Response of stiffened plated structures under stochastic excitation

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 71 (1988) 273-292 NORTH-HOLLAND

RESPONSE OF STIFFENED PLATED STRUCTURES UNDER STOCHASTIC EXCITATION

Abhijit MUKHERJEE and Madhujit ~UKHOPADHYAY ~e~~r~rnen~ of Naval Architecture, lndian institute of Technology, Kharagpur 721 302, India

Received 5 October 1986 Revised manuscript received 30 November 1987; 24 May 1988

A finite element algorithm for analysis of stiffened plates by normal mode approach under stationary random stochastic loading has been presented. The formulations for both spatially homogeneous and nonhomogeneous loads have been indicated. The isoparametric quadratic stiffened plate bending element employed has a unique feature that the stiffeners can be positioned anywhere within the plate element and they need not necessarily be on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, transverse shear deformation, and laminated materials. The formulation is applicable to thin as well as thick plates. A simple lumped load concept has been used for spatially homogeneous loading; for spatially nonhomogeneous loads, however, the isoparametric shape function and the Gaussian integration technique have been employed to form the generalized nodal cross spectral density matrix from the excitation. Displacement and stress response spectras for plates and stiffened plates under white noise and jet noise excitation have been evaluated. The results have been presented along with those of previous investigators. The perform- ance of the element has been found to be very good under ali circumstances.

Notation

a

b

PI [Cl

D XF

PI VI

E

m

plate dimension in longitudinai direction/inner radius of circular plate,

cross-sectional area of the stiffener,

plate dimension in transverse direction/ inner radius of circular plate,

strain matrix, damping matrix, flexural rigidity of the plate, Et3/

(12(1 - V”)), rigidity matrix, matrix of shape functions, Young’s modulus /expected

value, force vector,

ii S

ISI sx, SY

system admittance matrix, polar moment of inertia of the

stiffener, Jacobian matrix, Jacobian determinant, stiffness matrix, diagonal entry in mass matrix, shape function, displacement vector in principal

coordinates, arbitrary mode number, correlation between two points, cross spectral density between

two points, cross spectral density matrix, shear rigidities, plate thickness,

00457825/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

274 A. Mukherjee, M. Mukhopadhyay, Stiffened plates under stochastic excitation

u, v, w

V

displacements in the plane of the plate,

displacements of any point in the direction of the orthogonal coordinate system,

lateral deflection of the plate, matrix for eigenvectors, distance of a point in v-direction

on stiffener cross-section, vector of nodal displacements, rotations of the middle surface

normals, shear rotations of the middle sur-

face normals, nondimensional element coordi-

nates, Poisson’s ratio.

w circular frequency, p mean value of a system.

Subscripts

element, force, plate, pressure, arbitrary node, stiffener, global coordinates, displacement field, isoparametric coordinates, stress resultants.

1. Introduction

Many of the engineering structures are subjected to dynamic loads which are random in nature. Examples can be found in case of wave loads acting on ships, submarines, and offshore platforms, wind and earthquake loads acting on a high-rise building, suspension bridges and cooling towers, jet and rocket noise exciting aircraft structures and missiles, and dynamic loads generated on highways and runways due to their unevenness due to passage of vehicles and aeroplanes, respectively. The analysis of these structures in a deterministic sense does not portray their true behaviour. A clearer perspective can be obtained if the problem at hand is treated as stochastic which essentially consists of the definition of the environment (i.e. development of excitation spectra) and basing on it the determination of structural response (i.e., development of response spectra).

Einstein was the first to treat Brownian motion as a random process in 1905. The application of this concept to structural problem was due to Van Lear and Uhlenbeck [l]. The random response of structures has been reviewed by Lin [2], Lin and Donaldson [3], Olson and Lindberg [4], and Yang and Kapania [5].

Research on random vibration of stiffened plates has mainly been reported in the literature on aeronautical engineering. Transfer matrix technique was developed by Lin [2] for the random response analysis. An account of the application of the finite element method for the random response studies of structures before 1972 was given by Olson and Lindberg [8] and Olson [9]. Among the earliest investigators to apply finite element method to random analysis were Jacobs, Lagerquist and Gloyna [6] and Newsom, Fuller and Sherrer [7]. Olson and Lindberg [4, 81 and Olson [9] performed a series of theoretical and experimental investigations on the random response of stiffened panels under jet noise excitation. A triangular plate element with a compatible beam element whose stiffness in bending was infinity was initially used [S]. The formulation was later modified to accommodate the bending of the stiffeners [4].

A. Mukherjee, M. ~akhopadhyay, Stiffened plates under stochastic excitation 275

Their theoretical results showed good agreement with experimental values. Though the finite element method is indeed very versatile for the random analysis of

continuous structures, the main problem associated with it is in performing a set of integrations of the spatially nonhomogeneous excitation spectra to form the nodal force spectral density matrix.

Olson and Lindberg [4] introduced a linear polynomial function of excitation over the element to perform the integration in closed form. Dey [lo] assumed that the same random pressure is acting over the whole contributing area with full correlation. This assumption allowed the lumping of pressure around a node to form a nodal force spectral density matrix. Yang and Kapania [5] employed the displacement shape functions of the elements to form such matrix for spatially nonhomogeneous pressure spectra which freed the analysis from approximation involved in [4, lo].

In this paper a finite element algorithm for random vibration analysis of stiffened plates has been presented. The isoparametric stiffened plate bending element used in the formulation has a unique feature of accommodating the stiffeners anywhere within the element and they need not necessarily be on the nodal lines [ll, 121. The element is isoparametric quadratic which has advantages such as accommodating irregular boundaries, treating laminated materials, and accounting for shear deformation which extends its applicability to thick as well as thin plates. The present approach is more flexible than any other finite element modelling in that the mesh division is independent of the stiffener location.

Straight, curved, skew plates as also stiffened plates have been analyzed for random response. Some of the results are presented for the first time. The nodal force cross spectral density matrix for spatially nonhomogeneous pressure excitation has been generated by Gauss integration in the same way as has been done for stiffness matrix. Comparison of results wherever possible indicated good agreement.

2. Proposed analysis

The equation of dynamic equilibrium for an elastic system undergoing small displacements can be written as

where [MI, [Cl, and [lv] are the overall mass, damping, and stiffness matrices, respectively, and {F(t)} is the time-dependent random force vector. (6) represents the generalized displacement vector. Taking Fourier transform of (2.1) yields

([K] + iw[C] - w*[MJ){6} = {F} , (2.2)

Solution of (6) is obtained by inverting the admittance matrix ([-rU] + iw[C] - u2[M]) at each step of frequency. Though the method is well suited for arbitrary nonsingular damping, it requires a very high computation time. The modal decomposition approach eliminates such inversion and facilitates the computation to a great extent. This necessitates the damping matrix to be proportional which holds well for structural systems. In the present formulation

276 A. Mukherjee, M. Mukhopadhyay,

the modal decomposition approach has been performed using the stiffened plate element tion is discussed very briefly.

Stiffened plates under stochastic excitation

adopted. The solution for free vibration has been developed by the authors [ll, 121. The formula-

2.1. The isoparametric stiffened plate element

The evaluation of fundamental equations of the stiffened plates is based on the following assumptions:

(9 (ii)

(iii)

(iv)

The materials of the plate and the stiffener follow Hooke’s law. The bending deformation follows Mindlin’s hypothesis; therefore, the linear elements perpendicular to the middle plane of the plate before bending remain straight, but not necessarily normal to the middle plane of the plate after bending. The deflection of the points of the middle plane of the plate normal to the plane is small compared to the thickness of the plate. The common normal to the plate and the stiffener before bending remains straight after bending.

(v) The deflection in the z-direction is a function of x and y only.

(vi) The transverse normal stresses are neglected. The stiffness matrix for the stiffened plate is formulated from the contributions of the plate

and the stiffeners separately. The displacements at a node r of a quadratic isoparametric element (Fig. 1) are u,, u,, w,,

O,,, I!$, and at a point within the element

N-

(-1,O) -

‘9 1,l) 2(0,1) 3(1,1) 0 0

8 4(1,0) w

B

0 6 7(-1,-l) 6(0,-l) 5(1,-l)

(2.1.1)

Fig. 1. Isoparametric quadratic plate element. (a) Element coordinate. (b) Isoparametric coordinates.

A. Mukherjee, M. Mukhopadhyay, Stiffened plates under stochastic excitation 277

where [I] is a 5 x 5 identity matrix. IV, is the shape function. The formulation follows Mindlin’s hypothesis and shear strains are expressed as

(2.1.2)

where 0, and $, are the rotations of the midsurface normal (Fig. 2). The field variables for the plate

(;]=[:;q. (2.1.3)

Following the steps detailed in [12] a 40 x 40 stiffness matrix for the plate is obtained in the form

(2.1.4)

where [B] and [D] are the strain and rigidity matrices, respectively. The stiffness matrix for the stiffener is formed assuming the stiffener to be at any location

within the element (Fig. 3). The field variables for the stiffener along x-direction is written as

(2.1.5)

where y’ is the horizontal y distance from the centroidal axis.

.MIDPLANE

NORMAL TO MIDSURFACE

AFTER DEFORMATION I

i

Fig. 2. Deformation of plate cross-section.


I I t --PLATE

f ELEMEN? -_--------------q_--

I c

I

: 0 I I,

I

x”

I

i I

l f ! zx

Fig. 3. Plate element with the stiffener.

The strain displacement relationship of the stiffener is written in terms of the 8 nodes of the plate. Hence the contribution of the stiffener is reflected in all 8 nodes of the plate element which contain the stiffener. The detailed formulation is included in [12]. The case for stiffened plates where eccentricity of the stiffener has been neglected has been given in [ll].

A diagonal lumped mass matrix has been adopted for the plate and the stiffener. In this formulation the entries in all displacement degrees of freedom are proportional to that of the consistent mass matrix. Entries in all rotational degrees of freedom are zeroes. The expression for plate mass at a node Y is expressed as

m ‘P

,

where p is the mass density of the plate. The expression for stiffener mass at a node Y is

(2.1.6)

(2.1.7)

where p, is the mass density and A, is the cross-sectional area of the stiffener. The integrations involved in (2.1.4), (2.1.6), and (2.1.7) have been performed numerically

employing Gaussian integration techniques.

2.2. Solution procedure

The overall stiffness and mass matrices have been stored in a skyline storage form and solved for the free vibration by the simultaneous iteration technique of Corr and Jennings


[13]. The resulting natural frequencies and mode shapes have been used for subsequent response calculations under random loads.

2.3. System admittance matrix

Using the eigenvectors obtained from the free vibration analysis, the modal displacements can be expressed in terms of principal coordinates

(2.3.1)

where [X] is the matrix of eigenvectors and {q} is the matrix of principal coordinates. Substitution of (6) from (2.3.1) into (2.1) yields

(2.3.2)

The stiffness and mass matrices are orthogonal with respect to normal modes and these relations are given by

Pmw[Xl = r1 Gl 7 mwI[Xl = VI , (2.3.3)

where oi is the ith natural frequency and [I] an identity matrix. If we assume [X]‘[ C][X] to be diagonal, each modal equation of expression (2.3.2.) can be decoupled. The expression for the ith mode can be written as

1 4, + ciqi + qi = {Xi}‘{F(f)} .

wi

(2.3.4)

Taking Fourier transform of (2.3.4) for unit impulse applied for all degrees of freedom in succession,

M4) = 2 l ixi>’ (2.3.5)

-w+iwci+l “;

= ZZi(w>{Xi>t 7

where Hi(o) is the complex admittance of the ith mode and can be expressed as

Hi(w) = Of

0; - o2 + 2i5;.owi ’ (2.3.6)

where 5;. is the damping ratio of the ith mode. Using the relationship of (2.3.1), the complex admittance matrix for the whole system can

be written as

IHCo)l = [xlrHi(w)l Lx]’ ’ (2.3.7)


2.4. Cross spectral density matrix for generalized forces

The overall generalized nodal force vectors at an instant t for a distributed expressed as

where I./Ii is the Jacobian and NE indicates the number of elements. The correlation matrix for the generalized nodal forces between two instants,

pressure is

(2.4.1)

(2.4.2)

where E indicates the expected value. Using the relationship in (2.4.1),

NE r-t-

iRFFCtlT ‘211 = F I_/ [NlilJIi i=l C 1 J [NljIJIjE[Pi{51J1> ‘lJPj{527 7727 t211 d5 d77. i=l

(2.4.3)

Since the normal pressure spectra is assumed to be stationary, the correlation function [RFF(tl, t2)] is only a function of the time shift T or (t, - tl).

The cross spectral density matrix for generalized nodal forces can be formulated by taking the Fourier transform of (2.4.3),

(2.4.4)

where [S,,(r, , r2, o)] is the cross spectral density matrix of the normal pressure. Equation (2.4.4) shows that the integration over the whole domain is to be performed twice for the calculation of the generalized nodal force spectra from the generalized normal pressure spectra at each step of frequency. As such, this integration is tedious and a few simplifications have been made to bring down the complexity of the problem and the computation time. Based on the assumption of Thomson [14], the same cross spectral density of pressure acts over the whole contributing area. This assumption simplifies (2.4.4) as

[sFF(0)l = [ Zf? /I lINlilJli dt dT][S,,(w)l[ F \I LNljlJI j d5 d?l] 7 (2.4.5) i=l j=l

or

L%&J)l= [4[~&41M 7 (2.4.6)

where [A] is the diagonal matrix of areas associated with each node. However, for spatially nonhomogeneous spectras, the integration has to be performed by

A. Mukherjee, 119. M~khopadh~ay , Stiflened plates under stochastic excitation 281

Gaussian integration technique without simplifying assumptions such as linear variation of spectra over the element [4], etc.

2.4. Response spectra for displacements and stresses

The cross spectral density matrix for generalized displacements can be obtained from (2.3.7) and (2.4.4) as

K%,w1 = [~(~)l[~,,(~)ll~*~~)l f (2.4.7)

where ]JY*(w>] is the complex conjugate of the matrix [H(w)]. The relationship between the nodal displacements of an element and stresses at any point

within the element is expressed as

Hence the cross correlation matrix for stationary response for stresses between two arbitrary pairs of points rI and Q,

The cross spectral density matrix of generalized stresses can be obtained by Fourier trans- formation of (2.4.9),

(2.4.10)

3. Numerical results and discussion

Various stiffened and unstiffened plates under spatially homogeneous and nonhomogeneous pressure spectra have been analyzed. The normal mode approach has been adopted in all the examples.

3. I. Simply supported square plate under white noise

A simply supported square plate subjected to white noise has been solved by the present method (Fig. 4). The lowest four eigenvalues have been considered. The eigenvectors pertaining to the transverse displacements only have been retained and all other eigenvectors have been ignored for the response calculations, The problem is solved with various mesh divisions to test the convergence characteristics. The power spectral densities of displacement of plate centre for frequency o = 0 and o = 20.0 have been presented in Tables 1 and 2. In Tables 1 and 2, the transverse displacements at the nodes inside the plates are considered as active degrees of freedom. The results have been compared with those of [IO]. It is observed that the present element performs marginally better than the triangular plate bending element (TUBA-3) used in [IO].

282 A. Mukherjee, M. ~~kho~adhyay, Stiffened plates under stochastic excitation

(0)

I =1-o

DXF= 1‘0

,,=0.3

P=l'O

sx=35000

(b)

Fig. 4(a). 4 X 3 mesh division (23 active degrees of freedom). (b) 4 X 4 mesh division (33 active degrees of freedom).

Table 1 Power spectral density for displacement at the plate centre for simply supported square plate at w = 0.0 (Fig. 4)

Degrees Mesh Complex matrix Normal mode Present Analytical of freedom division method [ 101 method [lo] analysis solution [ 1.51

9 4x4 1.355 x 10-S 1.354 x 1o-5 23 4x3 1.5812 x lo-’ 25 6x6 1.502 x lo-’ 1.501 x 1o-5 1.714 x 1o-5 33 4x4 1.6864 x 1o-5 49 8x8 1.559 x lo-” 1.558 x lo-’

Table 2 Power spectral density for displacement at the plate centre for simply supported square plate at w = 20.0 (Fig. 4)

Degrees Mesh Complex matrix Normal mode Present Analytical of freedom division method [lo] method [lo] analysis solution [15]

9 4x4 1.35262 x 1O-3 1.3526 x W3 23 4x3 1.5585 x 1O-3 25 6x6 1.48406 x 1O-3 1.4839 x 1O-3 1.731 x lo-’ 33 4x4 1.6566 x 1o-3 49 8x8 1.5393 x 1o-3 1.5393 x 1o-3

3.2. Rectangular and skew stiffened plates under white noise

In this example a series of rectangular and skew plates with various skew angles have been subjected to white noise. The plates have all simply supported edges. The case of stiffened skew plates with a central stiffener is also considered for all the plates. The properties of the plates and the stiffeners have been described in Fig. 5. The method described in Section 3.1 has been adopted. In each case the lowest five modes have been employed for the solution. In the case of skew plates, however, the whole plate has been modelled due to the absence of symmetry. A 6 x 6 mesh has been adopted for the solution. The natural frequencies obtained from the free vibration analysis have been presented in Table 3. It may be noted that the natural frequencies of the stiffened plate are higher than the bare plate except for the second


c- 1'0 -I t = 1'0 DXF = 1.0

2, = 0‘3

P= 1'0 s,= 35000

A,= 0.1

I,= 10’0

J,= 2’5

Fig. 5. A skew plate with a stiffener (rp is the skew angle).

Table 3 Natural frequencies in rad/s for simply supported plates (Fig. 5)

Plate without stiffener Plate with stiffener

Skew angle Skew angle

Mode 0” 30” 45”

1 19.732 25.340 37.085 2 98.568 52.940 67.300 3 98.568 73.161 105.261 4 179.312 86.562 111.197 5 256.682 124.499 158.493

0 30” 45”

37.053 41.340 54.150 96.353 66.273 91.225

135.113 99.091 128.148 238.704 99.464 137.342 240.425 125.903 168.290

a Symmetric modes only.

and fifth modes of the rectangular plate. The mode shapes in the second and fifth modes for both rectangular stiffened and unstiffened plates are similar. Whether this discrepancy for the case of the rectangular plate is due to the arbitrary selection of the stiffener properties cannot be explained for certain. The displacement spectral densities for different frequencies at the points shown in Fig. 5 have been presented in Table 4. It is observed that the value, of the spectral density of the displacement at o = 0 decreases with increasing skew angles.

However, this trend is not observed for o = o, or o = w1 as expected, since the value of w1 is different for plates having different skew angles. Further, Table 4 indicates that the peak is very sharp at w = o, as against w = wl. This is due to resonance at the fundamental mode. This is, however, expected as the plate becomes stiffer with the increase of skew angle. As expected the displacement spectral density values for the stiffened plates have also been observed to be lower than the plates without the stiffener. In the case of plates without


Table 4 Displacement spectral densities for plates (Fig. 4)

Point

Plates without stiffener Plates with stiffener

Skew angle Skew angle

0” 30” 45” 0” 30” 45”

A o = 0.0 1.2522 x lO-5 4.6084 x lO-‘j 9.7026 x lo-’ 0 = w, 1.3853 x 10” 2.9309 x lo5 w = 6J1 0 = 25.0 w = 37.0

0.001119 0.04612

B w = 0.0 1.6864 x lo-* 6.2137 x 1O-6 w = ut 1.9166 x 1o’O 0 = w, w = 20.0 0 = 25.0

0.001613 0.001532

1.3407 x 1o-6 4.2769 x lo5

w = 37.0 0.06729

7.2415 x lo-’ 3.7869 x lo6

0 = 37.0 0.01539

6.8327 x lo-’ 3.3270 x lo6

6J = 37.0 0.09808

4.9899 x 1O-7 7.8923 x 10’

w = 41.0 0.001812

4.9023 x lo-’ 7.2362 x lo9

0 = 41.0 0.002022

1.4231 x lo-’ 2.0955 x lo8

w = 54.0 0.005680

1.6287 x lo-’ 1.8232 x 10’

w = 54.0 0.004452

stiffeners the values at the plate centre (point B) are larger than the values at point A. In the case of stiffened plates, because the stiffener passes through the point B, the cross spectral densities of displacement at A were higher.

3.3. Curved plates under white noise

The element introduced here has the ability to accommodate irregular boundaries. This phenomenon has been employed for the solution of random vibration problems of curved plates. The plate problem solved here is illustrated in Fig. 6. The annular sector plate has two

b = 2-O 5x= 35000

a= 1'0 A,= O,.I t = 1'0 I s=O.O416?

DxF=I'O J,= 0'04167

p=r.o

Fig. 6. A curved plate with stiffeners.


Table 5 Natural frequencies in rad/s for simply supported curved plates (Fig. 6)

Mode Plate without stiffener Plate with stiffener

1 5.221 10.5976 2 16.212 18.7175 3 32.710 36.0705 4 43.149 63.0357 5 62.780 66.8646

Table 6 Displacement spectral densities for curved plates (Fig. 6)

Location Plate without Plate with

w stiffener stiffener

A 0.0 2.624 x 1O-3 5.5671 x lo-’ 6J = w, w = 5.0 w = 11.0

0.4078 0.01135 w1 3.502 x 10’ 3.9562 x 10’

0.0 7.4718 x 10m4 6.2873 x lo-’ w f 0, w = 5.0 0 = 11.0

0.101 0.00955 WI 8.527 x 10’ 3.4230 x 10’

c 0.0 1.3746 x 1O-4 4.4443 x 1om5 w = WI 6J = 5.0 0 = 11.0

0.1388 0.00584 Wl 9.302 x 10’ 2.1363 x 10’

radial edges simply supported while the other edges are free. Due to the symmetry along the central radial line only one half of the plate was modelled. The five lowest modes have been considered and the eigenvalues for those modes have been presented in Table 5. The values of displacement cross spectral densities obtained at several points on the plate under white noise have been presented in Table 6. It may be noted that in case of bare plates the response at w = 0 decreases gradually from A to C, whereas, at w = o1 and w = ol, the response at C is more than that of B. The mode shape at fundamental frequency (ol) indicates a higher amplitude at C than B. This has been reflected in the displacement spectral densities,

3.4. Five-bay integrally stiffened clamped plate under jet noise

The five-bay integrally stiffened clamped plate analyzed by Olson and Lindberg [4] is considered in this example. The details of the plate and loading have been given in Fig. 7. The output of the eigenvalue analysis, the natural frequencies, have been presented in Table 7. The lowest 14 natural frequencies have been considered. The values agree very well with the theoretical and experimental results of Olson and Lindberg [4]. For the calculation of natural frequencies one quarter of the plate has been modelled with a 2 x 5 mesh. The conditions of symmetry and antisymmetry have been imposed along the two centre lines of the plate in four consecutive runs to accommodate all four possible types of modes for the full plate. The number of modes considered in each run has been indicated in Table 8. The eigenvectors for


J. ,,,,,,,,,, / ,,,,,,,,, / ,,,,

)I

, .--t_._.--.* ._._ ~I - * .-_e._--__--. 1 2 CQ

RB 3

/

/////////////////L ,//,r X

L STIFFENER 0.25*x 0.5* 211’ _1

@pp(cx,Y,W) = 4500/A (45002 + 80002+ ca2 1

[45002+~8000-w?][150d+~8000+4je

-[o-oslii/-o~o37/~c~~]

t = b.05” E = 107 psi

p = O-000262 lb sec21in4

y= 0.3

Fig. 7. Five-bay integrally stiffened panel.

Table 7 Natural frequencies in Hz for five-bay integrally stiffened panel

Mode

Symmetry Present

x Y Experiment [4] Theory [4] analysis

1 s S 609 623.5 619.8 2 A S 634 630.98 629.36 3 S s 6.51 638.65 629.60 4 A S 669 673.78 659.13 5 S S 682 673.79 660.00 6 S A 897 915.47 898.9 7 A A 910 920.53 901.7 8 S A 917 927.71 905.4 9 A A 928 935.04 910.6

10 S A 945 935.35 910.8 11 A S 1175 1160.8 1160.8 12 S S 1245 1210.5 1263.2 13 A S 1330 1288.4 1270.6 14 S S 1429 1394.6 1265.5

Table 8 Number of nodes considered in the free vibration of tive-bav stiffened panei

Boundary condition

x Y Number of modes

symmetry symmetry antisymmetry antisymmetry

symmetry antisymmetry symmetry antisymmetry


the quarter plate are then expanded for the full plate and used as the input to subsequent response analysis. If only the cross spectral density function for transverse displacements is of interest, the eigenvectors pertaining to transverse displacement may be retained ignoring others. If however the spectra for element stresses are also required, all the eigenvectors are to be considered. The pressure spectral density considered for the problem (jet noise) is spatially nonhomogeneous. Integrations as stated in (2.4.4) have been performed to generate the cross spectral density matrix for nodal forces. The integrations if performed at each frequency step consume a considerable amount of computer time. Instead, the whole range of frequencies can be divided into a few groups. The force spectra is calculated for one representative frequency in each group and kept constant over all the frequencies in the group. In this problem it was possible to classify groups of five frequencies as the plate consists of five equal bays. The central frequency of each group has been taken for the calculation of the force spectra [4]. Following similar steps, the force spectra has been calculated at frequencies of 650 Hz, 925 Hz, and 1300 Hz. To confirm that no significant error is introduced through such approximation, an analysis in a smaller scale was performed with calculation of force spectra at each step. No significant improvement was observed. However, such approximation was possible due to the nature of pressure spectra and structural configuration of the problem and cannot be adopted for a general case.

The displacement spectras for the centres of the bays 1, 2, and 3 have been presented in Figs. 8-10 along with the theoretical and experimental curves of [4]. The agreement among the curves is good up to a frequency of 1000 Hz. Though the peak at a frequency of 650 Hz is predicted accurately, the analysis has failed to predict the second peak at 1350 Hz. The failure is due to the consideration of only 14 modes. Olson and Lindberg [4] have considered 80

LEGENL,

PRESENT ANALYSIS

----- THEORETICAL [4] ‘,_:,:‘.:’ ,-,~,.:i(~~:~ ~i:~~,~~~j:i~~~::.~~~_ E X P E R I M E N T A L [ 4 ]

7.3X104 c I

I

1000 2000

FREQUENCY- HERTZ

Fig. 8. Displacement spectra at centre of bay 1.

I 3000

288 A. Mukherjee, M. Mukhopadhyay , Stiffened plates under stochastic excitation

PRESENT ANALYSIS

THEORETICAL [4]

EXPERIMENTAL [4]

0 I I

1000 2000

FREQUENCY- HERTZ


I 3000

7ak-4 -

LEGEND

PRESENT ANALYSIS ----- THEORETICAL [4] ‘+.‘,:,:- s..‘:.’ EX,,ER,MENTAL [4] _L,. ,c:,. . ...’ ,i:;:,:

N .- u-l a

r?

(Y c .-

--& Lo

u-l

7’3 xd2 - ..i

I I I

0 1000 2000 3000

FREQUE NCV- HERTZ



8.5x107

N ._

z. g 8.5~10’

LEGEND

PRESENT ANALYSIS ----- THEORETICAL [4] ;?$‘$$‘r:+~:.~ .! .,+ ;<,,:: ,,,;~;t,:w::.!‘- E X P E RI M E N TA L [ 41

-’ 8’5x10 0 1000 2000

FREQUENCY-HER12

Fig. 11. Stress spectra at A (Fig. 7).

________ -_. -._- 3000

LEGEND

- PRESENT ANALYSIS

----- THEORETICAL [4]

~~~~~~~~~ EXPERIMENTAL [4]

I- I

1000 2000

FREQUENCY-HERTZ

Fig. 12. Stress spectra at B (Fig. 7).

I 3000

290 A. ~ukherjee, hf. ~ukhopud~ya~, Stj~e~ed plates under stochastic excitation

modes and their prediction agreed well with experiment up to 2OOOHz. However, the effectiveness of the method is demonstrated through the accurate prediction of the first peak.

The cross spectral densities of stresses at different points have been presented in Figs. 11 and 12. They also showed similar nature as in the case of displacements.

4. Computer program

The problem discussed here is very involved computationally and a careful programming is

necessary to keep the solution efficient. The computer program developed by the authors for the solution of structural dynamics problems under spatially nonhomogeneous pressure spectra

(i> (ii)

(iii)

(iv) (4

has the following features. Automatic mesh generation. Generation of element stiffness and mass matrices for the plate and the stiffeners. Assembly of overall stiffness and mass matrices in skyline form. Fitting of the boundary conditions. Solution for eigenvalues and eigenvectors and storing them in low-speed magnetic memory.

(4

(y

(T? 1x

(4

(xi)

(xii)

Expanding the eigenvectors for the full structure considering symmetry and antisymmetry if necessary. Orthonormalizing the eigenvectors. Generation of modal admittance matrix. Generation of system admittance matrix and storing them in low-speed magnetic memory. Generation of nodal force spectral density matrix from pressure spectra and storing it in low-speed magnetic memory. Generation of nodal displacement spectra from nodal force spectra and system admittance matrix. Generation of stress spectra from nodal displacement spectra.

It is to be noted that the cross spectral density matrix of nodal forces and the system admittance matrix are symmetrically populated square matrices of the same order as the total

Table 9 CPU time requirements in different examples

Structure Mesh

division Analvsis CPU time in s

Skew plate Curved plate Five-bay stiffened plate

6x6

3x3

2x5

4x 10

eigenvaiue analysis displacement spectra eigenvalue analysis

displacement spectra eigenvalue analysis

expansion 10.8 displacement spectra 290.0

stress spectra 1200.0

405.0 38.0

170.0 5.0

450.0 (4 runs)


degrees of freedom of the system. Hence it is not practicable to store them in the computer core memory, instead a low-speed storage device (disk or tape) can be used. The output from free vibration run can also be stored in auxilliary storage to introduce the restart feature. The program was run in a Burroughs B 6700 computer and the CPU time requirements at different stages have been indicated in Table 9.

5. Conclusions

An isoparametric stiffened plate bending element has been employed for the random vibration analysis of stiffened plates. The element can accommodate irregular boundaries and laminated materials. The formulation considers shear deformation and is applicable to the analysis of both thick and thin plates. The formulations for both spatially homogeneous and nonhomogeneous pressure spectra have been presented. Several examples of plates under white noise and jet noise have been considered.

A detailed study of an integrally stiffened plate under jet noise has been undertaken. The results reveal the importance of higher modes to accurately predict the responses at high frequencies.

Acknowledgment

This work has been supported financially by the Ministry of Transport, Department of Surface Transport (Shipping Wing), Government of India. All the computer programs were developed and run on the Burroughs B-6700 computer of Regional Computer Centre, Calcutta.

References

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[2] Y.K. Lin, Probabilistic Theory of Structural Dynamics (Robert E. Crieger Publishing Co., Huntington, NY, 1976).

[3] Y.K. Lin and B.K. Donaldson, A brief survey of transfer matrix technique, J. Sound Vibration 10 (1969) 103-143.

[4] M.D. Olson and G.M. Lindberg, Free vibrations and random response of an integrally stiffened panel, Aeronautical Rept. LR-544, National Research Council, Canada, 1970.

[6] T.Y. Yang and R.K. Kapania, Finite element random response analysis of cooling towers, AXE J. Engrg. Mech. 110 (1984) 589-609.

[6] L.D. Jacobs, D.R. Lagerquist and F.L. Gloyna, Response of complex structures to turbulent boundary layers, AIAA Paper No. 69-20, 1969.

[7] CD. Newson, J.R. Fuller and R.E. Sherrer, A finite element approach for the analysis of randomly excited complex elastic structures, AIAA/ASME Structures Meeting, Palm Springs, CA, 1967.

[8] G.M. Lindberg and M.D. Olson, Vibration modes and random response of a multi-bay panel system using finite elements, Aeronautical Rept. LR-492, National Research Council, Canada, 1967.

292 A. Mukherjee, M. Mukhopadhyuy, Stiffened plates under stochastic excitation

[9] M.D. Oison. A consistent finite element method for random response problems, Comput. & Structures 2 (1972) 163-180.

[lo] S.S. Dey, Finite element method for random response of structures due to stochastic excitation, Comput. Meths. Appl. Mech. Engrg. 20 (1979) 173-194.

[ll] A. Mukherjee and M. Mukhopadhyay, Finite element free vibration of stiffened plates, Aeronautical J. (1986) 29’7-306.

[12] A. Mukherjee and M. Mukhopadhyay, Vibration characteristics of ho~zontally curved stiffened plates, Rept. No. MTSi8514, Department of Naval Architecture, Indian Institute of Technology, Kharagpur, India, 1985.

[13] R.B. Corr and E. Jennings, A simultaneous iteration algorithm for solution of symmetric eigenvalue problem, Internat. J. Numer. Meths. Engrg. 10 (1976) 647-663.

[14] W.T. Thomson, Continuous structures excited by correlated random forces, Internat. J. Mech. Sci. 4 (1962) 109-114.

[1_5] S.S. Dey, Finite eiement method for random response of structures due to stochastic excitation, ISD Rept. 221 (1973) 297-314.

Response of stiffened plated structures under stochastic excitation

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