beams 503 0022–460X/95/130503 + 11 $08.00/0
7 1995 Academic Press Limited
504
. . . .
was proposed which also takes account of the shear lag effect. However this method requires one to pre-define the appropriate warping function and it seems difficult to apply to complex section profiles. In the present paper a general method for free vibration analysis of thin walled members is proposed, based on reference [7]. For generality, a number of kinds of interpolation function are tried, and finally two kinds of function are proposed: i.e., the local linear function and the B3 spline function. They are used to interpolate the longitudinal displacement along the cross-section and along the longitudinal axis, respectively. Therefore, in the use of this method, human judgement is not required. In this sense, the method is a systematic one for free vibration analysis which can be applied to thin walled members of arbitrarily shaped cross-section. In tube-in-tube structures, the inner tube is connected by the floor with the outer tube so that the rigid section assumption can also be applied and this has been widely used in tall building analysis. Therefore tube-in-tube structures can also be analyzed by the present method. Four numerical examples are presented to demonstrate the convergence, accuracy and versatility of the proposed method. Open, closed, closed with outer flanges and tube-in-tube sections are included in the examples. The results compare well with those of references [8, 9] and of a general finite element package COSMOS [2].
2. VARIATIONAL PRINCIPLE FOR FREE VIBRATION OF THIN WALLED MEMBERS
An arbitrary cross-section prismatic thin walled member is shown in Figure 1, in which the z-axis is parallel to the member axis (a list of notation is given in the Appendix). The principle is based on the following two assumptions. (i) The cross-section can be regarded as rigid in its own plane. (ii) sn and ss (the stresses along the normal direction n and the tangential direction s, respectively) are much smaller than the longitudinal stress sz , such that, by Hooke’s law ez = (sz /E) − n{(ss /E) + (sn /E)} 2 sz /E.
Figure 1. (a) The thin walled member geometry; (b) thin walled member cross-section geometry.
(1)
-
505
The variational principle for free vibration of an elastic member can be expressed by the functional [10] v 2 = st(P/T)
(2)
where P is the amplitude of the possible strain energy induced in vibration, T is the amplitude of possible kinetic energy in vibration, and st means that the value of v 2 should take the stationary value of P/T. Hence d(P) − v 2d(T) = 0.
(3)
These two energy terms can be expressed by four independent displacement variables, w, u, vt and vn as
g $g H
P=
( 12 Eez2 t + 12 Ggzs2 t) ds + 12 GJd
Ss
0
g $g 0 1
1w t ds + 1z
H
=
1 2
E
Ss
0
2
g 0 1 2
G
Ss
0 1% 2
du ds
1
dz
0 1%
1w 1vt du + t ds + 12 GJd 1s 1z dz 2
2
dz
(4)
(in equation (4), Jd is equal to zero in the case of a closed cross-section),
g $g H
T = 12
re
tw 2 ds +
Ss
0
g
tvt2 ds +
Ss
g
%
tvn2 ds dz.
Ss
(5)
By taking the first variation of the functionals in equation (3) with respect to the variables w, u, vt and vn and using the standard procedures of the calculus of variations (such as integration by parts), the following governing differential equations and natural boundary conditions can be obtained: Et 1 2w/1z 2 + G(1/1s)[t(1w/1s + 1vt /1z)] + v 2retw = 0,
(6)
G(1 w/1s 1z + 1 vt /1z ) cos a + v re (vt cos a − vn sin a) = 0,
(7)
G(1 w/1s 1z + 1 vt /1z ) sin a + v re (vt sin a + vn cos a) = 0,
(8)
g0
(9)
2
2
2
G
Ss
2
2
2
2
2
1
1 2w 1 2v d2u + 2t rt ds + GJd 2 + v 2 1s 1z 1z dz
g
re t(vt r + vn l) ds = 0,
Ss
G(1w/1s + 1vt /1z)n,e tne xne = 0.
(10)
At the two ends z = 0 and z = H,
g
Ss
g 0 G
Ss
E
1w tds = 0, 1z
g 0 G
Ss
1
1w 1vt + sin a tds = 0, 1s 1z
1
1w 1vt + cos a tds = 0, 1s 1z
g 0 G
Ss
1
1w 1vt du + r tds + GJd = 0. 1s 1z dz
(11, 12)
(13, 14)
3. METHOD OF ANALYSIS
The Galerkin method is adopted in the analysis. For convenience, instead of vt and vn , vcx and vcy are used in the present analysis. The relationships between them in matrix form are vt = {h1 (s)}T{vc (z)},
vn = {h2 (s)}T{vc (z)},
(15a, b)
. . . .
506 where
{h1 (s)}T = [cos a(s), sin a(s), r(s)], {h2 (s)}T = [−sin a(s), cos a(s), l(s)],
(15c)
{vc (z)}T = [vcx (z), vcy (z), u(z)].
(15d, e)
In the Galerkin method, the basic displacements are expressed approximately as series of selected functions: i.e., w = {f(s)}T{w(z)},
vcx (z) = {y(z)}T{b1 },
vcy (z) = {y(z)}T{b2 },
u(z) = {y(z)}T{b3 },
(16–19)
where {f(s)}T = [f1 (s), f2 (s), . . . , fn (s)],
{w(z)}T = [w1 (z), w2 (z), . . . , wn (z)],
{y(z)}T = [y1 (z), y2 (z), . . . , ym (z)], {bj }T = [bj1 , bj2 , . . . , bjm ] wk (z) = {g(z)}T{ak }
(j = 1, 2, 3),
(k = 1, 2, . . . , n),
{g(z)} = [g1 (z), g2 (z), . . . , gm (z)],
{ak }T = [ak1 , ak2 , . . . , akm ].
T
In short form,
6 7 $
{w} [N1] 0 = {vc } 0 [N2]
%6 7
{a} , {b}
(20)
where {y(z)}T 0 K L G G G {y(z)}T G , G T G k 0 {y(z)} l3 × 3
T 0 K {g(z)} L T G G , {g(z)} [N1] = G G . . G G . T l k 0 {g(z)} n×h
(21, 22) {a} = [{a1 }T, {a2 }T, . . . , {an }T]T,
{b} = [{b1 }T, {b2 }T, {b3 }T]T,
(23, 24)
The functions fi (s) (i = 1, . . . , n), yp (z) and gp (z) ( p = 1, . . . , m) can be any functions series satisfying the displacement boundary conditions [10]. By using the equations (15a–e) and (16), the equations of the problem can be rewritten in matrix form as follows: The governing equations are
$
0 G[D']
E[A] 0
$
+ v 2re
[A] 0
%6 7 $
w0 0 −G[C] + v0c G[C]T 0
0 [D0]
%6 7 $
w' −G[B] + v'c 0
0 0
%6 7
w = 0. vc
%6 7 w vc
(25)
The natural boundary conditions at the two ends z = 0 and z = H are
$
E[A] 0 0 G[D']
%6 7 $
w' 0 + v'c G[C]T
%6 7
0 0
w = 0, vc
(26)
-
507
where [A] =
g
{f}{f}Tt ds,
[B] =
Ss
g
d{f} d{f}T t ds, ds ds
Ss
[C] =
g
Ss
d{f} {h1 }Tt ds, ds (27a–c)
[D1 ] =
g
{h1 }{h1 }Tt ds,
[D2 ] =
Ss
g
{h2 }{h2 }Tt ds,
[D'] = [D] + Jd
Ss
$
0
0
1
%
,
(27d–g) [D0] = [D1 ] + [D2 ]. (The primes denote differentiation with respect to z.) These equations show that the longitudinal (w) and lateral (vc ) vibrations are coupled. Often used boundary conditions, at z = 0 or z = H are as follows: for a fixed end {w(z)} = {0}, {vc } = {0}; for a simply supported end {w'(z)} = {0}, {vc } = {0}; for a free end {w'(z)} = {0}, [D']{v'c } + [C]T{w} = {0}. The Galerkin residual equation of the problem can be derived as
g$ H
0
+
[N1] 0
0 [N2]
$
−G[B] 0 0 0
% 6$ T
E[A] 0
%6 7
0 G[D']
$
w [A] + v 2re vc 0
%6 7 $
w0 0 + v0c G[C]T
0 [D0]
%6 77 w vc
−G[C] 0
%6 7 w' v'c
dz = 0.
(28)
By using the boundary conditions and equation (20), the eigenproblem equation can be finally obtained as
$
[K11 ] [K21 ]
[K12 ] [K22 ]
%6 7 $
{a} [M11 ] − v2 {b} 0
%6 7
0 [M22 ]
{a} = 0, {b}
(29)
or, in short form, [K]{c} − v 2[M]{c} = 0,
(29a)
where
g$ % $ % H
[K11 ] = E
0
[A]
dN1 dz + G dz
g$ % $ % g H
[K22 ] = G
T
dN1 dz
0
dN2 dz
T
dN2 dz, dz
[D']
g
H
[N1]T[B][N1] dz,
[K12 ] = G
g
H
[N1]T[C]
0
H
[K21 ] = [K12 ]T,
[M11 ] = re
[N1]T[A][N1] dz,
0
(30a)
0
[M22 ] = re
g
$ %
dN2 dz, dz
(30b, c)
H
[N2]T[D0][N2] dz.
0
(30d–f) The stiffness matrix [K] and the mass matrix [M] are both symmetric and positive definite. Equation (29) shows that the problem is a typical eigenproblem and can be solved by standard eigenvalue solvers. For the results presented in this paper, an eigenvalue solver based on generalized Jacobi method [11] was used. The eigenproblem can be simplified for cross-sections with one or two axes of symmetry. In the cases of two axes of symmetry, the two directions of the bending vibration modes and torsional modes are uncoupled, while in the cases of a single axis of symmetry, only the bending modes along that axis are separated. In cases of two axes of symmetry, the matrices [A], [B], [C], [D1 ] and [D2 ] (see equations (27a–e)) are equal to four times the
508
. . . .
integration along one-quarter of the cross-section, so that only one-quarter of the original cross-section needs to be calculated, saving computational effort. Similar conditions also occur in the case of cross-sections with a single axis of symmetry. 4. SELECTION OF DISPLACEMENT FUNCTION
From equations (16)–(19), one can see that only the functions fi (s) and yi (s) need to be used in the analysis. Combinations of these two sets of functions can represent arbitrary longitudinal warping displacements, lateral deflections and twist angles. The shear lag phenomenon can then be worked out in the analysis. There are many kinds of functions which are suitable for the analysis. The function series suggested in reference [7] and adopted in the present analysis as fi (s) is the local linear function, because of its simplicity and effectiveness. For the function yi (z), trigonometric, spline and polynomial functions have been tried. Note that they are required to satisfy the displacement boundary conditions only. Finally, the B3 spline function [12] was selected. 5. NUMERICAL EXAMPLES
Four cantilever thin walled members of different cross-sections have been studied. Static analysis [6, 7] shows that the shear lag effect is important only when the H/B ratio (height to breadth ratio of the thin walled member) is small; in these cases, the results differ significantly from those of classical theory. Therefore, in order to demonstrate the shear lag effect, all the examples to be presented have a small H/B ratio. In general, the results
Figure 2. Thin walled cross-sections for the examples. Dimensions in m. (a) Example 1, H = 15 m, t = 0·4 m, E = 3 × 1010 N/m2, E/G = 2·3, re = 2500 kg/m3; (b) Example 2, H, E, E/G, re as (a) t = 0.2 m; (c) Example 3, H, t, E, E/G, re as (b); (d) Example 4, H = 20 m, t = 0·2 m, E = 2·6 N/m2, E/G = 2·6, re = 2400 kg/m3.
-
509
T 1 Convergence study of the rectangular box section Mode order 1 2 3 4 5 6 7 8 9 10
Circular frequencies ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV 4 × 8† 6×8 12 × 6 12 × 8 12 × 12 12 × 16 63·25 100·9 186·9 261·0 362·8 369·2 552·6 562·2 765·6 821·4
61·49 100·9 186·9 246·3 362·8 369·2 520·3 562·2 765·6 767·9
61·06 100·1 186·8 242·8 362·8 363·6 512·3 561·6 752·3 755·9
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 762·2 754·6
61·06 100·1 186·8 242·8 362·8 363·6 512·3 561·6 752·4 755·4
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 752·2 755·4
† n × m denotes n segments along the cross-section and m segments along the longitudinal direction.
compare well with those of other similar theories [8, 9] and of a general finite element package COSMOS [2]. In all of these methods, the rigid section assumption is made and the shear effect is taken into account. Example 1. A rectangular box section shown in Figure 2(a) was investigated. The data from the convergence study are listed in Table 1. The first 11 frequencies as given by various methods are listed in Table 2. For investigating the influence of the rigid section assumption and the number of degrees of freedom (DOF’s) required for acceptable accuracy of a general finite element package, some results obtained by using COMSOS are listed in Table 3. It can be observed that (i) for the first two frequencies, the rigid section assumption has very little influence, but great influence on the high frequencies, and (ii) the number of DOF’s required in using COSMOS is much greater than for the present method. Example 2. An open I section shown in Figure 2(b), which was originally studied by Capuani et al. [8], was studied. The first 14 frequencies as obtained by various methods T 2 Results for Example 1 Mode order
Present theory
1 2 3 4 5 6 7 8 9 10 11
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 752·2 754·6 930·0
Circular frequencies ZXXXXXCXXXXXV COSMOS [2] Ref [9] 61·73 100·8 188·8 245·3 366·8 363·9 516·1 566·7 749·2 761·1 932·0
61·1 189·0 244·1 515·6 568·9 760·3 928·6
Vibration mode shape† BY BX T BY A BX BY T BX BY BY
† BX, vibration in x direction; BY, vibration in y direction; T, torsional vibration; A, longitudinal vibration.
. . . .
510
T 3 Results of the study of the rigid section assumption and COSMOS finite element package number of DOF’s required for a box shaped thin walled member
Mode order 1 2 3 4 5 6 7 8 9 10 11
Circular frequencies ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV With RSA† Without RSA† ZXXXXXXXCXXXXXXXV ZXXXXCXXXXV DOF’s DOF’s Mode DOF’s Mode 1080 1950 shape 1080 shape† 61·79 100·9 189·1 247·2 366·2 366·9 524·4 569·9 757·7 774·7 941·5
61·73 100·8 188·8 245·3 366·8 363·9 516·1 566·7 749·2 761·1 932·0
BY BX T BY A BX BY T BX BY BY
59·86 99·81 115·5 163·7 184·1 196·5 226·2 262·9 268·0 270·3 290·4
BY BX T BY BY BY T BY T BY BY
† the rigid section assumption. ‡ see Table 2 for key.
are tabulated in Table 4. Because of the shear lag effect is not taken into account in the Vlasov theory, large differences among the corresponding predicted frequencies can be found in Table 4. When the height of this member is increased to 90 m (H/B = 15), the shear lag effect will no longer be important. This case has also been investigated, and no difference can be observed between the first ten frequencies obtained from the Vlasov theory and the ones obtained from the present method.
T 4 Results for Example 2
Mode order 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Circular frequencies ZXXXXXXXXXXXCXXXXXXXXXXXV Present theory COSMOS [1] Ref [8] Vlasov 57·94 59·65 76·40 215·3 254·4 336·9 362·8 438·2 551·7 630·0 731·3 769·8 853·5 1088·0
† Table 2 for key.
57·77 60·12 76·83 216·5 251·5 333·7 366·9 437·5 540·9 626·4 719·0 766·2 827·3 1097·0
57·7
240·4
540·5
832·7
62·17 74·16 81·86 350·7 365·0 410·6 888·8 1029·0
Vibration mode shape† T BY BX BY T BX A BY T BY BX BY T A
-
511
T 5 Results for Example 3
Mode order
Circular frequencies ZXXXXXXXXCXXXXXXXXV Present theory COSMOS [2] Ref [8]
1 2 3 4 5 6 7 8 9 10
52·00 84·91 127·3 206·1 341·3 362·8 382·2 426·5 622·6 723·1
52·31 85·25 127·7 204·1 337·3 366·8 377·5 418·8 593·3 710·1
47·3 130·4 170·8 294·0 421·7 481·9
Vibration mode shape† BY, T BX BY, T BY, T BX A BY, T BY, T BY, T BX
† See Table 2 for key.
Example 3. A cross-section composed of a rectangular box section with two outer flanges, as shown in Figure 2(c), was studied by Capuani et al. [8], and was studied again by the present method and by COSMOS [2]. The first ten frequencies are listed in Table 5. The COSMOS and present method results are close to each other and larger differences can be noted between these and Capuani’s results. The reason for this may be that Capuani’s method depends significantly on the functions selected; therefore the results may be worse for some irregular cross-sections. Example 4. A tube-in-tube section as shown in Figure 2(d), which is widely used in tall buildings, was studied. The first 17 frequencies are compared with those from the finite element package COSMOS in Table 6. It is of interest to note that the frequencies of pure longitudinal vibration computed by the present method are identical to those of classical theory, as expressed by the formula fi = z(E/re )(2i − 1)/4H, where i is the mode number. T 6 Results for Example 4
Mode order 1&2 3 4&5 6 7 8&9 10 11 & 12 13 14 15 16 & 17
Frequencies (×105) ZXXXXXXCXXXXXXV Present theory COSMOS [2] 5·976 22·10 27·28 41·14 41·14 60·05 66·29 94·16 110·5 123·4 123·4 127·0
† See Table 2 for key.
6·219 22·15 27·84 43·11 43·11 60·71 66·19 93·50 109·5 128·7 128·7 125·3
Vibration mode shape† BX BY T BX BY A (int. tube) A (ext. tube) BX BY T BX BY T A (ext. tube) A (int. tube) BX BY
512
. . . . 6. CONCLUSIONS
The present study has shown that the proposed method is applicable to thin walled members of any cross-section shape. The numerical examples have fully demonstrated the accuracy, versatility and fast convergence of the method. The method is simple and general, and requires much less computational effort and time than the standard finite element method. It can be further improved by choosing more suitable shape functions along the profile of the section, and should be very useful in practical design, especially at its preliminary stage. REFERENCES 1. V. Z. V 1961 Thin-walled Elastic Beams (English translation, National Science Foundation, Washington, D.C.). London: Oldbourne Press. 2. M. L Cosmos/M User Guide. Structural Research and Analysis Corporation. 3. S. T, D. H. Y and W. W J. 1974 Vibration Problems in engineering. New York: John Wiley. 4. J. M. G and Y. K. L 1958 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 373–378. Coupled vibrations of thin-walled beams of open cross section. 5. C. F. G 1940 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics A-97, A-105. The normal modes of vibrations of beams having non-collinear elastic and mass axes. 6. K. K. K and Y. K. C 1989 Journal of Engineering Mechanics 115, 2271–2286. Mixed variational formulation for thin-walled beams with shear lag. 7. W. Y. L and X. S. W 1991 Proceedings of the Asian Pacific Conference on Computational Mechanics, Hong Kong. A semi-discrete method for shear lag analysis of thin-walled members. 8. D. C, M. S and F. L 1992 Earthquake Engineering and Structural Dynamics 21, 859–879. A generalization of the Timoshenko beam model for coupled vibration analysis of thin-walled beams. 9. F. L and M. S 1991 Thin-walled Structures 11, 375–407. The shear strain influence on the dynamics of thin-walled beams. 10. H. C. H 1989 Variational Principle in Elasticity and Application. Beijing: Scientific Publishing House. (In Chinese). 11. Z. H. C 1983 Matrix eigenvalue Problem. Shangai Science and Technology Publishing House. (In Chinese). 12. W. Y. L, Y. K. C and L. G. T 1986 Journal of Engineering Mechanics 112(1), 43–54. Spline finite strip analysis of general plates. APPENDIX: NOTATION s c Ss l r a t H Jd E G re xne
curvilinear co-ordinate along the contour of the cross-section shear center of the cross-section the whole cross-section distance from s to r , positive along positive direction of s, r being a vector from the shear center c perpendicular to the tangent at point s length of r s , distance from c to the tangent at point s, positive when r × ds is along the positive z-axis direction angle between the x-axis and the tangent at s, positive in anticlockwise direction =t(s), thickness of wall at s height (length) of the member Saint-Venant’s torsional constant (for open section only) Young’s modulus shear modulus density of material +1 when positive direction of s in the wall (e) connected at node n points away from the node n; −1 when positive s in the wall (e) points toward the node n; 0 when the wall (e) does not join at n
- nz v w(z, s) vt vn vcx , vcy u
513
+1 for z = H and −1 for z = 0 angular frequency of vibration longitudinal displacement displacement of point s along the tangent direction; positive when along positive direction of s displacement of point s along the normal direction, positive when v t × v n is along positive z-axis direction x and y components of shear center displacement vc twist angle of the cross-section, positive in anticlockwise direction
A DISPLACEMENT VARIATIONAL METHOD FOR FREE VIBRATION ANALYSIS OF THIN WALLED MEMBERS W. Y. L W. K. H Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hong Kong (Received 24 June 1993, and in final form 17 January 1994) By introducing an important modification into Vlasov theory [1], a general displacement variational method is proposed for free vibration analysis of thin walled members to take into account the shear effect. The differential equations obtained are solved by the Galerkin method. The method can be applied to arbitrary cross sections including open and closed section thin walled structures and tube-in-tube structures. The results compare well with those of other authors and of a general finite element program COSMOS [2] in which the rigid section assumption and shear effect have also been considered. The method requires less computer time than the standard finite element method and even the finite strip method.
1. INTRODUCTION
Thin walled members have been widely applied in engineering structures: e.g., box girder bridges, and shear wall and core wall structures in tall buildings. These structures have to resist dynamic loads such as wind, traffic and earthquake loadings, so that the understanding of the dynamic behavior of the structures becomes increasingly important. Efforts have been made by many authors to include the shear effect and the non-linear distribution of warping displacement in the analysis of thin walled structures, since they become more and more important nowadays. Timoshenko [3] first provided a beam model accounting for the effect of shear strain. Timoshenko [3], Gere [4] and Garland [5] analyzed the coupled vibration of an open section beam by considering the transverse motion of the centroid and rotation about the shear center. Vlasov [1] considered the warping effect and introduced rotational and warping inertia terms in flexural torsional vibration analysis. However, none of the models mentioned, due to their somewhat inaccurate assumptions, can take account of the shear lag phenomenon, which has been recently noted as an important factor in thin walled structure analysis. The Vlasov theory is a systematic theory for thin walled member analysis. In many cases, when the shear lag effect is small, the theory gives reasonable results. His theory has two basic assumptions: i.e., (i) the rigid section assumption and (ii) the assumption that the shear deformation along the central line of the cross-section is zero (in an open cross-section) or constant (in a closed cross-section). Many previous works have shown that Vlasov’s second assumption, which makes the theory incapable of accounting for the shear lag effect, is inaccurate. Abandoning this second assumption, Koo and Cheung [6] proposed a mixed variational principle, and Li and Wu [7] proposed a general displacement variational method for the analysis of static problems. In these two methods, the shear lag effect has successfully accounted for. In references [8] and [9] a model for vibration analysis of thin walled beams 503 0022–460X/95/130503 + 11 $08.00/0
7 1995 Academic Press Limited
504
. . . .
was proposed which also takes account of the shear lag effect. However this method requires one to pre-define the appropriate warping function and it seems difficult to apply to complex section profiles. In the present paper a general method for free vibration analysis of thin walled members is proposed, based on reference [7]. For generality, a number of kinds of interpolation function are tried, and finally two kinds of function are proposed: i.e., the local linear function and the B3 spline function. They are used to interpolate the longitudinal displacement along the cross-section and along the longitudinal axis, respectively. Therefore, in the use of this method, human judgement is not required. In this sense, the method is a systematic one for free vibration analysis which can be applied to thin walled members of arbitrarily shaped cross-section. In tube-in-tube structures, the inner tube is connected by the floor with the outer tube so that the rigid section assumption can also be applied and this has been widely used in tall building analysis. Therefore tube-in-tube structures can also be analyzed by the present method. Four numerical examples are presented to demonstrate the convergence, accuracy and versatility of the proposed method. Open, closed, closed with outer flanges and tube-in-tube sections are included in the examples. The results compare well with those of references [8, 9] and of a general finite element package COSMOS [2].
2. VARIATIONAL PRINCIPLE FOR FREE VIBRATION OF THIN WALLED MEMBERS
An arbitrary cross-section prismatic thin walled member is shown in Figure 1, in which the z-axis is parallel to the member axis (a list of notation is given in the Appendix). The principle is based on the following two assumptions. (i) The cross-section can be regarded as rigid in its own plane. (ii) sn and ss (the stresses along the normal direction n and the tangential direction s, respectively) are much smaller than the longitudinal stress sz , such that, by Hooke’s law ez = (sz /E) − n{(ss /E) + (sn /E)} 2 sz /E.
Figure 1. (a) The thin walled member geometry; (b) thin walled member cross-section geometry.
(1)
-
505
The variational principle for free vibration of an elastic member can be expressed by the functional [10] v 2 = st(P/T)
(2)
where P is the amplitude of the possible strain energy induced in vibration, T is the amplitude of possible kinetic energy in vibration, and st means that the value of v 2 should take the stationary value of P/T. Hence d(P) − v 2d(T) = 0.
(3)
These two energy terms can be expressed by four independent displacement variables, w, u, vt and vn as
g $g H
P=
( 12 Eez2 t + 12 Ggzs2 t) ds + 12 GJd
Ss
0
g $g 0 1
1w t ds + 1z
H
=
1 2
E
Ss
0
2
g 0 1 2
G
Ss
0 1% 2
du ds
1
dz
0 1%
1w 1vt du + t ds + 12 GJd 1s 1z dz 2
2
dz
(4)
(in equation (4), Jd is equal to zero in the case of a closed cross-section),
g $g H
T = 12
re
tw 2 ds +
Ss
0
g
tvt2 ds +
Ss
g
%
tvn2 ds dz.
Ss
(5)
By taking the first variation of the functionals in equation (3) with respect to the variables w, u, vt and vn and using the standard procedures of the calculus of variations (such as integration by parts), the following governing differential equations and natural boundary conditions can be obtained: Et 1 2w/1z 2 + G(1/1s)[t(1w/1s + 1vt /1z)] + v 2retw = 0,
(6)
G(1 w/1s 1z + 1 vt /1z ) cos a + v re (vt cos a − vn sin a) = 0,
(7)
G(1 w/1s 1z + 1 vt /1z ) sin a + v re (vt sin a + vn cos a) = 0,
(8)
g0
(9)
2
2
2
G
Ss
2
2
2
2
2
1
1 2w 1 2v d2u + 2t rt ds + GJd 2 + v 2 1s 1z 1z dz
g
re t(vt r + vn l) ds = 0,
Ss
G(1w/1s + 1vt /1z)n,e tne xne = 0.
(10)
At the two ends z = 0 and z = H,
g
Ss
g 0 G
Ss
E
1w tds = 0, 1z
g 0 G
Ss
1
1w 1vt + sin a tds = 0, 1s 1z
1
1w 1vt + cos a tds = 0, 1s 1z
g 0 G
Ss
1
1w 1vt du + r tds + GJd = 0. 1s 1z dz
(11, 12)
(13, 14)
3. METHOD OF ANALYSIS
The Galerkin method is adopted in the analysis. For convenience, instead of vt and vn , vcx and vcy are used in the present analysis. The relationships between them in matrix form are vt = {h1 (s)}T{vc (z)},
vn = {h2 (s)}T{vc (z)},
(15a, b)
. . . .
506 where
{h1 (s)}T = [cos a(s), sin a(s), r(s)], {h2 (s)}T = [−sin a(s), cos a(s), l(s)],
(15c)
{vc (z)}T = [vcx (z), vcy (z), u(z)].
(15d, e)
In the Galerkin method, the basic displacements are expressed approximately as series of selected functions: i.e., w = {f(s)}T{w(z)},
vcx (z) = {y(z)}T{b1 },
vcy (z) = {y(z)}T{b2 },
u(z) = {y(z)}T{b3 },
(16–19)
where {f(s)}T = [f1 (s), f2 (s), . . . , fn (s)],
{w(z)}T = [w1 (z), w2 (z), . . . , wn (z)],
{y(z)}T = [y1 (z), y2 (z), . . . , ym (z)], {bj }T = [bj1 , bj2 , . . . , bjm ] wk (z) = {g(z)}T{ak }
(j = 1, 2, 3),
(k = 1, 2, . . . , n),
{g(z)} = [g1 (z), g2 (z), . . . , gm (z)],
{ak }T = [ak1 , ak2 , . . . , akm ].
T
In short form,
6 7 $
{w} [N1] 0 = {vc } 0 [N2]
%6 7
{a} , {b}
(20)
where {y(z)}T 0 K L G G G {y(z)}T G , G T G k 0 {y(z)} l3 × 3
T 0 K {g(z)} L T G G , {g(z)} [N1] = G G . . G G . T l k 0 {g(z)} n×h
(21, 22) {a} = [{a1 }T, {a2 }T, . . . , {an }T]T,
{b} = [{b1 }T, {b2 }T, {b3 }T]T,
(23, 24)
The functions fi (s) (i = 1, . . . , n), yp (z) and gp (z) ( p = 1, . . . , m) can be any functions series satisfying the displacement boundary conditions [10]. By using the equations (15a–e) and (16), the equations of the problem can be rewritten in matrix form as follows: The governing equations are
$
0 G[D']
E[A] 0
$
+ v 2re
[A] 0
%6 7 $
w0 0 −G[C] + v0c G[C]T 0
0 [D0]
%6 7 $
w' −G[B] + v'c 0
0 0
%6 7
w = 0. vc
%6 7 w vc
(25)
The natural boundary conditions at the two ends z = 0 and z = H are
$
E[A] 0 0 G[D']
%6 7 $
w' 0 + v'c G[C]T
%6 7
0 0
w = 0, vc
(26)
-
507
where [A] =
g
{f}{f}Tt ds,
[B] =
Ss
g
d{f} d{f}T t ds, ds ds
Ss
[C] =
g
Ss
d{f} {h1 }Tt ds, ds (27a–c)
[D1 ] =
g
{h1 }{h1 }Tt ds,
[D2 ] =
Ss
g
{h2 }{h2 }Tt ds,
[D'] = [D] + Jd
Ss
$
0
0
1
%
,
(27d–g) [D0] = [D1 ] + [D2 ]. (The primes denote differentiation with respect to z.) These equations show that the longitudinal (w) and lateral (vc ) vibrations are coupled. Often used boundary conditions, at z = 0 or z = H are as follows: for a fixed end {w(z)} = {0}, {vc } = {0}; for a simply supported end {w'(z)} = {0}, {vc } = {0}; for a free end {w'(z)} = {0}, [D']{v'c } + [C]T{w} = {0}. The Galerkin residual equation of the problem can be derived as
g$ H
0
+
[N1] 0
0 [N2]
$
−G[B] 0 0 0
% 6$ T
E[A] 0
%6 7
0 G[D']
$
w [A] + v 2re vc 0
%6 7 $
w0 0 + v0c G[C]T
0 [D0]
%6 77 w vc
−G[C] 0
%6 7 w' v'c
dz = 0.
(28)
By using the boundary conditions and equation (20), the eigenproblem equation can be finally obtained as
$
[K11 ] [K21 ]
[K12 ] [K22 ]
%6 7 $
{a} [M11 ] − v2 {b} 0
%6 7
0 [M22 ]
{a} = 0, {b}
(29)
or, in short form, [K]{c} − v 2[M]{c} = 0,
(29a)
where
g$ % $ % H
[K11 ] = E
0
[A]
dN1 dz + G dz
g$ % $ % g H
[K22 ] = G
T
dN1 dz
0
dN2 dz
T
dN2 dz, dz
[D']
g
H
[N1]T[B][N1] dz,
[K12 ] = G
g
H
[N1]T[C]
0
H
[K21 ] = [K12 ]T,
[M11 ] = re
[N1]T[A][N1] dz,
0
(30a)
0
[M22 ] = re
g
$ %
dN2 dz, dz
(30b, c)
H
[N2]T[D0][N2] dz.
0
(30d–f) The stiffness matrix [K] and the mass matrix [M] are both symmetric and positive definite. Equation (29) shows that the problem is a typical eigenproblem and can be solved by standard eigenvalue solvers. For the results presented in this paper, an eigenvalue solver based on generalized Jacobi method [11] was used. The eigenproblem can be simplified for cross-sections with one or two axes of symmetry. In the cases of two axes of symmetry, the two directions of the bending vibration modes and torsional modes are uncoupled, while in the cases of a single axis of symmetry, only the bending modes along that axis are separated. In cases of two axes of symmetry, the matrices [A], [B], [C], [D1 ] and [D2 ] (see equations (27a–e)) are equal to four times the
508
. . . .
integration along one-quarter of the cross-section, so that only one-quarter of the original cross-section needs to be calculated, saving computational effort. Similar conditions also occur in the case of cross-sections with a single axis of symmetry. 4. SELECTION OF DISPLACEMENT FUNCTION
From equations (16)–(19), one can see that only the functions fi (s) and yi (s) need to be used in the analysis. Combinations of these two sets of functions can represent arbitrary longitudinal warping displacements, lateral deflections and twist angles. The shear lag phenomenon can then be worked out in the analysis. There are many kinds of functions which are suitable for the analysis. The function series suggested in reference [7] and adopted in the present analysis as fi (s) is the local linear function, because of its simplicity and effectiveness. For the function yi (z), trigonometric, spline and polynomial functions have been tried. Note that they are required to satisfy the displacement boundary conditions only. Finally, the B3 spline function [12] was selected. 5. NUMERICAL EXAMPLES
Four cantilever thin walled members of different cross-sections have been studied. Static analysis [6, 7] shows that the shear lag effect is important only when the H/B ratio (height to breadth ratio of the thin walled member) is small; in these cases, the results differ significantly from those of classical theory. Therefore, in order to demonstrate the shear lag effect, all the examples to be presented have a small H/B ratio. In general, the results
Figure 2. Thin walled cross-sections for the examples. Dimensions in m. (a) Example 1, H = 15 m, t = 0·4 m, E = 3 × 1010 N/m2, E/G = 2·3, re = 2500 kg/m3; (b) Example 2, H, E, E/G, re as (a) t = 0.2 m; (c) Example 3, H, t, E, E/G, re as (b); (d) Example 4, H = 20 m, t = 0·2 m, E = 2·6 N/m2, E/G = 2·6, re = 2400 kg/m3.
-
509
T 1 Convergence study of the rectangular box section Mode order 1 2 3 4 5 6 7 8 9 10
Circular frequencies ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV 4 × 8† 6×8 12 × 6 12 × 8 12 × 12 12 × 16 63·25 100·9 186·9 261·0 362·8 369·2 552·6 562·2 765·6 821·4
61·49 100·9 186·9 246·3 362·8 369·2 520·3 562·2 765·6 767·9
61·06 100·1 186·8 242·8 362·8 363·6 512·3 561·6 752·3 755·9
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 762·2 754·6
61·06 100·1 186·8 242·8 362·8 363·6 512·3 561·6 752·4 755·4
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 752·2 755·4
† n × m denotes n segments along the cross-section and m segments along the longitudinal direction.
compare well with those of other similar theories [8, 9] and of a general finite element package COSMOS [2]. In all of these methods, the rigid section assumption is made and the shear effect is taken into account. Example 1. A rectangular box section shown in Figure 2(a) was investigated. The data from the convergence study are listed in Table 1. The first 11 frequencies as given by various methods are listed in Table 2. For investigating the influence of the rigid section assumption and the number of degrees of freedom (DOF’s) required for acceptable accuracy of a general finite element package, some results obtained by using COMSOS are listed in Table 3. It can be observed that (i) for the first two frequencies, the rigid section assumption has very little influence, but great influence on the high frequencies, and (ii) the number of DOF’s required in using COSMOS is much greater than for the present method. Example 2. An open I section shown in Figure 2(b), which was originally studied by Capuani et al. [8], was studied. The first 14 frequencies as obtained by various methods T 2 Results for Example 1 Mode order
Present theory
1 2 3 4 5 6 7 8 9 10 11
61·06 100·1 186·8 242·8 362·8 363·6 512·2 561·6 752·2 754·6 930·0
Circular frequencies ZXXXXXCXXXXXV COSMOS [2] Ref [9] 61·73 100·8 188·8 245·3 366·8 363·9 516·1 566·7 749·2 761·1 932·0
61·1 189·0 244·1 515·6 568·9 760·3 928·6
Vibration mode shape† BY BX T BY A BX BY T BX BY BY
† BX, vibration in x direction; BY, vibration in y direction; T, torsional vibration; A, longitudinal vibration.
. . . .
510
T 3 Results of the study of the rigid section assumption and COSMOS finite element package number of DOF’s required for a box shaped thin walled member
Mode order 1 2 3 4 5 6 7 8 9 10 11
Circular frequencies ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV With RSA† Without RSA† ZXXXXXXXCXXXXXXXV ZXXXXCXXXXV DOF’s DOF’s Mode DOF’s Mode 1080 1950 shape 1080 shape† 61·79 100·9 189·1 247·2 366·2 366·9 524·4 569·9 757·7 774·7 941·5
61·73 100·8 188·8 245·3 366·8 363·9 516·1 566·7 749·2 761·1 932·0
BY BX T BY A BX BY T BX BY BY
59·86 99·81 115·5 163·7 184·1 196·5 226·2 262·9 268·0 270·3 290·4
BY BX T BY BY BY T BY T BY BY
† the rigid section assumption. ‡ see Table 2 for key.
are tabulated in Table 4. Because of the shear lag effect is not taken into account in the Vlasov theory, large differences among the corresponding predicted frequencies can be found in Table 4. When the height of this member is increased to 90 m (H/B = 15), the shear lag effect will no longer be important. This case has also been investigated, and no difference can be observed between the first ten frequencies obtained from the Vlasov theory and the ones obtained from the present method.
T 4 Results for Example 2
Mode order 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Circular frequencies ZXXXXXXXXXXXCXXXXXXXXXXXV Present theory COSMOS [1] Ref [8] Vlasov 57·94 59·65 76·40 215·3 254·4 336·9 362·8 438·2 551·7 630·0 731·3 769·8 853·5 1088·0
† Table 2 for key.
57·77 60·12 76·83 216·5 251·5 333·7 366·9 437·5 540·9 626·4 719·0 766·2 827·3 1097·0
57·7
240·4
540·5
832·7
62·17 74·16 81·86 350·7 365·0 410·6 888·8 1029·0
Vibration mode shape† T BY BX BY T BX A BY T BY BX BY T A
-
511
T 5 Results for Example 3
Mode order
Circular frequencies ZXXXXXXXXCXXXXXXXXV Present theory COSMOS [2] Ref [8]
1 2 3 4 5 6 7 8 9 10
52·00 84·91 127·3 206·1 341·3 362·8 382·2 426·5 622·6 723·1
52·31 85·25 127·7 204·1 337·3 366·8 377·5 418·8 593·3 710·1
47·3 130·4 170·8 294·0 421·7 481·9
Vibration mode shape† BY, T BX BY, T BY, T BX A BY, T BY, T BY, T BX
† See Table 2 for key.
Example 3. A cross-section composed of a rectangular box section with two outer flanges, as shown in Figure 2(c), was studied by Capuani et al. [8], and was studied again by the present method and by COSMOS [2]. The first ten frequencies are listed in Table 5. The COSMOS and present method results are close to each other and larger differences can be noted between these and Capuani’s results. The reason for this may be that Capuani’s method depends significantly on the functions selected; therefore the results may be worse for some irregular cross-sections. Example 4. A tube-in-tube section as shown in Figure 2(d), which is widely used in tall buildings, was studied. The first 17 frequencies are compared with those from the finite element package COSMOS in Table 6. It is of interest to note that the frequencies of pure longitudinal vibration computed by the present method are identical to those of classical theory, as expressed by the formula fi = z(E/re )(2i − 1)/4H, where i is the mode number. T 6 Results for Example 4
Mode order 1&2 3 4&5 6 7 8&9 10 11 & 12 13 14 15 16 & 17
Frequencies (×105) ZXXXXXXCXXXXXXV Present theory COSMOS [2] 5·976 22·10 27·28 41·14 41·14 60·05 66·29 94·16 110·5 123·4 123·4 127·0
† See Table 2 for key.
6·219 22·15 27·84 43·11 43·11 60·71 66·19 93·50 109·5 128·7 128·7 125·3
Vibration mode shape† BX BY T BX BY A (int. tube) A (ext. tube) BX BY T BX BY T A (ext. tube) A (int. tube) BX BY
512
. . . . 6. CONCLUSIONS
The present study has shown that the proposed method is applicable to thin walled members of any cross-section shape. The numerical examples have fully demonstrated the accuracy, versatility and fast convergence of the method. The method is simple and general, and requires much less computational effort and time than the standard finite element method. It can be further improved by choosing more suitable shape functions along the profile of the section, and should be very useful in practical design, especially at its preliminary stage. REFERENCES 1. V. Z. V 1961 Thin-walled Elastic Beams (English translation, National Science Foundation, Washington, D.C.). London: Oldbourne Press. 2. M. L Cosmos/M User Guide. Structural Research and Analysis Corporation. 3. S. T, D. H. Y and W. W J. 1974 Vibration Problems in engineering. New York: John Wiley. 4. J. M. G and Y. K. L 1958 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 373–378. Coupled vibrations of thin-walled beams of open cross section. 5. C. F. G 1940 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics A-97, A-105. The normal modes of vibrations of beams having non-collinear elastic and mass axes. 6. K. K. K and Y. K. C 1989 Journal of Engineering Mechanics 115, 2271–2286. Mixed variational formulation for thin-walled beams with shear lag. 7. W. Y. L and X. S. W 1991 Proceedings of the Asian Pacific Conference on Computational Mechanics, Hong Kong. A semi-discrete method for shear lag analysis of thin-walled members. 8. D. C, M. S and F. L 1992 Earthquake Engineering and Structural Dynamics 21, 859–879. A generalization of the Timoshenko beam model for coupled vibration analysis of thin-walled beams. 9. F. L and M. S 1991 Thin-walled Structures 11, 375–407. The shear strain influence on the dynamics of thin-walled beams. 10. H. C. H 1989 Variational Principle in Elasticity and Application. Beijing: Scientific Publishing House. (In Chinese). 11. Z. H. C 1983 Matrix eigenvalue Problem. Shangai Science and Technology Publishing House. (In Chinese). 12. W. Y. L, Y. K. C and L. G. T 1986 Journal of Engineering Mechanics 112(1), 43–54. Spline finite strip analysis of general plates. APPENDIX: NOTATION s c Ss l r a t H Jd E G re xne
curvilinear co-ordinate along the contour of the cross-section shear center of the cross-section the whole cross-section distance from s to r , positive along positive direction of s, r being a vector from the shear center c perpendicular to the tangent at point s length of r s , distance from c to the tangent at point s, positive when r × ds is along the positive z-axis direction angle between the x-axis and the tangent at s, positive in anticlockwise direction =t(s), thickness of wall at s height (length) of the member Saint-Venant’s torsional constant (for open section only) Young’s modulus shear modulus density of material +1 when positive direction of s in the wall (e) connected at node n points away from the node n; −1 when positive s in the wall (e) points toward the node n; 0 when the wall (e) does not join at n
- nz v w(z, s) vt vn vcx , vcy u
513
+1 for z = H and −1 for z = 0 angular frequency of vibration longitudinal displacement displacement of point s along the tangent direction; positive when along positive direction of s displacement of point s along the normal direction, positive when v t × v n is along positive z-axis direction x and y components of shear center displacement vc twist angle of the cross-section, positive in anticlockwise direction