Pergamon

Computers d Structures Vol. 50. No. 4. pp. 471-480, 1994 Coowinhc @=!J 1994 Elscvier S&ma Ltd

Printodin &ea~Britain. All ri&s mrvcd 004s7949/94 s6.00 + 0.00

VIBRATION ANALYSIS OF STIFFENED PLATES

C. J. CHEN, W. Lru and S. M. C&RN Department of Civil Engineering, Chung Cheng Institute of Technology, Tahsi, Taoyuan 33509,

Taiwan, R.O.C.

(Received 11 November 1992)

Abstract-A spline compound strip method has been presented for the free vibration analysis of stiffened plates. The plate was discretized and modelled as strip elements. The displacement function of a strip element has been expressed as the product of the conventional transverse shape functions and longitudinal cubic B-splines. The flexural, torsional, and axial effect of the stiffeners towards the strip were derived and incorporated into a direct methodology. The theory has been illustrated with several examples including one-directional stiffened plates and cross stiffened plate. The results of the present analysis were adequately put into comparison with those from other previous methods.

INTRODUCITON

This analysis isconcemed with the free vibration analysis of stiffened plates. Such plates are of practical interest as a consequence of their application in aircraft structures, ship superstructures, highway bridges, launching struc- tures for rocket, and buildings and other structures.

A number of earlier investigations have focused on the vibration analysis of the stiffened plate structures. The stiffeners were assumed by the orthotropic plate approximation to be dispersed within the plate, and the stiffened plate was idealized as an orthotropic plate [l, 21. A stiffened plate was converted by the grillage approximation into a series of intersecting beams [3,4]. The plate and the stiffener were separately considered while still, however, maintaining compati- bility between them. Several methods, i.e. the Rayleigh-Bitz method [5,6], the matrix method [7-91, the finite difference method [lO-131, the finite element method [ 14,151, and the finite strip method [ 161 have all been proposed to solve the stiffened plate problem.

In this paper, the spline compound strip method is derived for the free vibration analysis of eccentrically stiffened plates. The spline compound strip method can be considered as a special form of the tinite element method owing to the plate being discretized in one direction, i.e. the plate is divided into finite strips. The

Bitz-Galerkin method, with B-splines as trial functions, is applied in another direction. The compound strip is a substructing technique which allows for stiffener el- ements to be attached to a strip. The presence of the stiffener in a strip is incorporated through means of in- ternally constraining the stiffener displacement fields to the strip displacement fields. Also, the flexural stiffness and mass of the stiffeners, the torsional stiffness and associated mass effect of the stiffeners, and the axial stiff- ness and associated mass effect are derived and assem- bled toward the relative strip into a direct methodology.

PROPERTIES OF BSPLINE

The spline function chosen in this paper to represent the displacement along the length of the strip is the cubic B-splines (shortened as B,-spline). The length of the strip is divided into n sections. By using the Bay- leigh-Bitz method, the displacement is taken as the summations of (n + 3) local B, -splines by

n+l

s(Y)= C ci4i(Y)9 i--l

where r#+ is a local B3 -spline and ci is a coefhcient to be determined. A local B, -spline is a piecewise polynomial that has an inherent C* continuity. Also, each B, -spline is non-zero only over four sections. A standard B,-

spline can be expressed as 0; S <yi-2

6 -Y&2)'

(Yi+l -Yi-*)(Yi-Yi_*)(Yi_, -Yi_2) ’ yi-2 ” <“-’

1 (Y,+2-S)’

(Yi+2-Yi-l)(Yi+2-Yi)(Yi+2-Yi+I)’ yi+“s <yi+2

0; _Yi+2 <S. (2)

471

412 C. J. CHEN et al.

ii.,

0 1 2 3 4 5 6

(a)

0 1 2 3 4 5 6

(cl Fig. 1, Amended cubic B-splines with a (a) free edge, (b) simply supported, and (c) clamped boundary

knot.

Equation (2) is directly derived from the divided difference of Green’s function [17]. However, the direct evaluation of the B-spline from eqn (2) has to be performed with caution as a consequence of the loss of significance possibly occurring during the computation of the various difference quotients. Special provisions also have to be made in the case of repeated points owing to the denominator in eqn (2) being zero. The difficulties mentioned above are avoided in this paper through usage of the recurrence relation previously developed by deBoor [18], which requires no special arrangements in case of repeated knots and does not suffer from an unnecessary loss of numerical precision. Two

N [A,1 0 0 0

0 N,M,l 0 0

0 0 N3 [4wil N4[40,1

approaches existing in modifying the B3-splines for the sake of satisfying the prescribed boundary conditions, which are: modifying the local splines that include the boundary knot [18, 191, and choosing the multiplicity of the boundary knot [20]. B3-splines are modified in this paper by choosing the latter approach. Amended B,-splines with a free edge, simply supported, and clamped boundary knot are shown in Fig. 1.

DISPLACEMENT FUNCTION

The displacement of a strip (Fig. 2) is expressed as the product of the conventional transverse shape functions Ni and longitudinal B3-splines as follows:

N2[4ujl 0 0 0

0 N214ujl 0 0 (3) 0 0 Ns [4wjl N6[4ojl

11%) J

Vibration analysis of stiffened plates

/ a

B3-splinefunction /

nodallinei

Fig. 2. A typical strip element.

in which

N,=l-r N2 =r

N, = 1 - 3r2 + 2r3, N, = x(1 - 2r + r*)

N, = (3r* - 2r’), N6 = x(r* - r), (4)

where r = x/b.

The terms ]4uil* I&& ]4vil, Marl, ]&J ]&jl, t&l, [&,I are row matrices and each matrix has (n + 3) B,-splinesasdefinedby[~]=[rp_,,cp,,rp,,...,cp,_,, rp,, qn+ ,I, in which pi is a B,-spline which can be directly modified for adaption at various prescribed boundary conditions at the ends of strip, and n is the number of sections. Also {I+}, {ui}, {w,}, {EJ,}, {nj}, {v,}, {w]} and {O,} are the corresponding column matrices of displacement coefficients. The subscripts i and j denote nodal lines i and j, respectively.

The presence of stiffeners in a strip element is incorporated by explicitly constraining the stilfener dis- placement fields to the relative strip displacement fields.

The displacement field, d,, for an xdirectional stiffener is coupled to the strip element displacement field according to

where e,, is the distance from the plate midplane to the centroid of x-stiffener, and the comma denotes differentiation. Similarly, they -stiffener displacement field is taken as

where e,, is the distance from the plate midplane to the centroid of y-stiffener.

STIFFNESS AND MASS MATRICES

The dynamic problem found in free vibration analysis of elastic structures is formulated by apply- ing D’Alembert’s principle of dynamic equilibrium. Application of this principle demonstrates an inertia force equal to the mass times the acceleration being assumed to act on the structure in the direction of negative acceleration. The equilibrium equation for a structure without damping is consequently

413

where {a} is the time-dependent displacement vector; (6) is the acceleration and the dot denotes time differentiation; [M] is the global mass matrix; and (K] is the stiffness matrix.

The following assumption can be made for fne vibration, which is

J(t)=6 sinot and g(r)= -o*sinot. (8)

Substitution of eqn (8) into eqn (7) gives the general- ized eigenvalue problem

(WI - ~WI)@) = P% (9)

where o is a natural frequency, and {S} is the corresponding mode shape.

STIFFNEss AND MASS MATRICES OF A STRIP

The strain energy of flexural and membrane deformation of a strip is

CJ=; {u}‘{c}dxdy, J‘s

(10)

where

The stress matrix and strain matrix can be related to the displacement vector (6) by

{cl =PlW

and

Ic) = VW] = PIWI~~.

The kinetic energy of a strip is

T-fp 111

{u}*dV,

where

If I= {i I= Wt@(O~.

(13)

(14)

(19

(16)

474 C. J. CHEN ef al.

Substituting eqns (13), (14) and (16) in eqns (10) and (15) and applying Hamilton’s principle yield stiffness matrix [K,] and mass matrix [M,] of a strip. The matrix [K,,] is the same as that shown in (211. The matrix [M,] is shown as follows

W,l= Ptb

where p is the density of the plate, and t is the thickness. The non-zero elements of mass matrix are shown in Appendix 1.

. Mp,, 0 0 0 M,,,, 0 0 0

Mpz 0 0 0 Mpx 0 0

Mp,, Mpu 0 0 Mpu M,,xI

Mp44 0 0 MN7 448 Mpss 0 0 0

sym.

(17)

STIFFNESS AND MASS MATRICES OF STIFFENERS

An x-stiffener is attached to the strip and oriented in the transverse direction at the local coordinate y,. The strain energy for an x-stiffener of flexural, torsional, and axial effect then becomes

where E,,A,, EJ,, and G,,J, are the respective axial, flexural, and torsional rigidities of the x-stiffener. Minimization of the strain energy with respect to the displacement coefficients yields the stiffness matrix, [K,,], for the x-stiffener

K, I= (19)

where the non-zero elements of [K,] are shown in Appendix 2. The kinetic energy for an x-stiffener due to deflection (w,,), rotation (aw,/ax and dw,,/ay), and axial

deformation (u,,) is

where m, is distributed mass per unit of length: Imy is the mass moment of inertia with respect to the y-axis; and I, is the polar mass moment of inertia with respect to the x-axis. Minimization of the kinetic energy with respect to the time-displacement parameters gives the mass matrix, [M,,], for the x-stiffener

?M -11 0 Mm,, Mm14 Mxs,, Mm 0 Mm,,

0 0 0 000 0

M xs33 Mu34 Mu35 0 Mxr37 Mm,,

[Mm1 = M x&4 M ma 0 Mx,.,, M,

M xs55 0 M.xs57 M.xs5,

0 0 0 sym.

M Mrs,~ .rrn M .x&3

(21)

Vibration analysis of stiffened plates 475

where the non-zero elements of [M,,] are shown in where the non-zero elements of [M,,J are shown in Appendix 3. Appendix 5.

A similar procedure is used for developing the The stiffness and mass matrices of the stiffeners are stiffness and mass matrices for the y-stiffener. A directly applied toward the strip stiffness and mass y-stiffener is assumed here to be attached to the strip matrices, respectively, prior to assembly. and oriented in the longitudinal direction at the local coordinate x,. The strain energy of flexural, torsional, and axial effect then becomes EXAMPLES

Example 1

The simply supported stiffened plate (Fig. 3) of

au 2 ( >I Aksu and Ali [lo] has been analyzed by the proposed

+ E,, A, ys ay

dy, (22) method. The results are presented in Table 1 and put into comparison with those from other previous

where Eys A,, E,,sZ,,s, and G,,J,,* are the respective methods [8, 10, 11, 131. The results obtained by axial, flexural, and torsional rigidities of the y- Long [8] are given in column 1. In-plane displacement stiffener. Minimization of the strain energy with in the stiffening direction is included in this analysis,

respect to the displacement coefficients yields the in-plane inertia is, however, excluded. Results in

stiffness matrix, [K,,J, for the y-stiffener I

WyJ =

00 0 000 0 0’ K Ys= Kysu Kys24 0 Kyszi Kysn ‘ha

K ys33 Kysu 0 0 Kysv Kysss K 0 0 Y* Kys47 Kys,a

00 0 0

sym. K Yfi K &a Yd7

K 437, ys77

K YS88

(23)

where the non-zero elements of [K,,s] are shown in Appendix 4.

The kinetic energy for a y-stiffener due to deflec- tion (w,,), rotation (aw,,/ax and aw,,/ay), and axial deformation (v,,) is

where my is distributed mass per unit of length; Zmy is the polar mass moment of inertia with respect to y-axis; and Z, is the mass moment of inertia with respect to the x-axis. Minimization of the kinetic energy with respect to the time-displacement parameters gives the mass matrix, [My,], for the y-stiffener

r 00 0 0

column 2 are provide-d by Aksu and Ali [IO]. Both in-plane displacement and in-plane inertia are, however, included in this study. Results in other columns are obtained by including in-plane displace- ment and inertia in both directions. The results obtained from the present analysis are adequately compare with those from [l l] but with smaller total degrees of freedom.

Example 2

A stiffened plate with all dimensions identical to those of Example 1, but having different boundary conditions, has been analyzed. The effect of neglect- ing the in-plane displacements has also been studied. T’he resulting natural frequencies are presented in Tables 2 and 3, and put into comparison with those

00 0 0 1 M ys22 My,3 Mya., 0 Mya Mys27 Mya

M ys33 M ys,s, 0 M,,,, Mm37 Mys3s

[My,1 = M Y* 0 Myse Mys47 Mysa

00 0 0’ (25)

C. J. CHEN et af.

I- 203mm*

s

Fig. 3. Centrally stiffened plate.

l.29 11711 tnnl libl

Fig. 4. A double ribbed square plate.

Table 1. Natural frequencies (Hz) of a simply supported plate having a centrally spaced stiffener

Method Stiffness Finite diff. Finite diff. Semi-analytical Finite element Present method 1101 ill] finite diff. [13] [15] 4 sets of analysis

Mode [81 8x limesh 9x 13mcsh 3Omeshx lterm 5xSmesh 7xSmesh

: 224.0 273.6 238.82 267.02 254.94 269.46 231.26 260.32 257.05 272.50 245.17 277.25 3 484.9 483.23 511.64 489.23 524.70 511.42

Table 2. Natural fra@teneics (Hz) of a centrally stiffened C-C-t-S rectangular plate

Mode

In-plane dispIacements coltside&

1 2 3

Semi-analytical Imite diff.

method [13]

347.73 357.41 648.92

Method

Finite element method [lS]

373.41 379.45 717.02

Present analysis

375.76 381.46 696.24

In-plane displacements 357.41 379.45 381.46 n&=md : 382.24 395.30 394.2 1

3 714.90 745.19 700.64

C: Clamped edge. S: Simply supported.

Table 3. Natural frequencies (Hz) of a eentrally stiffened s-s-s-C rectangular plate

Method Semi-analytical finite dilTerenee Finite element

Mode method [I31 method 1151 Present analysis

In-plane displacements I 292.01 298.74 307.60

considered 2 300.18 315.99 317.47 3 570.01 616.68 611.40

In-plane displacements : 292.0 1 298.74 307.60 negtectGd 321.92 329.50 328.28

3 639.29 625.57 649.11

C Clamped edge. S: Simply supported.

Vibration analysis of stiffened plates 477

Table 4. Natural frequencies (Hz) of a double rib&i ckmped square plate

Method Olson and Hazeil[l4] Finite ekment

Mode Experimental Theoretical analysis [Is] Present analysis

1 1204 QOQ 1272.3 965.3 lE 1283.0 918.3

3 1319 1364.3 1396:4 1371.1 4 1506 1418.1 1481.0 1435.5 5 1560 1602.9 1629.9 1592.5 6 1693 1757.1 17M.5 1719.7

: 1807 1962 2015.4 1854.1 1930.9 1979.2 1861.9 1997.6 9 2052 2109.4 2261.1 2055.2

10 2097 2253.1 2115.0 2459.2 11 2410 2453.8 2596.6 2538.2 12 2505 2566.3 2788.3 2554.9 13 2618 2624.2 3024.8 2663.6 14 2631 2729.6 3077.5 3004.1 15 2467 2731.9 2542.1 3102.1 16 2964 2915.4 3329.0 3206.0 17 3169 3180.1 3254.9 18 3135 3242.0

ZE:

3641:5 3255.3

19 3120 3279.1 3418.4 20 3251 3313.3 3687.5 3948.2 ; 3745 3446 3412.3 3635.6 4168.0 3997.8 3967.5

4063.1 4019 4059.4 4364.2 4147.5 4053 4135.1 4175.7 4192.1

Mode

Table 5. Natural frequencies (Hz) of a cross-stiffened plate

Method Unstiffened plate Stiffened plate

Exact solution Finite difference [211 Present analysis method 1111 Present analvsis - _ .

1 30.79 30.76 129.08 139.29 2 76.97 76.95 158.10 163.18 3 123.16 123.12 182.75 175.61

from [13] and [15]. A suitable agreement among the results between the present analysis and finite element analysis [15] is observed. In case of the C-C-C-S plate, the second natural frequency of the plate (when in-plane displacements are considered) is interesting noted here to be identical to the first natural frequency obtained by neglecting them. In case of the S-S-S-C plate, the comparison is also interesting in that neglecting the in-plane displacements does not change the first mode. The phenomena mentioned above have been previously reported and also explained in [ 13,151. Neglecting the in-plane displace- ments of the stiffener is expectedly indicated by the results of present analysis shown in Tables 2 and 3 to increase the natural frequencies of the plate.

Example 3

A double ribbed square plate having all edges clamped (Fig. 4), as previously analyzed and tested by Olson and Hazel1 [14], has been analyzed by the proposed method. The stiffeners (2.29 x 17.8 mm2) have been placed at one third distance from both the edges parallel to the y-axis. The natural frequencies obtained by the proposed method are presented in Table 4 and compared with the results from [14, 151.

A sufficient agreement among the results from the present analysis and 1141 is observed.

Example 4

The cross-stiffened plate system of Aksu [I I] has been analyzed by the proposed method (results are shown in Table 5). This system consists of a 635 mm square plate (2.54mm thick), simply sup- ported on all edges and has two integral stiffeners (12.7 x 22.22 mm’) placed on the axis of symmetry. In column 1 and column 2, the natural frequencies of the unstiffened plate are respectively obtained by the exact solution [2l] and proposed method. Excellent agreement is observed. Columns 3 and 4 show the natural frequencies of the stiffened plate in which in-plane displacements and inertia in both directions are included in the analysis. Results obtained from Aksu [II] and the proposed method are in adequate agreement.

SUIWMARV

A spline compound strip method was developed for the free vibration analysis of the stiffened plate. The presence of the stiffener in a strip was incorpor- ated by an explicit method. The stiffness and mass

478 C. J. CHEN et al.

matrices of the stiffeners were directly added into the plate stiffness and mass matrices at the element level. This substructure has consequently been termed here as the compound strip. The results of numerical examples that were given in this paper were in good agreement with the results from experiment, the finite difference and the finite element methods. However, a significant savings in computer time and data preparation effort has heen achieved by using the proposed procedure whenever the finite difference and finite element procedures have heen compared.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

REFERENCES

S. F. Ney and G. G. Kulkami, On the transverse free vibration of beam-slab type highway bridges. J. Sound Vibr. 21, 249-261 (1972). C. S. Smith, Bending, buckling and vibration of orthotropic plate-beam structures. J. Ship Res. 12, 249-268 ( 1968). C. Qmidvaran, Free vibration of grid stiffened plates. J. Sound Vibr. 19, 463-472 (1971). T. Balendra and N. E. Shanmugam, Free vibration of plates structures by grillage method. J. Sound V&r. 99, 333-350 (1985). T. Mixusawa, T. Kaxita and M. Naruoka, Vibration of stiffened skew plates using B-spline functions. Comput. Struct. 10, 821-826 (1979). R. B. Bhat, Vibrations of panels with nonuniformly spaced stiffeners. J. Sound Vibr 84, 449-452 (1982). B. R. Long, Vibration of eccentrically stiffened plates. Shock. Vibr. Bull., U.S. Naval Res. Lab., Proc. 33, Part I, 45-53 (1968). B. R. Long, A stiffness-type analysis of the vibration of a class of stiffened plates. J. Sound Vibr. 16, 323-335 (1971). Y. K. Lin and B. K. Donaldson, A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels. J. Sound Vibr. 10, 103-143 (1969). G. Aksu and R. Ah. Free vibration analysis of stiffened plates using finite difference method. J. Sound Vibr. 48, 15-25 (1976). G. Aksu, Free vibration analysis of stiffened plates by including the effect of inplane inertia. J. appl. Mech., Trans. ASME 49, 206-212 (1982). M. Mukhopadhyay, Vibration and stability analysis of stiffened plates by semi-analytic finite difference method, Part I: consideration of bending displacements only. J. Sound Vibr. 130, 27-39 (1989). M. Mukhopadhyay, Vibration and stability analysis of stiffened plates by semi-analytic finite difference method, Part II: consideration of bending and axial displacements. J. Sound Vibr. 130, 41-53 (1989). M. D. Olson and C. R. Haxell, Vibration studies on some integral rib-stiffened plates. J. Sound Vibr. 50, 43-61 (1977). A. Mukherjee and M. Mukhopadhyay. Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30, 1303-1317 (1988). R. S. Srinivasan and K. Munaswamy, Dynamic response analysis of stiffened slab bridges. Compur. Slruct. 9, 559566 (1978). L. L. Schumaker, Spline Funcfion: Basic Theory. John Wiley, New York (1981). Y. K. Cheung, S. C. Fan and C. Q. Wu, Spline finite strip in structure analysis. Proc. Int. Conf. on Finite Element Methods, pp. 704-709. Shanghai. Science Press, Beijing and Gordon & Breach, New York (1982).

19. S. C. W. Lau and G. J. Hancock, Buckling of thin flat-walled structures by a spline finite strip method. Thin -walled Struct. 4, 269-294 (1986).

20. H. Antes, Bicubic fundamental splines in plate bending. Int. J. Numer. Meth. Engng 8, 503-511 (1974).

21. C. J. Chen, W. Liu and S. M. Chern, Torsional analysis of shear core structures with openings. Compuf. Strucr. 41, 99-104 (1991).

APPENDIX 1

The inner product of B-spline matrices is allowed here to be represented by

([hlr> LJLI) = s

’ MmlT4.1 dv. 0

The non-zero elements of [M,,] are then expressed as follows:

M,,, = $&,I’~ [&,I)

Mp22 = f([d,ilr, [&,I)

M,,, = $%#U, [&,I) 22

M,,34 = zj$([&lr, [&I)

Mfl = &b2([do,lr~ [&,I)

M,,, = d([eU’, [&,I)

MISS = a([d,lr* [dujl)

M,n26 = @#~,il~~ &,I)

Mp66 = f([d,jl9 [d,I)

Mn,, = %([d,,l*> [&,I)

Mp4, = &b([&,lr> @,I)

Mp,, = !$([d,jlr> [d-j])

M,,,s = $+([&,lr~ [&,I)

Mti8 = ~~2([~o,1r> [&,I)

M,,s = %b(le%,lr~ %,I)

Mpss = &b’(I+0,19 Idojl).

APPENDIX 2

Vibration analysis of stiffened plates 419

480 C. J. CHEN et al.