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Journal of‘Sound and Vibration (1977) 54(4), 537-547
Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 SNH, England (Received 12 January 1977, and in recised.form 2 June 1977) A flat, rectangular finite strip element and a compatible thin-walled, open section beam element are used to predict the vibration characteristics of flat skin-stringer structures of riveted construction. Strip only idealizations and strip plus thin walled, open section beam idealizations have been used in the analyses. The effects of neglecting skin membrane action and varying the effective width of stringer-skin attachment are investigated. 1. INTRODUCTION years, considerable interest has been shown in the analysis of curved skin-stringer structures, such as those found in aircraft fuselage construction. This interest has been prompted by the concern over the response of such structures to acoustic excitation and the resulting sound radiation and fatigue problems. A fuselage structure essentially consists of a cylindrical shell stiffened by circumferential frames and longitudinal stringers. Tests on a full scale structure [I] showed that adjacent panels across a frame vibrate independently of one another, with the frames acting as rigid boundaries. Thus most attempts at theoretical analysis have considered part of the fuselage structure between two adjacent frames. lnitial attempts at analyzing skin-stringer structures neglected the effect of curvature. In keeping with this, the present paper is concerned with flat skin-stringer arrays. Lin [2] first developed a method for predicting the limiting frequencies of the major response bands. In the initial attempts at predicting the natural frequencies and modes of vibration finite difference techniques were used [3-61. The next method to be utilized was the transfer matrix method [6-91). A muchmodified version ofthis, which can be referred to as the dynamic stiffness matrix method [lo, 1 I], has also been used. In recent years, the finite element method of analysis has received a considerable amount of attention. This method has also been used to analyze skin-stringer arrays [12-l 61. Although very versatile, it does suffer from the fact that very often a large number of degrees of freedom is required to describe the behaviour of a structure accurately. In many situations, the geometric and material properties of a structure do not vary in one direction. Cheung [ 17-- 191 has made use of this fact in developing a more efficient finite element method for this class of structure. He has named it the finite strip method. It can be regarded as an approximate version of the dynamic stiffness method referred to above. It does have the advantage though of leading to a linear algebraic eigenvalue problem which can be solved more efficiently. The finite strip method is more efficient than the standard finite element method since it involves fewer degrees of freedom. In this paper, the finite strip method is used to determine the vibration characteristics ot flat skin-stringer arrays of riveted construction. The standard flat strip, which includes both bending and membrane action, as developed by Cheungis used. In addition, a new compatible. thin-walled, open section beam element is developed. 537 In recent
Analyses of skin-stringer structures are carried out with the use of strip only idealizations and strip plus thin walled, open section beam idealizations. The effects of neglecting skin membrane action and varying the effective width of stringer-skin attachment are investigated.
2. FINITE 2.1.
Consider the rectangular flat strip element shown in Figure 1. The kinetic and strain energies for an isotropic plane stress element are T = +ph [ (a2 + ti’) dA,
A” 1
Eh __I u=2(1 -9)
[u: + c: + 2~24,u, + +( 1 - v) (u, + z#] dA, (2) I’ A where u and v are the displacements in the x and y directions, E is Young’s modulus, v is Poisson’s ratio, p is the density, h is the plate thickness, A is the area of the element, ( ), ( )y, (s) = a( )/ax, a( )/ay, a( )/at and t denotes time.
Figure I. Rectangular flat strip element.
The in-plane
M fi= c
by functions
of the form
1 dF,(x) ----
u and v are represented LN(JJ)J{4”hn7
1’= c
F*(x) LN(Y)J
where 2, and F,(x) are the characteristic numbers and functions, respectively, of a uniform beam. These functions are chosen since they satisfy the boundary conditions along x = 0 and x = 1 where I is the length of the strip. The shape functions ~N(y)l are given by LN(Y)l
= L( 1 - 4 r?l,
where q = y/b, where b is the width of the strip. Also (qJT = LU, u,l,
= L”1 “21.
Substituting expressions (3) into equations (1) and (2) results in the required inertia and stiffness matrices. Expressions for the elements of these matrices are given in references [18] and [19]. The kinetic and strain energies for an isotropic plate bending element are T= +ph /-,2 dA,
cJ=)DJ[w:, + ‘6 A
where D = Eh2/ 12( 1 - v’).
+ 2”‘4’, wBy+ 2( i - v) w&,1&I,
(6) (7)
The normal
w is represented
by a function
of the form (8)
FfJ.4 LN(.Y)! hv>w
w= : ttl=l
In this case the shape functions beam : namely, LN(J)1
= L( 1 -
~N(y)j are taken to be the same as the functions 3q2 + 2q3) b(q - 2rj2 + q3) (3r2 - 293) b(-$
for a uniform
+ q3)].
Also GLJr = LWl Ql w2 021, (IO) where 13= awjay. Substituting expression (8) into equations (6) and (7) results in the required inertia and stiffness matrices. Again, expressions for elements of these matrices are given in references [18] and [19]. In the following applications the membrane and bending matrices were combined to produce a single element with the following degrees of freedom : LU, 2.2.
L)2 w2
a thin walled, open section beam element whose cross-section
is shown in Figure 2.
0 +
C t
Figure 2. Cross-section centre.
of a thin-walled, open section beam. A, Attachment
It is assumed that the element is attached to the neighbouring The kinetic energy of such an element is given by
point; C, centroid;
0, shear
plate along a line through
where U, v and w are the displacements of the attachment point A in the x, y and z directions, p is the density, 2 is the length of the element, A is the area of cross-section, aY and a, are the co-ordinates of the centroid of the cross-section and IYy,ZYzand I, are the second moments of area of cross-section about axes through A.
540 According to the theory developed section beam element is given by 2U=
by Vlasov [20], the strain energy of a thin walled, open
I‘u, O, dx + El, j L& dx + 0
2EI,= / r, wXXdx + El, j w:, dx + 2EJ, ( L’, 8, dx + 2EJ, j w, O, dx + i, 0 0 0
+ EJ, f O;, dx + GJ 10: dx, 0 0
where E is Young’s modulus, G is the shear modulus, J is the Saint Venant torsion constant, S, and S, are the first moments of area of cross-section about axes through A and S, J, J,= and J, are warping constants. The first moments of area are given by S, = a,A,
The warping
may be evaluated
s, = -c, s, + C, S, Jon = -cz I,z + cy Iyy -
S,=aZA. by using the following
(13) expressions
J, = -C, I, + C, I,* - o, S, J, = -C: J, + C, J, - wA S, + Jo,
WA sz,
where C, and CZ are the co-ordinates of the shear centre 0 of the cross-section, oA is the value of the warping function at the attachment point and Ju, is the warping constant. In evaluating w, and J, the pole of o is taken to be at the shear centre 0 and its origin is chosen to make the mean warping of the cross-section zero. Forrmulae for Jo, for typical cross-sections may be found in references [21]-[24]. The displacements U, u and w along the attachment line are represented by functions of the form U=
M 1 dF,(x) 7 7 c In=, &n
w = ; F,(x) w, m=l
1’= z Fm(x)Um, m-l 0=
2 ,7,(x) 8,. rn=l
In this form they are compatible with the displacement functions (3) and (8). Substituting expressions (15) into equations (11) and (12) results in the required inertia and stiffness matrices referred to the following degrees of freedom: LUUW~~,. The expressions for the elements of these matrices are given in the Appendix. 2.3.
The method of transforming the above element matrices into a common set of co-ordinate axes a.nd of subsequently assembling the transformed matrices is identical to the method used for plane frames. The solution for the natural frequencies and modes of vibration is obtained by any of the standard procedures used in finite element analysis. It should be noted that in the case of the boundaries x = 0 and x = 1 both being simply supported the coupling matrices between the degrees of freedom for different values of m are identically zero. In this case, the analysis may be performed with only a single term used in each of the series of equations (3), (8) and (15). The analysis is then repeated for different values of m. This is the only case considered in this paper.
54 1;
References [17] and [19] contain results for a simply supported square plate which was analyzed with the use of eight strips. A cpmparison with the exact solution shows the results to be highly accurate. In order to determine the minimum number of strips required to represent each panel in the following examples, a rectangular plate, having the same dimensions as a single bay in one of the examples, was analyzed. The plate has dimensions 20.828’ cm x 50.8 cm and a thickness of 0.1016 cm. The following material constants were used in all the examples presented in the section: E = 7.24 x 10” N/m2, v = 0.3, p = 2795.67 kg/m3. The plate was analyzed with, respectively, 4, 6 and 8 strips parallel to the long edge being used. The convergence of the results for increasing the number of strips is shown in Table 1. TABLE
(Hz) of a simply supported,
rectangular plate
Number of strips Mode (I,]) (12) (1,3) (1,4)
Exact solution
66.17 236.28 521.31 926.48
66.17 236.16 519.98 919.40
66.16 236.09 519.29 915.78
4 66.18 236.97 528.51 1014.7
3nly the modes having one half-wave in the long direction were considered. In the followin!,: examples of stiffened plates, the modes to be considered consist of combinations of individual panel modes which have one half-wave in each direction. It can be seen from Table I that sufficient accuracy will be obtained if each panel is represented by four strips. 3.2.
In order to check the method, the six bay skin-stringer structure previously considered in references [2], [3] and [6-81 was analysed. This structure consists of a skin 50.8 cm x 124.8 cm x 0.1016 cm which is stiffened by seven identical top-hat section stringers placed at equal distances (20.8 cm), parallel to the short side. The physical properties of the stringers are given in Table 2 in the column marked “Stringer A”. TABLE
Physical properties
Physical parameter A (cm*)
I, (cm? IYZ(cm? I=, (cm4) uy (cm) uL (cm) S, (cm? JWY(cmj) J,; (cm’) J, (cm”) J (cm”)
of stringers
Stringer A 1.485 745 I
0.0 3.455 0.0
1.829 0.0 0.7196 o-o 4.428 9.419 x 1,0--S
Stringer B 0.3782 0.7168 0.5128 0.4470 0.8077 1.080 0.1252 0.04863 0.1743 0.04958 6.381 x 1o-4
M. PETYT TABLE 3 (Hz) of a six bay,
skin-stringer structure
Mode I 2 3 4 5 6
Limiting frequencies (2) 98.9
Finite difference (3)
Transfer matrix (7)
100.3 104.7 111.4 116.3 124.4 131.1
99.9 103.1 108.2
Finite strip 99.9 103.1 108.4 115.4 122.9 129.1
115.0 122.1 127.9
Each individual bay was represented by four strip elements parallel to the stringers and each stringer by one thin-walled, open section beam element, making a total of 3 1 elements. In order to be consistent with previous analyses, the membrane displacements of the skin were taken to be zero. The long edges of the structure were assumed to be simply supported. The first group of natural frequencies, which correspond to modes having one half-wave in the direction of the stringers, is presented in Table 3. This table also contains the limiting frequencies, as predicted by the method of reference [2], and the natural frequencies predicted by finite difference and transfer matrix techniques. There is close agreement among all the results. The mode shapes also agree with those presented in reference [7]. 3.3. INVESTIGATIONOF THE EFFECT OF VARIOUSSIMPLIFYING
Previous analyses, such as finite difference and transfer matrix methods, have introduced various simplifying assumptions in order to make a solution possible. With the finite strip method it is possible to relax many of these assumptions with very little additional effort. In this section, the results of some typical investigations are presented. Two of the most common assumptions are that the membrane displacements in the skin are zero, and the cross-sections of the stringers do not distort in their plane. The existence of stringers, attached to one side of the skin, introduces coupling between the bending and membrane actions of the skin. This can be included in the finite strip model by retaining the membrane action described in section 2.1. Cross-sectional distortion of the stringers can be introduced by modelling the stringer as an assemblage of finite strip, plate elements rather than a thin walled, open section beam. This approach has already been used for integrally stiffened plates in references [IO] and [ 181.
0.0248 m Figure 3. Geometry of a one bay stiffened plate.
As a check on this approach, the one bay stiffened plate shown in Figure 3 was analysed. The thickness of all the members is 1.181 mm. This structure was represented by eight strip elements, four for the plate and two for each stringer, one for the web and one for the flange. The natural frequencies are compared with the measured ones (in parentheses), which have been obtained from reference [25], in Table 4. Results have been obtained for 171= I, 2 and 3 half waves in the direction of the stringers. Only the plate modes were measured, but the finite strip results can be seen to be in very good agreement. TABLE 4 Natural frequencies
Plate sym Stringer sym Stringer asym Plate asym
qf a
one bay, skin-stringer structure
136.5 (134.5) 146.2 166.8 305.4 (312.5)
223.6 (219.5) 415.0 429.6 604.2 (569.5)
252.5 (255.2) 803.9 829.4 663.5 (650.4)
The next structure to be considered was a five bay skin-stringer structure. The dimensions of the bays and the properties of the stringers were taken from reference [26]. The structure., therefore, consists of a skin 22.86 cm x 57.15 cm x 0.071 cm which is stiffened by six identical z-section stringers placed at equal distances (11.43 cm), parallel to the short side. The details of the stringers are given in Figure 4.
Figure 4. Details of a stringer cross-section.
Initially, each bay was represented by four strip elements parallel to the stringers and each stringer by three strip elements (to represent one web and two flanges). One flange element coincided with one skin element to give a composite element 0.142 cm thick. This composite element was first joined to the adjacent plate elements by means of rigid links as indicated in Figure 5(a). The long edges of the skin were assumed to be simply supported. The natural frequencies for the first group of modes are given in row 1 of Table 5. The mode shapes along the length of the plate parallel to the simply supported edges are shown in Figure 6. The mode shape in the perpendicular direction is one half sine wave. This analysis involved 64 linear constraints for the complete structure. These can be avoided if the off-set of the composite element and the stringer web element are taken to be zero as shown in Figure 5(b). The results obtained with this configuration are shown in row 2 of Table 5. The differences between the two sets of results are small. Keeping the same idealization, the analysis was repeated with the membrane action of the plate removed. The frequencies obtained are shown in row 3 of Table 5. Again, there is very little change. The mode shapes obtained from these two analyses indicate negligible stringer cross-sectional distortion. Next the structure was analyzed with the use of four strip elements for each bay and one thin walled, open section beam element for each stringer. The stringer data used is labelled
5. Details
of stringer
(a) x, nodes
(b) with zero off-set:
TABLE 5 Naturalfregurncies
(Hz) of ajve
skin-stringer structure Mode no. Analysis no.
1 2 3 4 5 6
255 253 253 260 260 164
210 267 268 268 268 173
296 294 294 280 281 199
328 326 328 294 295 237
358 355 358 305 307 281
5~ Figure
6. Modes
of vibration
of a five bay flat panel.
“Stringer B” in Table 2. The frequencies are given in row 4 of Table 5. This analysis was also repeated with the skin membrane action removed. These results are shown in row 5. Again the change in frequencies is only small. Comparison of rows 2 and 3 with rows 4 and 5 shows that the major discrepancy is in mode 5, the stringer bending mode. The main difference in the two sets of analyses is that for analyses 2 and 3 it was assumed that the top flange of the stringer remained in full contact with the skin, whilst in analyses 4 and 5 it was assumed that the stringer made only a line contact. with the skin. Reference [9] has indicated that if the cross-sectional area of one-third of the skin between stringers is significant when compared with the stringer cross-sectional area., then the stringer should be assumed to make full contact with the skin. In the present example this is not the case and so the results given in rows 4 and 5 are considered to be the correct ones. Finally, an analysis was performed to demonstrate the importance of including the effect of warping restraint in the beam analysis in section 2.2. The last analysis was repeated with the warping constants S, Jwy, JwZ and J, set to zero. The results obtained are given in row 6 of Table 5. Comparison with row 5 shows large discrepancies between the two sets of frequencies, particularly in the lower mode frequencies which involve torsion of the stringers.
4. CONCLUSIONS It has been shown that the finite strip method can accurately predict the natural frequencies and modes of vibration of flat skin-stringer arrays of riveted construction. The elements used consist of a previously published flat plate element together with a compatible thin walled, open section beam element which has been presented in this paper. Both plate strip only and plate strip with beam idealizations have been used. The choice between these two depends upon the amount of skin which can be assumed to be in contact with the stringer. Although the stringers couple the bending and membrane action of the skin, neglecting the membrane action of the skin has little effect on the natural frequencies of the modes considered. It has been found that the finite strip method is more efficient than the finite difference, transfer matrix and finite element methods previously used. The examples presented also demonstrate that the finite strip method is also more versatile than these since less simplifying assumptions need to be made.
ACKNOWLEDGMENTS The author would like to thank the Procurement Executive, Ministry of Defence for the financial support of this project, and also his colleague A. Y. Abdel-Rahman for his assistance in carrying out the computation involved.
REFERENCES 1. B. L. CLARKSON and R. D. FORD 1962 Journal oj’the Royal Aeronautical Society 66, 31-40. The response of a typical aircraft structure to jet noise. 2. Y. K. LIN 1960 Journal of Applied Mechanics 27, 669-676. Free vibrations of continuous skinstringer panels. 3. Y. K. LIN, I. D. BROWN and P. C. DEUTSCHLE1964 JournalofSoundand Vibration 1, 14-27. Free vibration of a finite row of continuous skin-stringer panels. 4. W. J. TRAPP and D. M. FORNEY (editors) 1965 Acoustic Fatigue in Aerospace Structures. Syracuse University Press. 5. T. WAH 1965 in Acoustic Fatigue in Aerospace Structures (editors W. J. Trapp and D. M. Forney), 299-310. Lateral oscillations of sheet-stringer panels. Syracuse University Press.
6. Y. K. LIN 1965 in Acoustic
Fatigue in Aerospace Structures (editors W. J. Trapp and D. M. Forney), 163-184. Dynamic characteristics of continuous skin-stringer panels. Syracuse University Press. 7. C. A. MERCER and C. SEAVEY 1967 Journal of Sound and Vibration 6, 149-162. Prediction of natural frequencies and normal modes of skin-stringer panel rows. 8. Y. K. LIN and B. K. DONALDSON 1969 Journal of Sound and Vibration
10, 103-143. A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels.
9. J. P. HENDERSONand A. D. NASHIF 1971 American Society of Mechanical Engineers Paper No. 7 l- Vibr-101. The effect of stringer width and damping on the response of skin-stringer structures. 10. W. H. WITTRICK and F. W. WILLIAMS 1972 Proceedings of IUTAM Symposium 1970 on High Speed Computing of Elastic structures, Li&e. University of Likge. Natural vibrations of thin, prismatic, flat-walled structures. 11. F. W. WILLIAMS 1972 Journal of Sound and Vibration 21, 87-106. Computation of natural frequencies and initial buckling stresses of prismatic plate assemblies. 12. G. M. LINDBERGand M. D. OLSON 1967 National Research Council of Canada, Aeronautical Report LR-492. Vibration modes and random response of a multi-bay panel system using finite elements. 13. R. N. YURKOVICH, J. H. SCHMIDTand A. R. ZAK 1971 Journal of Aircraft 8, 149-155. Dynamic analysis of stiffened panel structures. 14. M. D. OLSON and G. M. LINDBERG1971 Journal of Aircraft 8,847-855. Jet noise excitation of an integrally stiffened panel. 15. F. F. RUDDER 1972 NASA CR-1959. Study of effects of design details on structural response to acoustic excitation. 6.1-6.12. Accurate 16. G. M. LINDBERG 1972 Symposium on Acoustic Fatigue AGARD-CP-113, finite element modelling of flat and curved stiffened panels. 17. Y. K. CHEUNG and M. S. CHEUNG 1971 Journal of Engineering Mechanics Division, Proceedings Society of Civil Engineers 97, 391-411. Flexural vibrations of rectangular and other polygonal plates. 18. M. S. CHEUNG and Y. K. CHEUNG 1971 Journal of Sound and Vibration 18, 325-337. Natural of the American
vibrations of thin, flat-walled structures with different boundary conditions. 19. Y. K. CHEUNG 1976 Finite Strip Method in Structural Analysis. Oxford: Pergamon Press. 20. V. Z. VLASOV 1961 Thin-waIledElastic Beams. Second edition, translated from the Russian by the Israel Program for Scientific Translations for the N.S.F. and the Dept. of Commerce, U.S.A., Office of Technical Services, Washington, D.C. 21. J. H. ARGYRIS and P. C. DUNNE 1952 Part 2 of StructuralPrinciples and Data. London: Sir Isaac Pitman & Sons, Limited, fourth edition. Structural analysis. 22. S. P. TIMOSHENKOand J. M. GERE 1961 Theory of Elastic Stability. New York: McGraw-Hill Book Company, Inc., second edition. 23. K. ZBIROHOWSKI-KOSCIA1967 Thin Walled Beams, from Theory to Practice. London: Crosby Lockwood and Sons Limited. 24. J. T. ODEN 1967 Mechanics of Elastic Structures. New York: McGraw-Hill Book Company, Inc. 25. F. GIANNOPOULOS1974 M.Sc. Dissertation, University of Southampton. Effect of cross-sectional distortion of stiffeners on the natural frequencies of stiffened plates. 26. R. D. FORD 1961 University of Southampton A.A.S.U. Report Number 181. The response of a model structure to noise-Part 1.
The elements
of the upper triangle
m II(l/X,)~,pA,
of the inertia
m22 = 11 PA + 12 PI,
m23 = I1 PI,
m33 = I1 PA + I2 PI,
m34 = ay II PA,
matrix are given by m13=-(~/k,)az~~~A, m24 = -a z I 1, m44 = I, p(l,
+ I,).
The elements of the lower triangle of the stiffness matrix are given by k, = EJWZJL, k, = -EAdUU
k, = -(l/Am) I4 EAay, 4,
k3z = E&z 14,
k, = -E&,(1/&,) 14, ka = EJ,I,
k, = EI, I.+,
k, = El, 14, k, = EJ, Ia, k, = EJ, I4 + GJI,.
In the above r, = J!Fi(x) dx = I/2, 0 dx = (mn)‘/21,
dx = (mr)4/21-‘.

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