g. s. Palani, n. r. Iyer and t. v. s. r. Appa Rao - Finite Element Analysis of Eccentrically

Compurers & S~rucrures Vol. 43, No. 4, pp. 651-561. 1992 004%7949/92 $5.00 + 0.00 Printed in Great Britain. 0 1992 Pergamon Press Ltd

AN EFFICIENT FINITE ELEMENT MODEL FOR STATIC AND VIBRATION ANALYSIS OF ECCENTRICALLY

STIFFENED PLATES/SHELLS

G. S. PALANI, N. R. IYER and T. V. S. R. APPA RAO

Structural Engineering Research Centre, Council of Scientific and Industrial Research, Taramani, Madras-600 113, India

(Received 30 April 1991)

Abstract-Two isoparametric finite element models (QSSSI and QL9Sl) for static and vibration analysis of plates/shells with eccentric stiffeners have been proposed. The eight-noded QS8Sl and the nine-noded QL9Sl models have been arrived at by appropriately combining serendipity or Lagrangian plate/shell elements with the three-noded isoparametric beam element employing suitable transformations for eccentricity. Transverse shear deformations are included in the formulation making the models applicable for both moderately thick and thin plates. Numerical studies have been conducted for static and vibration analysis of concentrically and eccentrically stiffened plates to determine the efficacy of the proposed models. Vibration analysis has been carried out using four mass lumping schemes. The proposed QL9Sl model with consistent diagonal mass lumping scheme has been found to be more efficient than the QS8Sl model and the other models available in the literature for static and vibration analysis (in the higher frequency/modes range) of plates/shells with eccentric stiffeners.

1. INTRODUCTION

Many industrial structures such as those used for aerospace, marine and offshore applications are, gen- erally, made up of stiffened plate/shell panels. The design of these structures involves detailed analysis for static and dynamic responses. In order to model these complex structures, it is necessary to identify and adopt suitable analytical/numerical methods which will be reliable as well as economical in representing the structural behaviour of the stiffened plate/shell panel components. It may also be noted that in cases of dynamic response analysis of ship hull structures subjected to propeller-induced excitations, frequencies for higher modes of vibration have to be computed. Therefore, the methodology of analysis and the finite element model chosen must also guar- antee satisfactory performance in this domain. A number of analytical/numerical models have been proposed in the literature. Among these, the performance of simple models like the grillage model [ 1,2], and orthotropic model [3,4] are not satisfactory for solving generalized stiffened plate/shell problems. A more accurate model is achieved by representing the plate and stiffeners separately and by main- taining compatibility between the two. Different analytical methods and numerical techniques such as the Rayleigh-Ritz method [S], finite differences method [6], constraint method of analysis [7], semi- analytic finite difference method [8,9], transfer matrix, finite strip, wave approaches [lO-161, and finite element method [ 17-22,3 l-341, have been used to solve the problem. A more complete review of the static and dynamic behaviour of stiffened plates can

be seen in [23-251. Among all the numerical methods, the finite element method has been found to be reasonably accurate with less complexity [17-22, 31-341 to model stiffened panels. The finite element models used in [17-221 employ plane stress elements in the formulation and therefore, obviously cannot represent the true bending behaviour of stiffened plates. Hence, the basic idea of the stiffened element was further extended to produce isoparametric stiffened plate finite element models [21,22] based on Mindlin’s theory [26] which accounts for transverse shear deformations. The effect of each stiffener and the associated generalized strains were expressed [21,22] by using the plane element (eight- noded serendipity) shape functions and their derivatives.

Gupta and Ma [27] and Balmer [28] investigated the use of linear shape functions for approximating axial displacement in beam finite elements, used as eccentric stiffeners with linear transformation of the nodal displacements. They found that this leads to an inconsistency which can result in large errors in deflections and stresses of stiffened plates. Miller [29] introduced additional degrees of freedom (DOF) for these elements to circumvent the problem. Miller also found that the inconsistency introduced by using the first degree polynomial representation for axial dis- placment in beam finite element leads to rather large errors in the prediction of dynamic behaviour of stiffened plates [30] and the added DOF reduces this error considerably. Deb and Booton [31], in their model FEM(Ml), overcame the problem of inconsistency by assuming quadratic shape functions for all displacements of the stiffeners.

651

652 G. S. PALANI et al.

Recently, Deb and Booton [31], Thompson et these four mass lumping schemes. Though there is no al. [32] and Satsangi and Mukhopadhyay [33] used an significant difference in the frequencies obtained by eight-noded model for static analysis of eccentrically using all these four mass lumping schemes for the first stiffened plates. Mukherjee and Mukhopadhyay [34] few modes (up to three), the consistent diagonal extended this eight-noded model for vibration analy- lumping scheme performs better than the others for sis of eccentrically stiffened plates. All these models higher modes. It has also been observed that the use eight-noded serendipity elements as the plate performance of the QL9Sl model is much better element and assume that the stiffeners follow the than the QS8Sl model, especially for higher modes of same displacement field as that of the plating. Use of vibration. Moreover, the QS8Sl model has been plate element shape functions for stiffeners involves found to be stiffer for stiffened plates with more large computational effort. Whereas the use of beam number of constraints (like clamped boundary con- element shape functions for the same is not only a ditions), resulting in higher frequencies. The present simplified approach but also requires lesser compu- investigation considers the stiffeners to be located tational effort. Deb and Booton [31] used this ap- along plate/shell element boundaries. Since the per- proach to derive the FEM(MI) model (three-noded formance of QL9Sl model with consistent diagonal stiffener element combined with eight-noded plate mass lumping scheme is superior even for higher element) for static analysis of eccentrically stiffened modes of vibration than others, this model is rec- plates and showed that this approach gives quite ommended not only for general use, but also for good results. dynamic response analysis of complex structures.

However, the eight-noded plate element mentioned above has been found to lock in shear for thin plates for certain boundary conditions and mesh configur- ations even when reduced integration (RI) [35] and selective reduced integration (SRI) techniques [36,37] are used. In addition, it has also been reported earlier [36], that this element is comparably more sensitive to element shape distortions and element aspect ratios. Hence, the application of this element for analysis of complex structures is questionable. Whereas the performance of the nine-noded Lagran- gian plate/shell element with either RI or SRI has been proved to be near optimal [35,36] even for very thin plates. Hence, a need was felt to develop a stiffened plate/shell element model using a nine- noded plate/shell element and a three-noded isoparametric stiffener element. This paper presents the development of finite element model and the studies conducted to evaluate its performance in static and vibration analysis of eccentrically stiffened plates/shells. The QL9Sl model is based on a nine- noded plate/shell element and a three-noded isoparametric stiffener element. A model (QSSSl), similar to that of Deb and Booton’s FEM(M1) model [31], based on an eight-noded plate/shell element and a three-noded isoparametric stiffener element was also formulated and used in the studies conducted. The above models are based on Mindlin’s theory which enables these models to be used for both very thin and moderately thick plates. Since the elements have been formulated using the isoparametric concept they can be used in the analysis of complex structures with irregular boundaries. Performance studies of the two models, QL9Sl and QS8S1, for static and vibration analysis of stiffened plates with different boundary conditions have been carried out. Four mass lumping schemes namely, (i) consistent mass matrix (CMM), (ii) consistent diagonal lumping (CDL), (iii) proportional lumping (PL), and (iv) equal lumping (EL) have been used to derive element mass matrices. Vibration studies have been conducted by using all

2. FORMULATION

The equation of equilibrium of an elastic system in motion, undergoing small displacements can be writ- ten in matrix form as

WI VI + [Cl{S I+ Pa6 1 = {W)}. (14

To compute the natural frequencies of the system by considering the eigenvalue problem, the above equation may be simplified as (neglecting damping effects)

WI - 4v){~ 1 = PI? (lb)

where w = natural frequency in rad/sec and {A 1 = corresponding modal vector or eigenvector.

In eqns (la) and (lb) [Ml, [C] and [K] are the overall mass, damping and stiffness matrices, respectively and {a}, {s} and (6) are the generalized acceleration, velocity and displacement vectors, respectively. {F(t)} is the generalized force vector. The overall matrices can be formed by assembling the element matrices by using well-known pro- cedures [38].

The formulation of element stiffness and mass matrices for stiffened plate/shell models are discussed below.

2.1. QL9Sl model

This model has been derived by combining the nine-noded Lagrangian element (Fig. la) with the three-noded isoparametric stiffener element employing suitable transformation for eccentricity.

2.1 .l. Assumptions. In addition to the assumptions for Mindlin plates[26], the following assumptions are also made in the formulation:

1. The common normal to the plate and the stiffener before bending remains straight after bending.

Static and vibration analysis of stiffened plates/shells 653

2. Stiffener section is symmetric about the middle plane of the web.

3. Bending of stiffeners in the plane of the plate is negligible.

4. Stiffeners are located only along the boundaries of plate elements.

2.1.2. Plate/shell and sttflener element formulation. The element stiffness and mass matrices of stiffened plate/shell models consist of the contri- bution of plate/shell elements and that of stiffener elements. Detailed formulation of element stiffness and mass matrices for eight- and nine-noded plate/shell and three-noded beam elements have been presented in [36]. Only a brief account of the formulation of the element stiffness and mass matrices of plate and stiffener elements and the transformation required to take into account the eccentricity of the stiffeners is presented in the following.

2.1.2.1. Element st@ness and transformation matrices. The subscriptp, xs and ys, in the following, respresent plate, X-stiffener (parallel to 5 axis) and Y-stiffener (parallel to q axis), respectively. Whereas the subscripts ‘,x’ and ‘,y’ denote the partial derivative of the function with respect to x and y, respectively.

(a)

Parent element

The displacement field for the plate elements (Fig. 1) (with five DOF at each node) can be expressed as

where nP is the number of nodes for each plate element (Fig. l), i.e. eight or nine, and N(<, n)i are the element shape functions [36, 381 and I, is a 5 x 5 identity matrix.

The generalized strain-displacement relationship for the plate elements can be expressed as

tp = % Bidpi = BpSp, (3) i-l

Parent element

Fig. 1. Eight- and nine-noded elements: (a) nine-noded Lagrangian element; (b) eight-noded serendipity element.


and the generalized stress-strain relationship for the nodal DOF can be transformed to plate nodal DOF, plate elements can be expressed as i.e.

aP = D,c, = D,B$,. (4) a,, = T,, 6, and 6,, = T$,, . (10)

The displacement field for a three-noded X-stiffener Using the principle of virtual work, the equilibrium

element (with four DOF at each node) can be equations can be expressed as

expressed as NXS

r KS 1 iJc;crp dA + 1

s ac~cr,, dx

I

where N(r) are the element shape functions [36,38] and I., is a 4 x 4 identity matrix.

Substituting eqns (3), (4), (6), (7) and (10) into (11),

The generalized strain-displacement and stress- the equilibrium equations can be expressed in terms

strain relations for the X-stiffener elements can be of plate nodal displacement variables as

expressed as

where

25 &S,‘[B,TD,B,]s, dA

(6) A NW

+ 1 as%[T~B~D,B,,T,,IG, dx (7)

I I

N,C +C

s asp’[T,‘,B,‘,D,,,B,,T,,16, dy

I

ct, = {%x -hLY -qwu?~.rs- WYA. N =

f.r adpTN(L v)*q dA + ff a~;N(t, ~1% (12)

In a similar way, the generalized strain-displacement A

(B,.,) and stress-strain (D,,) relations for Y-stiffener N NV, elements can also be expressed. The elements of the * aS;[K,]6, dA + c ~~;[K,,]cf, dx

matrices BP, DP, B,,, D,, B,., and DyJ are given fs A

explicitly in [39]. The stiffener nodal DOF can be transformed to the +5

1 as%]K,,]d, dy

plate nodal DOF using the relationship as shown I

below. For an X-stiffener (Fig. 2) N =

fs a~;W, rtYq dA

= up - h kp A

U,,

(13)

Expressing in matrix form

(8) where [K,] = j,BiD,B, dA = plate element stiffness matrix, [K,,] = T,S[j, Bz>D,B, dx]T,, = X-stiffener element stiffness matrix, [K,,,] = T.j[i,B:D,,B,, dy] T,., = Y-stiffener element stiffness matrix, q = uni-

Similarly, the transformation matrix Tys for Y- form loading acting over the element, p = concen- stiffener elements can be derived. Hence stiffener trated loads or moments and N,, N,, and NyS are

Static and vibration analysis of stiffened plates/shells 65.5

Plate mid-surface 1

Neutrat axls of ’ stiffener

PLate mid-surface

I Neutral axls of stiffener

Sectlon A- A

Fig. 2. Plate with an eccentric X-stiffener.

the number of plate, X- and Y-stiffener elements, respectively.

The stiffness matrices for the plate elements have been evaluated using the SRI technique consisting of a 3 x 3 Gauss rule for bending and a 2 x 2 Gauss rule for shear. While RI technique of 2 x 2 Gauss rule for both bending and shear has been used to evaluate the element stiffness matrices in case of stiffener elements.

2.1.2.2. Element mass matrix. Four different mass lumping schemes have been used to conduct the vibration analysis. These are (i) consistent mass matrix (CMM), (ii) consistent diagonal lumping (CDL), (iii) proportional lumping (PL), and (iv) equal lumping (EL). These mass lumping schemes have been used for the plate and the stiffener elements to compute element mass matrices. Among these schemes CMM has been derived by assuming the variation of acceleration field over an element to be the same as the variation of displacement field over that element. Hence, the shape functions that describe the displacement field over the element have been used in deriving the kinetic energy and the element mass matrices. In the CDL scheme there are only diagonal elements and they are obtained by scaling the diagonal entries of CMM, and the total mass of the element is conserved. Proportional and equal lumping schemes are based on physical lumping of

the mass to the element nodes. Hence, the mass matrices obtained by PL and EL schemes are also diagonal matrices. A detailed formulation of all these schemes is presented in [36]. In the diagonal lumping schemes, the effect of rotary inertia has not been considered. The transformation matrices given in eqn (10) can be used when rotary inertia effects are to be included. However, this will result in a non-diagonal mass matrix. Approximate methods as discussed by Surana [40] can be used to diagonalize the mass matrix. This has not been attempted in the present investigation, as this leads to more computational effort with little improvement in accuracy.

2.2. QS8Sl model

This model has been formulated by combining the eight-noded serendipity element (Fig. 1 b) with the three-noded isoparametric stiffener element employing suitable transformation for eccentricity. The pro- cedures discussed in Sec. 2.1 are also used for this model to formulate the element stiffness and mass matrices. The procedure to formulate the stiffness and mass matrices for this model is same as that used for the model QL9S1, except that the shape functions of eight-noded serendipity element have to be used in evaluating the strain-displacement matrix [eqns (2) and (3)] and the element mass matrix of the plate element.


2.3. Extraction of eigenvalues

Equation (lb) is solved for the eigenvalues or natural frequencies (o in rad/sec) and for the corresponding eigenvectors or mode shapes {A}, by using the sub-space iteration technique[41]. A Sturm se- quence check has been performed to ensure that no eigenvalues are missed. The overall stiffness and mass matrices are assembled in banded form. The com- puter program in [41], for matrices stored in sky-line form, has been suitably modified to take banded matrices as input.

3. NUMERICAL EXAMPLES w - deflectton x IO’ In

Numerical studies have been conducted for both static and vibration analysis of plates with concentric/eccentric stiffeners to validate and to determine the efficacy of the finite element models (QSSSl and QL9Sl). Two example problems of static analysis and four examples problems of vibration analysis have been solved for which the results are available in the literature.

3.1. Static analysis

4

3 w

2

I

C

I’ L---J

a 1

u ,x ,4:,r;:-+‘~\ - / + q-+-t-f

t) a r, +yZ.5 In

I I 0 125 0 25 0 375

Example 3.1.1. A simply supported square plate with a centrally placed X-stiffener, as shown in Fig. 3, is analysed with 2 x 2 and 4 x 4 meshes considering quarter plate. Deflections along the centre lines of the plate are computed by using the proposed QS8Sl and QL9Sl models. The results are compared with those of [7] which were obtained by using constraint method analysis (CMA), for concentrically and eccentrically stiffened plates. These are shown in Fig. 4. The results obtained by the present analysis for both 2 x 2 and 4 x 4 meshes compare well with those reported in [7] in spite of the fact that the number of DOF for the present analysis (for 2 x 2 mesh) are less than those used by CMA of [7]. It may also be noted

Distance along centre he (In)

(b)

Fig. 4. Variation of deflection along the centre line for the case of stiffener as (a) concentric and (b) eccentric. - QL9Sl (4 x 4 mesh); +QSfBl (4 x 4 mesh); 0 QL9Sl

(2 x 2 mesh); 0 QS8Sl (2 x 2 mesh); 0 CMA (7).

that CMA uses a higher order polynomial than that used in the present analysis.

Example 3.1.2. A simply supported rectangular plate with centrally placed stiffeners, one in each direction as shown in Fig. 5, is analysed using 2 x 2 and 4 x 4 meshes considering quarter plate. Vari- ations of deflection along two sections of the plate are shown in Fig. 6. The results of the present analysis for both 2 x 2 and 4 x 4 meshes agree well with those obtained by using CMA in [7]. It may be noted that as in example 3.1.1, in this case also the number of DOF and the order of polynomial used in the present analysis are lesser than those used by CMA in [7].

k---IIn------- Fig. 3. Finite element mesh for static analysis of a simply supported square plate with a central X-stiffener. Plate and stiffener details: E = 17 x lo6 psi; Y = 0.3; a = b = 1.0 in; t = 0.01 in; udl = 1.0 psi; X-stiffener (S,T)

size = 0.01 x 0.1 in.

6r ty

0 0 125 0 25 0 375 0.5

Distance along centre line (In)

(a)

05

3.2. Vibration analysis Example 3.2.1. A clamped square plate with a

centrally placed Y-stiffener (Fig. 7) is analysed for free vibration. This problem was analysed earlier by Nair and Rao [42] using a high precision element STIFPTI, a lower order element STIFPT2, SAP IV and ASKA. Mukhopadhyay[g 91 used the semi- analytic finite difference method (SFDM) to analyse the same problem. Frequencies upto first six modes are obtained using the present finite element models for stiffened plates with concentric and eccentric stiffeners. The results are compared in Fig. 8. It can be seen from Fig. 8(a) that the frequencies obtained by using QL9Sl are in close agreement to those of


the high precision element STIFP’Tl. Frequencies obtained by using QSSSl are found to be higher. This may be due to the locking of the eight-noded plate element for clamped boundary conditions as reported in [36]. Results of the same analysis with four mass lumping schemes are presented in Table 1. It can be observed from Table 1 that all the four mass lumping schemes give frequencies of the same order for the first few modes, whereas for higher modes consistent diagonal lumping (CDL) and proportional lumping (PL) schemes exhibit better performance than that achieved by other schemes (including consistent mass matrix (CMM) approach).

Example 3.2.2. A simply supported cross-stiffened square plate studied by Aksu [43] as shown in Fig. 9(a) has been analysed by using the present models with a 4 x 4 mesh for quarter plate. The results for the case with concentric stiffeners are plotted in Fig. 9(b). The results obtained by using QL9Sl agree well with those of finite difference method (FDM) [43] and SFDM [8]. Frequencies obtained by using QSBI are aiso found to be in good agreement. This behaviour may be due to the lesser locking of the eight-noded element with less number of constraints for simply supported boundary conditions The results obtained by using the different mass lumping schemes for eccentrically stiffened

Y

1 ss r

ss--__-_ -c__s-___- I x

I I 1

f

‘S* I I I I I

ss

3 ;

- *X

t------_- 30 in -4

Fig. 5. Finite element mesh for static analysis of a simply supported rectangular plate with orthogonal stiffeners. Plate and stiffener details: E = 30 x lo6 psi; Y = 0.3; a = 30 in; b = 60 in; t = 0.25 in; udl = 10.0 psi; X-stiffener (S,)

size = 0.5 x 5.0 in; Y-stiffener (S,) size = 0.5 x 3.0 in.

0.12 r f Y

‘1 J 0 75 15.0 22 5 30.0

y(h)

(al

w - deflection x 10” in

W

0 75 15.0 22.5 300

y (in)

(b)

Fig. 6. Variation of deflection at x = 7.5 and x = 15.0 for the case of stiffener as (a) concentric and (b) eccentric. -QL9SI (4 x 4 mesh); +QS8Sl (4 x 4 mesh); Cl QL9Sl

(2 x 2 mesh); 0 QS8Sl (2 x 2 mesh); e CMA (7).

plates are presented in Table 2. It can be observed that the CMM scheme with rotary inertia effects shows only marginal improvement in the results compared to other schemes.

Example 3.2.3. Natural frequencies of a simply supported rectangular plate with a centrally placed

Y

I +-300mm-+

c L

:

E c c 0

is

t------_- 600 mm -=j

Fig, 7. Finite element mesh for vibration analysis of a clamped square plate with a central Y-stiffener. Plate and stiffener details: E = 6.87 x 104N/mm2; p = 2.78 x 10e6 kg/ mm-‘; Y = 0.34; t = I mm; 0 = b = 600 mm; Y-stiffener (S,)

size = 3.31 x 20.25 mm.


,+-____+__--7 T , .

I I I I I 2 3 4 5 6 7 6

Mode No.

(a)

f -frequency m Hz

409 I I I I I

2 3 4 5 6

Mode No.

(b)

Fig. 8. Frequencies for the first eight and six modes, respectively, for the case of stiffeners as (a) concentric and (b) eccentric. l QL9Sl; + QSSSI; n STIFPTI (42); 0 STIFPTZ (42); n SFDM (8,9); x SAPIV (42); + ASKA

(42).

Y-stiffener (eccentric) as shown in Fig. IO(a) have

been evaluated up to the first six modes using the present models (with a 3 x 2 mesh for quarter plate and using different mass lumping schemes). It can be observed from Fig. 10(b) that the frequencies obtained by using the present models are in good agreement with those obtained from FDM [44] and SFDM [9]. The frequencies up to the first six modes obtained by using the four mass lumping schemes are shown in Table 3.

Example 3.2.4. A rectangular stiffened plate with dimensions identical to those of Aksu and Ali [44], but with mixed boundary conditions (SSSC-three sides simply supported and one side clamped) is

analysed (with a 3 x 4 mesh for half plate) by using the present models. The same plate was analysed by Mukhopadhyay by using SFDM [9] and FEM [34]. The results of present analysis are presented in Table 4. It can be observed from Table 4 that the results of the present analysis agree well with those obtained by using SFDM [9]. The results obtained by using QL9Sl model are closer to the values of SFDM [9] than to the FEM values [34].

3.3. General observations

Performance of QS8Sl model (which is similar to that of Deb and Booton’s FEM(M1) model [31]) is very poor for structures which have many constrained DOF (like a clamped plate). The stiff behaviour of this model (refer to Fig. 8) for stiffened plate/shell structures which have many constrained DOF, may be attributed to the locking of the eight- noded plate/shell element. Even for plate/shell structures with simply supported boundary conditions, the QS8Sl model appears to be stiffer for higher modes of vibration resulting in higher frequencies. This can be observed in Tables 2-4. On the other hand the QL9Sl model exhibits better performance in all the numerical examples solved involving different boundary conditions. For the numerical example 3.2.4, frequencies obtained by using the QL9Sl model are closer to the values of SFDM [9] than that of FEM values given in [34]. This demonstrates the better performance of the QL9Sl model. It may also be noted that the stiffness and mass of stiffeners for the proposed models have been evaluated by using three- noded beam element shape functions. This reduces the computational effort significantly compared to the other models available in the literature [31-341, which use eight-noded plane element shape functions for the same. Among the four mass lumping schemes used for vibration analysis, the frequencies for higher modes obtained by using CDL scheme fall in between the frequencies obtained by using the other mass lumping schemes and are more closer to the results presented in the literature. While the frequencies obtained by using the CMM scheme are higher, the frequencies are lower for EL and PL schemes, as can be observed in Tables 14. This indicates that the CDL scheme gives better performance. It is observed

Table I. Natural freauencies in Hz of a clamped square plate with a central Y-stiffener (concentric)

Present study (4 x 4 mesh) - QL9Sl QS8Sl

Mode STIFPTl STIFPTZ SAP IV ASKA No. 1421 1421 t421 t421 CDL CMM EL PL CDL CMM EL PL

1 50.45 45.89 50.74 50.79 50.66 50.71 58.84 59.06 58.74 58.74 2 63.71 59.51 74.21 65.89 63.98 64.06 63.81 63.90 76.04 76.49 75.77 75.77 3 75.16 70.04 75.13 75.26 74.88 75.03 101.72 102.86 100.98 100.98 4 85.30 79.11 84.83 85.00 84.42 84.64 124.19 126.27 122.52 122.52 5 113.69 104.18 113.92 114.34 113.02 113.51 162.02 165.52 158.18 158.18 6 120.89 109.89 146.40 119.60 120.70 121.21 119.56 120.18 158.38 160.75 156.58 156.58 7 143.20 131.39 144.50 145.53 142.44 143.57 166.16 171.25 162.82 162.82 8 146.90 141.85 178.30 169.34 147.71 148.22 145.99 146.93 181.91 188.60 176.02 176.02

Static and vibration analysis of stiffest plates/shells 659

Y

I F-317.5 mm-+ SS

I I k I L I I 7 I

ss_-___s__-_+--_____ss E

i I 2

I (0

I

‘5

I I I I 1

ss

f

x

v SFDM (8)

. FDM (43)

t------- 635 mm ------/ Mode No.

(a) (b)

Fig. 9. Vibration analysis of a simply supported square plate with orthogonal stiffeners (concentric). (a) Finite element mesh; (b) frequencies for the first six modes. Plate and stiffener details: E =2.07 x IOsN/mm2; p = 7.83 x 10e6 kg/mm’; v ~0.3; A’- (S,) and Y- (S,) stiffener

size = 12.7 x 22.22 mm; a = b = 635 mm; I = 2.54 mm.

Table 2. Natural frequencies in Hz of a simply supported square plate with orthogonal stiffeners (eccentric)

Present study (4 x 4 mesh)

QL9SI QS8Sl

CMM CMM Mode No. CDL withi withoutS EL PL CDL with without EL PL

: 125.08 137.03 125.00 137.08 125.18 137.08 124.95 136.91 125.42 136.97 126.53 142.79 126.53 142.95 126.70 143.00 124.95 142.75 126.51 142.75 3 146.42 146.49 146.60 146.17 146.31 152.27 152.71 152.82 152.26 152.26 4 212.95 213.27 213.43 212.63 212.79 227.59 229.02 229.02 227.27 227.27

: 312.10 324.36 312.74 325.47 313.85 326.43 309.72 321.49 3ll.IS 323.24 329.13 372.10 333.59 379.90 334.70 380.54 309.72 368.44 328.02 368.44

t With rotary inertia effects. 1 Without rotary inertia effects.

Y

I=- 300 mm -4 I

ss I I I : T I /

ss , E I

ss 0

I 5 1% I I _.I._

SS x

t-_- 600 mm -4

(0)

i

El -X

,‘.J

___=*r_-_--r :::” 6t-

r ‘v , l QLSSI + QSBSI I

, / v SFDM (9)

A FDM (44) I I I I

3 4 5 6

Mode No.

(b)

Fig. 10. Vibration analysis of a simply supported rectangular plate with a central Y-stiffener (eccentric). (a) Finite element mesh and (b) frequencies for the first six modes. Plate and stiffener details: E = 2.07 x 10’ N/mm2; p = 7.83 x 10m6 kg/r&; v = 0.3; a = 6OOmm; b = 41Omm; t = 6.33 mm; Y-

stiffener (S,) size E 12.7 x 22.22 mm.


Table 3. Natural frequencies in Hz of a simply supported rectangular plate with a central Y-stiffener (eccentric)


QL9Sl QS8Sl Mode FDM SFDM No. 1441 [91 CDL CMM EL PL CDL CMM EL PL

1 249.19 244.26 254.91 255.13 254.49 254.8 1 255.16 256.24 255.16 255.76 2 262.16 263.88 264.20 263.40 263.12 264.83 265.95 265.15 265.15 3 508.71 523.94 526.64 521.39 523.14 528.71 535.08 528.55 528.55 4 538.86 539.85 543.35 535.24 538.10 561.34 513.59 560.86 560.86 5 570.32 511.25 581.39 569.93 574.39 627.22 644.42 620.55 620.55 6 162.14 787.34 800.23 110.19 781.77 195.62 827.61 784.63 784.63

Table 4. Natural frequencies in Hz of a SSSC rectangular plate with a central Y-stiffener (eccentric)


QL9Sl QS8Sl

CMM CMM Mode SFDM FEM No. [91 1341 CDL with? without1 EL PL CDL with without EL PL

1 292.01 298.74 293.32 293.80 293.96 292.53 293.00 300.96 302.71 302.87 301.12 301.12 2 300.18 315.99 309.56 309.87 309.87 308.44 309.08 316.88 318.31 318.47 316.40 316.40 3 570.01 616.68 594.12 596.83 596.83 590.31 592.69 618.95 627.71 628.34 616.73 616.73 4 620.86 625.16 625.80 609.56 616.25 652.38 673.54 674.18 645.05 645.05 5 654.60 659.70 659.70 640.44 648.56 714.45 741.83 742.30 698.37 698.53 6 802.46 813.82 815.83 783.36 795.93 845.91 884.74 886.65 825.22 825.22

t With rotary inertia effects. $ Without rotary inertia effects.

that less computational effort is required for CDL

scheme as compared to that of CMM scheme, both in computing the mass matrices and also in extracting the eigenvalues. It can be observed from Tables 2 and 4 that inclusion of rotary inertia in the CMM scheme results in only marginal improvement in the results. It may be noted that the performance of the CDL scheme is superior for evaluating higher modes of vibration.

4. SUMMARY AND CONCLUDING REMARKS

Two finite element models (QL9Sl and QSSSl) for static and vibration analysis of eccentrically stiffened plates/shells have been presented. Detailed numerical studies have been conducted to evaluate the efficacy of the two models. The following are the observations/conclusions based on the studies conducted:

1. The finite element models proposed in the present investigation are capable of representing the true behaviour of stiffened plate/shell panels, as they are modelled by employing plate and discrete stiffener elements.

2. The finite element models proposed in this paper are very simple in formulation. Moreover, the evalu- ation of the stiffness and mass matrices of the stiffeners by using three-noded beam element shape functions requires less computational effort compared to that required to evaluate the same by using plate element shape functions. This reduces the overall computational effort significantly, when large number of stiffeners are encountered.

3. Among the four mass lumping schemes considered, the consistent diagonal lumping (CDL) scheme has been found to be better in terms of convergence and efficiency, even for higher modes of vibration.

4. The frequencies obtained by using QL9Sl model with consistent diagonal lumping scheme fall in between those obtained by the other methods/models.

5. The QS8Sl model is similar to that of Deb and Booton’s [3 l] FEM(M 1) model. This model has been found to be stiffer than the QL9Sl model for stiffened plates which have more constraints on the boundary. The nine-noded model QL9S1, proposed in the present study, is found to perform better than the QS8Sl model and the other models available in the literature for static and vibration analysis (in the higher frequency range) of stiffened plates.

6. For analysis of complex structures like ship hulls, aircraft structures etc., QL9Sl model with consistent diagonal lumping scheme is recommended for modelling the stiffened plate/shell panels to evaluate the vibration and dynamic responses.

Acknowledgements-The authors wish to thank their col- league Mr J. Rajasankar for his valuable suggestions. This paper is published with the kind permission of the Director, Structural Engineering Research Centre, Madras.

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2.

REFERENCES

S. Kendrick, The analysis of a flat plated grillage. European Shipbuilding 5, 4-10 (1956). J. Clarkson, Tests of flat plated grillages under concentrated loads. Trans. R. Inst. Naval Arch. 101, 129-140 (1959).


3.

4.

5.

6.

7.

8.

9.

10.

II.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

H. A. Schade, The orthogonally stiffened plate under uniform lateral load. J. Appl. Mech., ASME 62, 143-146 (1940). H. A. Schade, The effective breadth of stiffened plating under bending loads. Trans. Sot. Naval Arch. Marine Engng 59, 403420 (1951). C. L. Kirk, Vibration of centrally placed stiffened rectangular plates. J. R. Aero. Sot. 65, 695697 (1961). T. Wah, Vibration of stiffened plates. Aero. Quart. 15, 285-298 (1964). M. P. Rossow and A. K. Ibrahimkhail, Constraint method analysis of stiffened plates. Cornput Struct. 8, 5160 (1978). M. Mudkopadhyay, 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 of 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). Y. K. Lin, Free vibration of continuous skin-stringer panels. J. Appl. Mech., ASME 27, 669676 (1960). Y. K. Lin, I. D. Brown and P. C. Deutsche, Free vibration of a finite row of continuous skin-stringer panels. J. Sound Vibr. 1, 14-27 (1964). Y. K. Lin and B. R. Donaldson, Brief survey of transfer matrix technique and special reference to analysis of aircraft panels. J. Sound Vibr. 10, 103-143 (1969). C. A. Mercer and C. Seavey, Prediction of natural frequencies and normal modes of skin-stringer panel rows. J. Sound Vibr. 6, 1499162 (1967). R. N. Yurkovich, Dynamic analysis of stiffened panel structures. J. Aircraft 8, 1499155 (1971). M. Petyt, Finite strip analysis of flat skin-stringer structure. J. Sound Vibr. 54, 537-547 (1977). G. Sen Gupta, Natural frequencies of skin-stringer structures using a wave approach. J. Sound Vibr. 16, 567-580 (1971). A. A. Tingler, A method for simulating concentrated forces and local reinforcements in stress computation. Mathematics of Finite Elements and Applications, pp. 963-970. Academic Press, London (1973). J. Colville and J. Abassi, Plane stress reinforced con- crete finite elements. J. Struct. Div., ASCE 100, 1067-1083 (1974). J. D. Davies, 0. F. Hughes and F. Mistree, Finite elements in the design of optimum ship structures. Nav. Arch. Report 76/2, University of New South Wales (1976). 0. F. Hughes, Ship Structural Design. John Wiley, Chichester (1983). M. Mukhopadhyay, Stiffened plate plane stress elements for the analysis of ship structures. Comput. Struct. 13, 563-573 (1981). M. Mukhopadhyay and S. K. Satsangi, Isoparametric stiffened plate bending element for the analysis of ship structures. Trans. R. Inst. Naval Arch. 126, 144-151 (1984). S. K. Satsangi and M. Mukhopadhyay, A review of static analysis of stiffened plates. J. Struct. Engng, Struct. Engng Res. Centre, Madras 15. 117-126 (1989). A. Mukherjee and M. Mukhopadhyay, A review of dynamic behaviour of stiffened nlates. Shock and Vibr. Digest 23, 3-8 (1986). _

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

M. Mukhopadhyay and A. Mukherjee, Recent ad- vances in the dynamic behaviour of stiffened plates. Shock and Vibr. Digest 27, 69 (1989). R. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31-38 (1951). A. K. Gupta and P. S. Ma, Error in eccentric beam formulation. Int. J. Numer. Meth. Engng 11, 1473-1477 (1977). H. A. Balmer, Another aspect of the error in eccentric beam formulation. Int. J. Numer. Meth. Engng 12, 1761-1763 (1978). R. E. Miller, Reduction of the error in eccentric beam modelling. Int. J. Numer. Meth. Engng 15, 575-587 (1980). R. E. Miller, Dynamic aspect of the error in eccentric beam modelling. Int. J. Numer. Meth. Engng 15, 1447-1455 (1980). A. Deb and M. Booton, Finite element models for stiffened plates under transverse loading. Comput. Struct. 28, 361-372 (1988). P. A. Thompson, P. Bettess and J. B. Caldwell, An isoparametric eccentrically stiffened plate bending element. Engng Comput. 5, 110-I 16 (1988). S. K. Satsangi and M. Mukhopadhyay, Finite element static analysis of girder bridges having arbitrary plan- form. Inr. Ass. for Bridge Struct. Engng 17, 65-94 (1987). A. Mukherjee and M. Mukhopadhyay, Finite element free vibration of eccentrically stiffened plates. Comput. Struct. 30, 1303-1317 (1988). E. D. L. Pugh, E. Hinton and 0. C. Zienkiewicz, A study of quadrilaterial plate bending elements with reduced integration. Int. J. Numer. Mefh. Engng 12, 10541079 (1978). G. S. Palani, N. R. Iyer and T. V. S. R. Appa Rao, Studies on the performance of isoparameteric finite elements for static and vibration analysis of beams and plates. Project Report No. SST-RR-88-2. Structural Engineering Research Centre, Madras (1989). T. J. R. Hughes, M. Cohen and M. Havour, Reduced and selective integration techniques in finite element analysis of plates. Nucl. Engng Design 40, 203-222 (1978). 0. C. Ziewkiewicz, The Finite Element Method. McGraw-Hill, New York (1977). G. S. Palani, N. R. Iyer and T. V. S. R. Appa Rao, Studies on the performance of isoparametric finite models for static and vibration analysis of eccentrically stiffened plates/shells. Project Report No. SST-RR-89- 3, Structural Engineering Research Centre, Madras (1990). K. S. Surana, Lumped mass matrices with non-zero inertia for general shell and axisymmetric shell elements. Int. J. Numer. Meth. Engng 12, 1635-1650 (1978). K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1976). P. S. Nair and M. S. Rao, On vibration of plates with varying stiffener length. J. Sound Vibr. 95, 19-29 (1984). G. Aksu, Free vibration analysis of stiffened plates by including the effect of in-plane inertia. J. Appl. Mech. 49, 206212 (1982). G. Aksu and R. Ah, Free vibration analysis of stiffened plates using finite difference method. J. Sound Vibr. 48, 15-25 (1976).

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g. s. Palani, n. r. Iyer and t. v. s. r. Appa Rao - Finite Element Analysis of Eccentrically