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Geometrically nonlinear analysis of composite stiffened plates using finite elements

B. Chattopadhyay Civil Engineering Department, Bengal Engirleering College, Howrah 711 IO_?, India

P. K. Sinha Aerospuce Engineering Department, Indian Institute of Technology, Kharagpplrr 721 302, Indiu

M. Mukhopadhyay Ocerul Eq.yirlrerirlg R NULYII ArchittJcnrre Department, Imlirrrl Institute of Techrlology, tiharugplrr 721 302, India

Composite stiffened plates have been analysed for large deflection using the finite element method. An eight noded isoparametric laminated stiffened plate bending element developed earlier has been used. The element has the capability of including transverse shear deformation for the plate and the stiffener and incorporating one or more stiffeners anywhere within the element. The finite element analysis has been made using Mindlin’s formulation and with the assumption of small rotation. The nonlinear equilibrium equations are solved by the Newton-Raphson iteration procedure. Plates and stiffened plates of isotropic and anisotropic material with different boundary conditions have been analysed and results have been compared with those available. Parametric studies have also been carried out.

INTRODUCTION

With the increased application of fibre reinforced composites in various fields, research on their behaviour in different structural forms has also increased. Most commonly used structural forms are plates and shells, used in aircraft and ships’ structures. Stiffened forms have been proved to be more efficient than the unstiffened ones as the former provide effective use of the material. Research on stiffened composite plates has been directed, so far, mostly towards buckling and post-buckling analysis. A few studies on the vibra- tion analysis have recently been found. It is observed that the flexural behaviour of composite stiffened plates has not been studied fully. Very few investigations that have been carried out so far are restricted to small deflections only. At

higher loads, the transverse deflection of plate is large compared to its thickness. The bending- stretching coupling comes into play and the load deflection behaviour ceases to be linear. A large deflection analysis gives accurate responses. For composite plates coupling effects arising from ply orientations make the matter more complicated. The large deflection flexural response of com- posite plates has been extensively studied in recent years.2

Even with its growing importance, studies on the large deflection behaviour of composite stif- fened plates are rather meagre. Turvey’* has analysed uniformly loaded ring-stiffened compo- site circular plates for large deflection. In his modelling, the effect of a stiffener has been incor- porated in the plate by considering statically equivalent local body forces acting on the plate.

107

108 El. Chattopadhyuy, P. K. Sinha, M. Mukhopadhyay

The large deflection equations have been solved by the finite difference scheme. Liao and Reddy” have investigated the large deflection behaviour of composite stiffened plates and shells by the finite element method. They have used degenerated three dimensional shell elements and associated curved beam elements which have been derived from the degenerated elements by imposing appropriate kinematic constraints. The incre- mental equations of motion developed using the principle of virtual displacement of a continuum and the total Lagrangian concept have been solved by the Newton-Raphson iteration proce- dure.

The experimental investigation on the tlexural behaviour of composite stiffened plate in the non- linear range has been carried out by Hyer rr a1.j They have used the STAGS (Almroth & Brogan’)

computer code to compare the experimental results with the analytical ones. STAGS models the stiffened plate as shell branches.

In the present paper, an eight-noded isopara- metric composite stiffened plate bending element developed earlier (Chattopadhyay et cd.“) has been used for the nonlinear analysis. The ele- ment’s ability to include transverse shear deformation for both the plate and the stiffener and to take into account any number of stiffeners within the element has been discussed by them. Nonlinear equilibrium equations have been derived using the principle of virtual work. The finite element formulation has been made using Mindlin’s hypothesis and with the total Lagran- gian concept. Nonlinear equilibrium equations arc solved by the Newton-Raphson iteraction pro- cedure.

Stiffened and unstiffened rectangular plates of isotropic and composite materials have been solved for large deflection flexure and results are compared with those available in the literature. Parametric studies have also been carried out.

FORMULATION

As stated earlier, an eight noded isoparametric element has been employed to model the stiffened plate. The element matrices of the stiffened plate element consists of the contributions of the plate and that of the stiffener. The formulations for the element has been presented by Chattopadhyay et

al.” and as such the same is not repeated here. Only the formulation for nonlinear equilibrium equations and their solutions has been presented

in this paper. The formulation is based on Mindlin’s hypothesis and with the assumptions of small rotation. Total Lagrangian approach has been adopted. Nonlinear equilibrium equations have been derived using the principle of virtual work.

Virtual work equation

For an elastic continuum undergoing small or large displacement, the equilibrium of internal and external forces can be expressed by the virtual work equation as:

{q}d{d}‘= d/c}‘@}-{R}d{6}“={0} !

(1) ,

where j 6 } = nodal displacements of the discretised continuum,

{ R ] = generalised external nodal forces, { II,) = Resultant of internal and external forces. {E} and {CT] represent strains and stresses as

usual. Now,

d{e}=[B]d{6} (2)

[B] is different from [I?] in the sense that [B] con- tains terms which are dependent on displacements when displacements are large.

Combining eqns ( 1) and (2) yields

{v}d{6}‘= d(6}“[~]7jo}-(R}d{6}‘={O} I I’

or,

Equation (3) is the nonlinear equation of equili- brium to be solved. The integrals are carried out element by element and the contributions to nodal equilibrium are summed in the usual manner. Therefore eqn (3) can represent the equilibrium equation either at element level or in the assembled form.

Variation of strain

(i) Plate element Due to the virtual displacements dj 6’) at the nodes, the variation of strain within an element is considered. {SF} are the displacements at the

Nonlinear analysis of composite stiffened plates 109

nodes of an element represented” by:

{S(‘}={S, &..... S,]

for an eight noded isoparametric plate bending element, where

kV=j%4 w, 8_,,? 8.A

The midplane displacement components at any point within the element are given by

‘U ZI we.,, 8,,}‘= i N,[Z,]{6}, I= 1

Considering first order nonlinearity, the strain- displacement relations are:

au av )‘,,,2=-+-+

ax2 ax,

SO, the generalised strain components are,

(4)

where {&::I is the inplane strain, expressed as,

{E,,] is the bending strain,

{E:] is the shear strain,

and finally the nonlinear strain ( E,,}~ is defined as,

1 aw 7

i ij z ax,

(8)

component of inplane

P-4

The superscript 0 represents the linear and L represents the nonlinear part of the strains. The stress-strain relationship for the linear part of the strains are same as those given by Chattopadhyay et al. 3

The strain components can be expressed as,

{&j=#+{&JL (10)

Considering the variation

d{E~=d(&}“+d(riL, W)

the variation of the nonlinear part of the strain component can b

19). k,J’- ofeqn (9)

(a

{%I’- =

(7)

where

aw ax,

0

aw ax2

aw ax, (Ii aw

ax,

le derived with the help of eqn

can be expressed as:

0

aw

ax, (12)

110 R. Chattopadhyay, I’. K. Sinha, M. Mukhopadhyay

are the displacement gradients with respect to lateral displacement w. Considering the variation of { E,,}~-, i.e.,

d{~,~}f-==;d[A]{B}+;[A]d{B} (13)

and using the special property (Zienkiewicz”) of [A ] and ( 19}, the above expression reduces to

d{E,J =]A]d@] (14)

( O} may be represented in terms of nodal displace- ments

iW=Kw”~ [G] =[C,G?.....GJ

where

El=

11.5)

(16j

I (17)

Therefore

d{8j=[G]d{&‘] (US)

Equation ( 14) can be written as

d( E,‘}‘. = [A ] [ G] d{ 6 “1 (1%

Again from vector representations of (E]!- as in eqns(5)and(lO),

dj E}‘, [Al [()I WI

FM =

itIN,.

ax,

0

aN, ax,

0

aN I

ax,

aN,

ax,

(24)

(25)

(26)

(ii) Stifjbner element For a stiffener orientated parallel to xI direction,

G,] d/6’}

the generalised strain components (Chatto- padhyay et 01.~) expressed in terms of the dis- placements of plate mid-plane are,

= [RI’, d{ d “1

Equation ( 11) can be written as

d{.z}=[B](‘d{6”}+[B]1. d(P)

iE/=[R](lj6’)+i[B](-j6}

(21)

(21a)

where [B]” can be represented as

(22)

where

[

P,J’ Nl - PI:!= [Ol PJ

[()I m.

L&lz’ Ml’ and P,32 are represented as follows:

(23)

Subscript st is used to indicate stiffener. Considering first order nonlinearity, the are

{%I =

I aw i- 2 ax,

0

0

0

\’ ’ /

strains

(28)

Nonlinear analysis of composite stifened plates 111

(29) (30)

\ / 1,

IR-P,)=- I&-,

P”

I 5, _ % Displacement

6 n+l * p’ Ko+ K, = Se cant

stiffness

Now, proceeding in a similar way as for the plate element, different matrices related to the variation of strain can be derived. The resulting matrices are given in the following, with subscript st to indicate the stiffener.

(31)

(32) t-

Fig. 1. The Newton-Raphson method. (33)

djc},, =[B]I:dj6’}+[B]i;dr6”} (34)

where [B]:, is given by Chattopadhyay et al.-’ and [RI’;, can be expressed as,

(35)

The expressions for the variation of strain derived in this section has been utilised in the formulation of the tangent stiffness matrix arising during solu- tion of the nonlinear equations discussed in the next section.

or, A@},,= -bW1~1,

or,A~6},,=[Kl;‘j-~iy,, From eqn (3) it can be understood that

(39)

which is the difference between external and internal load at point ~1. As the solution is approached, { ?,!I), tends to zero.’ The iteration process is terminated when

Solution of nonlinear equations

hmw ({R}7‘{R})"2

x 1OO~Toll (40) The nonlinear equilibrium eqn (3) is solved by the Newton-Raphson iteration procedure. In eqn (3) it+} is a nonlinear function of {S}. The solution process is illustrated in Fig. 1. If solution at n, (d,,}

is reached, an improved solution at M + 1, i.e., {S,,, ,), can be obtained using a curtailed Taylor series expansion of { p],

where Toll is the specified tolerance. In the pre- sent paper 1% tolerance has been adopted for all the problems solved.

Expression for tangent stiffness

A{%={@ Tangent stiffness, expressed in eqn (38), is derived by taking the variation of eqn (3) as,

14 r7+1 =k%+A{% (37)

where A{ d},, can be rewritten by substituting ’ d(a) (41)

(38) Now{a}=[D](c}andd{o}=[D]d{Ej. Using eqn (2 1 ),

112 B. Chattopadhyay, P. K. Sinha, M. illukhopadhyay

since

[B] =[R]” +[B]‘~ and d[B] = d[B]‘-. (43)

Therefore, combining eqns (32)-( 34) yields

divi= ! (@I ) ‘b} (1

+ I

([Bl”+[B]‘.)‘[D]([Bl” +[B]‘)d{d”} ,’

=([K],,+[K]“+[K]‘-)dj&‘}

=[K],.d{d’} (44)

where

[K]“= [Bjo7[D][Bl”dz~ I I’

(45)

[K]‘.= I

[Bl”‘[fl][B]‘~ dr, + [R]L’[D][B]” dv ,’ I I’

+ I

[B]“‘[D][B]‘- dz/ I’

(46)

(47)

where N, , N,, N(, are inplane stress resultants which can be obtained from the relation

(46

where {N}, {M}, A,, B,,, II,, are as defined by Chattopadhyay et al. 3

{E,,} and { chj are defined in eqns (5)-( 7) and (9).

Expression for internal force

Internal force j,.[f?]‘{ o} can be expressed in the following form using eqn (2 1 a)

([B]” +[B]‘)‘[D]([B]“+$[B]‘){&‘}dr~ I’

= [KW”}

where [K], is secant stiffness expressed as

[K],= ([B]“)‘[l)][B]“do+; 1’ I I I

,([H]“!‘[I)][R]‘.dr,

+ ([HI’.) ’ [D][B]” dz/ I

([H]‘.)‘[ll]‘(B]dz~ (49)

The relations are also true for the stiffener ele- ment if the appropriate matrices for the stiffener element given earlier are used.

Tangent stiffnesses and internal forces are cal- culated at element level and assembled after- wards.

NUMERICAL RESULTS

To show the accuracy and applicability of the present formulation, a few problems on bare and stiffened plates of isotropic and laminated aniso- tropic materials have been solved. The results have been compared with those available in the literature. For all the problems solved, applied loading is uniformly distributed and tolerance level is 1%. The stresses have been calculated at Gauss points and extrapolated to get them at nodes using bilinear extrapolation. 4 x 4 mesh divisions have been used for the quarter plate for all the problems except where specifically mentioned.

Bare isotropic plates

The case of isotropic unstiffened square plates with simply supported and clamped boundary conditions, analysed by Pica et al.” have been solved. Geometry of the plate is indicated in Ref. 8. Material properties are E = 2.0685 x 10” N/m’. Y = 03 16. The abbreviations used, i.e., LN, QS, etc. stand for different element types and are given in the paper of Pica et al.” The nondimen- sional parameters are W= w(O,O)/t, c, =&/ (Et’), o2 = o(x,y)u’/(E$), where i=x or y, and a(.~, y) is extreme fibre stress.

(u) C’lampeci square plute

Centre deflections, centre and edge stresses obtained by the present approach at different load levels along with those obtained by Pica et al.” and

Nonlinear analysis of composite stiffeenedplates 113

other researchers have been presented in Tables 1-3. The results are in good agreement.

Bare composite plates

(bj Simply supported square plate Results for centre deflection and centre stresses along with those obtained by Pica et al.” and other researchers have been presented in Tables 4 and 5. Results compare well.

A few of the composite unstiffened plates solved for large deflection by Reddy” have been analysed by the present approach. The plates are of cross-ply angle-ply and orthotropic construc- tion. Plate geometry and boundary conditions are as given by Reddy. ’ ’ Material properties are E_,/

Table 1. Clamped square plate - central deflections W

Load Q

Levy:

LN

Finite element (Pica et al.“) Present

QL QH RI BFS

17.7Y 0.237 0.2349 0.2368 0.236 1 0.2387 0,236 1 0.236 1 38.3 0.471 0~4705 0.4699 0.4687 0.4717 0.4685 0.4684 63.4 0.695 0.6990 0.6915 0.6902 0.6016 0.6900 0.6900 95 0.912 0.9191 0.9029 0.9015 @9008 0‘90 12 0.90 16

134.9 1.121 1.1318 1.1063 1~1050 1.1025 1.1047 1.1054 184 1.323 I.3350 1.3009 1.2997 1.2961 1.2992 1.3008 245 1.521 1.5347 1.4928 1.4916 1.4874, 1.4908 1.4928 318 1.714 1.7439 1.6786 1.6775 1.6774 1.676 1 1.6788 402 I902 1~9109 1.8555 1.8545 1.8529 1.8524 1.855’)

LN four noded linear element. QL quadratic Lagrangian element. QH quadratic Heterosis element (Huges & Cohen”). BFS Bogner-Fox-Schmit triangle (Bogner ef a/. I‘). RI Razzaque-Irons triangle (Razzaque’“).

Load Q

17.79 38.3 63.4 95

134.Y 184 245 318 402

Levy’

2.6 5.2 8.0

11.1 13.3 15,9 19.2 21.9 25.1

Table 2. Clamped square plate - centre streses 0, (0,O)

Finite element (Pica et al.“) Present

LN QL QH RI BFS

2.6274 2.6319 2.6144 2.6890 2.6328 2.6798 5.5435 5.48 16 5.4520 54140 5.4847 5.5898 8.5570 8.3258 8.2908 8.0205 8.3372 8.5022

11.575 11.103 11.066 10.521 11.127 11.347 14.567 13.827 13.789 12.971 13.886 14.137 17.493 16.497 16.456 15.390 16.552 16.873 20.446 19.225 19.178 17.885 19.296 19.659 23.605 2 1.994 21.938 20.438 22.08 1 22.488 26.318 24.780 24.713 23.020 24.88 1 24.660

Table 3. Clamped square plate - edge stresses U, (a/2,0)

Load Q

17.79 38.3 63.4 95

1349 184 245 318 402

Levy’

5.48 11.52 18.03 25.32 33.5 42.4 52.8 63.9 75.8

LN

3.3663 7.1122

11.198 15.639 20.46 1 25.597 31.172 37.459 43.121

Finite element (Pica et al.“)

QL QH RI

5.3163 5,299 1 5.2523 11.216 11.189 10.807 17.726 17.697 16.680 24.967 24.946 23.001 33-045 33.038 29.875 41.885 41.900 37.257 51.719 5 1.763 45.357 62.325 62.401 54.011 73.407 73.520 63.000

BFS

5.2915 11.137 17.556 24.649 32.485 40.96 1 50.25 1 60.097 70.190

Present

5.4218 11.418 18.018 25.333 33.452 42.322 52.109 62.619 73.522

114 R. Chattopadhyay, P. K. Sinha, M. Mukhopadhyay

Table 4. Simply supported square plate - central deflection W

Load

0

Rushton’” Finite clement (Pica et al.?

LN OS QL

0.3478 0.3480 0.8 184 0.8 185 14655 I.4657 2.3Y27 2.3032 3.8 124 3.x I34 fkO52 I 6053Y

9.16 36.6

136.5 586

2344 9377

0.335 0.353 I 0.8 18 O.K.303 1.47 1.4Y3 I 2.40 24380 3.83 3.8985 6.07 6.157 I

Present

QH

0.3478 0.3437 0.8 184 0~8094 1.4655 14537 2.3Y28 2.3145 3.8128 3.7851 h.0530 WOY 3

QS quadratic serindipity clcmcnt.

Table 5. Simply supported square plate - centre stresses (7, (0,O)

Load

U

Rushton’”

LN

Finite element (Pica et d.“)

OS QL OH

Present

9.10 246 2fl.354 2.6214 366 A.YO 7.1837 7W’h

146.5 14.5 15.263 13m1 586 304 31.3Y6 .30. I83

‘344 65.2 6842X 65673 Y377 l-15.3 155.24 I -IY.hh

2+JO2Y ‘60 16 2fi653 6.9 S26 Cc)787 7.1556

1_1+l35 l-l+l I 5 15.034 30. I X8 30.1 30 31~01 I 65.756 65661 67-480

1 IY.YY I-lY.87 IS.3.7ti

E,. = 40, G,,,/E, = O-6, C;,, /E!. = GJE, = 0.5, and I’,,.= 0.25. The.results of displacements at differ- ent load levels have been plotted in Fig. 2 along with those obtained by Reddy. The agreement among the results is excellent.

Isotropic stiffened plates

The problem of isotropic square plates simply supported at four edges and stiffened by central stiffeners, one in each direction, has been solved. Geometry and boundary conditions of the plate are given by Reddy. ” Material properties used by Reddy are E=2.07~ 10” N/m’ and ~=03. Figure 3 represents the results of central deflec- tions of stiffened and unstiffened plates at dif- ferent nondimensional load levels as obtained by the present method and those by Reddy. The results are in good agreement.

Rectangular plates, clamped along all the edges and having one central stiffener parallel to the x2 direction, have been solved by Koko and Olson6 using super elements. Using symmetry, half of the plate is modelled with 6 x 6 mesh divisions (Fig. 4). Geometry of the plate and the material pro- perties are available from Koko and Olson’@ paper. Figures 5a and 5b present the panel centre and stiffener centre deflections (Fig. 4) respect- ively at different load levels along with those by

I Present Reddy

11980) 2.5 Crossply A

Angleply ___ 0

2.0 Orthotropic -,-.- A

1.5 w

T; 1.0

0.5

a/h=100

0 0 100 200 300 LOO 500 6C

p=q (a/hIL IO-'/Eg

Fig. 2. Load detlection curves for four layer angle-ply (4S/ - 45/45/ - 45) and cross ply (O/90/0/90) squaw plates.

Koko and Olson.” Panel centre detlections agree well with those of Koko, whereas the stiffener centre deflections are slightly different with a maximum difference of 12%.

Stiffened composite plates

The results available on the analysis of rectan- gular composite plates stiffened by rectangular stiffeners is due to Liao and Reddy.” Problems solved by them have been solved by the present approach. Geometry and boundary conditions of

Nonlinear analysis of composite stiffened plates 115

280 - Stiffened

----Non-stiffened

,” a 210- Reddy 11990 1

0 0.L 0.8 1.2 l-6

Cent re def lectlon w /h

Fig. 3. Load deflection curves for stiffened and unstiffened square isotropic plate.

500 mm e cl

IOOOmm

Fig. 4. Mesh divisions for rectangular stiffened plate of Koko and Olson. A - Panel centre. B - stiffener centre.

the simply supported plates with central stiffeners, one along each orthogonal direction, are given by Liao and Reddy.” The material properties are E,/ E!,= 25, E,.= 7.031 X 10” N/m? v,~,= O-25, G,,./

’ E!,= 0.5, G,, - G,,., GJEy= 0.2.

(a) Cross-ply stiffened plute In Fig. 6: results of central deflections of the stif- fened plate at different load levels have been com- pared with those by Reddy. The results compared well.

(b) Angle-ply stiffened plate Geometry and material properties for this plate is the same as that of (a). Figure 7 presents the results of central deflections as obtained by the present method and those by Reddy. Present deflections are slightly lower than those by Reddy.

It can be observed from Figs 6 and 7 that the cross-ply stiffened plates are stiffer than angle-ply stiffened plates. Bending moments M,r, and ikZYz at

1.0

- Present

Oxtr/ (a)

I o- 5 10 15 2; i

4

!!

Panel centre displacement (mm) 5

1.0

- Present

O-8- o Koko (1991) o

1

i 1 Stiffenercentre displacement (mm)

Fig. 5. Load deflection curves for rectangular stiffened plate (isotropic).

1 - 120- - Present

“E o Liao 0. Reddy(19891

I I J 0 0.2 0.4 0.6 O-8 1.0

Central deflection, w (xl0mm)

Fig. 6. Load deflection curves for cross-ply stiffened plate.

the centre of the cross-ply and angle-ply stiffened plates are presented in Figs 8a and Eb, respect- ively. These results are not given by Reddy.

From the figures, the following observations can be made.

Bending moment M_,.2 of the cross-ply stiffened plate is higher than Mx,. This indicates that the structure is stiffer in the xz direction. This is due to the fact that the layers in the x2 direction are arranged in a particular sequence (preferred for

116 B. Chattopadhyay, P. K. Sinha, M. Mukhopadhyay

higher stiffnesses) compared to those in the x, direction. Moments IUY, and 111,? are almost the same at the centre of angle-ply stiffened plates. This is evident from the uniform stacking sequences in both the directions which give similar stiffness properties.

- 120 - Present

“E 0 Liao IL Reddy(1989)

2 100 0

,g

Ln 0

: 40 C .-

“0 0 -I

60 I/ 20-

0 I I I 1 0 0.2 o-4 0.6 0.8 1.0

Central deflection,w (xlOmm)

Fig. 7. Load deflection curves for angle-ply stiffened plate.

62.5

2 25.0

-G >

2 12.5

z 0 -I I 0

0 5 10 15

Moment Mxl x10 (N-mm)

62.5

J 5

Moment Mx2 x 10 I N -mm)

Fig. 8. Bending moments at centre of stiffened composite plate of Reddy (a) cross-ply (b) angle-ply.

Parametric study

The load deflection behaviour of eccentrically stiffened composite plates has been studied for different stiffener parameters. For the purpose of the study an eccentrically stiffened cross-ply square plate has been considered. The stiffeners are placed along the x2 direction. The parameters varied are the eccentricity and the number of the stiffeners. Figures 9a and 9b show the geometry, boundary conditions and stiffener positions. Plates with a single central stiffener are analysed

400 mm

(4

x21 I

u =v:w= e,,= ex2= 0

33.33 1-l 33.33 1_133.33_ ~u=v=w= ,- I- - I eq=ex2=o

g:3zy

(b)

Fig. 9. Geometry and boundary condition (BC-6 of Ref. 9) of the composite stiffened plate used for parametric study (a)

one stiffener and (b) two stiffeners.

Nonlinear analysis of composite stiflenedplates 117

using a quarter plate model and those with two stiffeners using a half plate model. The details of mesh divisions are presented in Figs 9a and 9b. The material properties are the same as those used by Liao and Reddy.x The results are pre- sented in Figs 1 Oa, lob, 11 and 12. Figures 1 Oa and 10b show the load deflection characteristics of the plate for different stiffener eccentricities, having one and two stiffeners respectively. From these figures it can be observed that the stiffness of the composite stiffened plate increases with the increase in the stiffener eccentricity. Also the non- linear effects are reduced with the increase in the number of stiffeners.

Figure 11 shows the load deflection curves for the plate with varying numbers of stiffeners having the same eccentricity. From this figure it can be observed that the increase in the number of stif- feners increases the stiffness of the composite plate. In Fig. 12, the deflection patterns of the stiffened plate along the centre line parallel to the x, direction is presented for a constant load level and varying number of stiffeners. This figure illu- strates the change in the deflection pattern with the increase in number of stiffeners.

CONCLUSIONS

Formulations for geometrically nonlinear analysis of laminated composite plates with laminated stif- feners has been presented in this paper. It is evident from the results that the stiffened compo- site plate bending element developed earlier (Chattopadhyay et d3) performs well in the nonli- near range for both isotropic and anisotropic stif- fened plates.

The formulation is simple and can be easily programmed.

For the type of stiffener presented in this paper, the ply orientations of the plate and the stiffener can be arranged properly to achieve desired stiff- ness properties in any direction. The + 45” orien- tations in the plate and the stiffener give similar stiffness properties in either of the orthogonal directions.

For a composite plate the stiffness can be increased either by increasing the number of stif- feners or by increasing the stiffener eccentricity. Though rectangular stiffened plates have been analysed in the present paper, stiffened plates of

Fig. 10.

- One stiffener

Deflection (mm)

-Two stiffeners

Deflection (mm)

Load-deflection curves for eccentrically stiffened composite plates.

0

s 1

v

es= 9.0mm

OV 0 1 2 3 4

Deflection (mm)

Fig. 11. Load-deflection curve for varying no. of stiffeners having same eccentricity.

Unstiffenedv

Fig. 12. Deflection shape of the composite stiffened and unstiffened plate along centre line parallel to X, axis.

* Distance along plate centre (mm)

ci

b 200 4001

11x R. Chattopodhyay, 1’. I(. Sinha. Ml. Mukhopadhyay

any arbitrary planform can be analysed. It can be shown (Chattopadhyay et ~1.~) that the stiffener formulation of the present element can be modi- fied to include general open-section stiffeners.

The present formulation can be extended to include nonlinear dynamic postbuckling and other aspects of analysis of composite stiffened plates.

REFERENCES

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