DSpace at VNU: Non-linear stability of stiffened laminated composite plates...

Today, analysis of linear laiditi composite plates has been studied by many authors.. However, the analysis of noj-liei laminated composite plates has received comparatively little a tte

V N U J O U R N A L O F S C I E N C E , M athem atics - Physics T.xx, N()4 - 2004

N O N - L I N E A R S T A B I L I T Y O F S T I F F E N E D

L A M I N A T E D C O M P O S I T E P L A T E S

K h u c V a n P h u

M ilitary Technical A ca d em y

A b s t r a c t T h is paper deals with the non-linear sta b ility of t h e stiffeiK'd lai.ilt( com posite p late s u b je c te d to biaxial loads Numerical results are presented for illutr ii theoritical an a ly sis of stiffened and unstiffened la m in a ted c o m p o s it e plates.

Key words Stiffened lam in a ted c o m p o site plate, Shap e m em ory alloys (SM A), stbity

1 I n t r o d u c t i o n

Stiffened lam inated composite plates are vised extensively in Naval, Aerospao.Vi tomobile applications and in Civil engineering,v.v Today, analysis of linear laiditi composite plates has been studied by many authors However, the analysis of noj-liei laminated composite plates has received comparatively little a tten tio n [3, 4, 5] sjedl] tor muilysis of noil-linear stiffened laminated composite plates and shells subjoti 1 compress l)i-axial loads This problem is studied in the present paper

2 G o v e r n in g e q u a t i o n s o f l a m i n a t e d p l a t e s

Let’s consider a rectangular stiffened laminated composite plate, ill which call y (

is a unidirectional composite m a teria l This plate is subjected to H uniform coin)(^i(

o i l Vcich edge, with result ants p , and Py respectively (Figure 1 ), where p x and 'y ai

arbitrarily but as the plate is working in the elastic stage, so that every stress tf‘ defined by every loading sta te respectively and doesn’t depend on the process >an Thus, we can put

Py = O P ,

Th(‘ Strain-displacement, relations in the non-linear theory are of the form

T y p e s e t by

33

34 K h u c Van P h i

d 2 w

kx =

d 2 IV

ky ~ ỡ y 2 ’

k Xy — — 2 7

Ỡ2 W

where a, V w are the m idplane displacements along the X , y and z axes respectively

7

S2

f

Zl

X

Y

ỌI

N

Fz# 1

Integrating the st.ress-stra.in equations through the thickness of plate we obtain the expressions for stress resultants and flexion moments:

N x — { A l l 4- E \ A \ / S \ ) e x 4- A i 2 £y 4- { E \ A \ / S i ) z \ k x 4- p ^ / s 1 ,

N y = ( A ‘22 4- E 0 A 2 / S 2 )Sy + A \ 2 EX + ( E 2A 2/ S 2) z 2ky -f P y / s o,

M e — (£>11 + f i i / i / 5 i ) f c « 4- D i2ky 4- ( E i A i / $ i ) z i £ x i ,

M y = ( D 2 2 + E 2 I 2 / s o) k y 4- D i o k x + ( E 2 A 2 / S2)z2Sy,

Mx y — Dị-t(\h’Xy i

where

- D i j ( i j = 1 , 2 and 6) are e x t e n d in g a n d b e n d in g stiffn e sse s o f th e plate

without stiffeners,

respectively,

- A\ Ao cire the section areas of the longitudinal and transversal stiffeners respec­ tively

- 1 1 In are the inertial moments of cross-section of the longitudinal and transversal

St iff oners, respectively,

- ,sI , s 2 are the distances between two longitudinal stiffeners and between two transversal stiffenors respectively,

- Z o are the distances from the mid-plane to the centroids of th e longitudinal and transversal stiffeners, respectively,

- P ' , P' are the recovery tensile force ill the SMA wires

N o n - l in e a r s t a b i li t y o f s t i f f e n e d l a m i n a t e d c o m p o s i t e p l a te s 35

- I r l \ are m e recovery rensiie iorce 111 rile OIVI/V VVIIt'b

The equilibrium equations of a plate according to [2] are

ONr , d N xy

O X a y

- —-7— 4- 2 - — - H T— 7 T 4- iV;r- —r -f XỊJ — + ivy — 5- + 7 :r — Ộ + i yTTT —

Substituting (2) and (3) into (4) after some operations we ob tain the equilibrium equations of the lam inated plate

, s d u ) d 2w , A A s d w d 2w d w d 2w

A \ / s i ) - — ——^ -f (A 12 + ^Gf)) — 7T“77— -Aogtj— T p r

-(Ao2 4- E o A - ỉ / s o ) ^ ^ + + (^12 + ^60)77-77 - (^2^2/^2)2:277-^ +

+ ( A22 4- E 0 Ả 2 / Sn) — 77 Õ" + (^12 + ^OGyTT- o 0 + ^ 6 6 o o —

-( D u + E i / i / S i ) ^ ^ + 2 -( Ơ 1 2 + 2D-(ifi) - ^ 2 Q^ 2 + -(£*12 + £ '2/ 2 / S2) ỹ ^ 4 ' _

- ^ 7 2 - ( p v + p y i ^ h f ~ ( E i A 1/ s 1) z 1^ - ( E 2A 2/ s2) z 2 ị ^ ~

- ( E l A l / s 1 ) z 1i^ ^ - ( E 2 A 2 / s 2 ) z2^ ^ - ụ A n +

- - ^ “T o ( ) - ^ 12T T h r - 0 ( ^ 2 2 + E 2A 2 s 2 ) ^ - y ( I

-2 ỡ y 2 V y r) 2 Ớ.7;2 V d y J 2 d y V d y )

2 Aq 6 —— 7-— ——— — (-All + E \ A \ / s 1) — — 2" — 2^4(56 n 0 0 ^12 fj o o

r-, A / X 9 v d 2 W

(Aoo -f- E o A o / s o ) —— _ 9 — 0-

ỠU d i r

d v d 2 w d v d w

12 dy Ox 2 " 66 Ỡ£ d x d i

36 K h u c V a n P h u

For a plate simply supported on all edges, the following boundary conditions are imposed

-f A t ed g es X = 0 and X = a

+ At edges y = 0 and y = b

The boundary conditions discussed here can be satisfied if th e buckling mode shape

is represented by

where

- a, ft : edges of plate ill X and y axial directions respectively,

- m, n : the numbers of halfwave in the X and y axial directions respectively

Substituting expressions (8) into the equilibrium equations (5) and applying the Galerkin procedi ice yield the set of three algebraic equations w ith respect to the amplitudes

u,nn, Vnnii w mn, where th e first tw o eq u a tio n s o f th is s y s t e m a re linear algebraic eq u ation s

foi umm V-mn

(l\ U tmi + O 2 V 11 U 1 — (l,\W mil 4- ( 1 4 W ~Lfl,

(9)

a !jUinn + fleKnn — ^ W i n n "f Cl%W^lxrx, Getting from (9) expression t/mn, Knn with respect to w mn and substituting into third equation of (5) we obtain a non-linear equation with respect to w •

a9W^lw + a i 0 W?nn 4- ( a n 4- XPx ) W inn = 0, (10)

where a, are coefficients which depend on the material, geometry and the buckling mode

shape

i A 1—' A / \ m2b

a i — ( A l l + E \ A \ / S \ ) — ■ — h A

a 0*2 = &5 = ( A \2 -f A ẽo )m n ,

_ / El 4 , \ m 3nb

— { E \ A \ / S \ ) z \

16

„ 2 _

r r a

66

b '

a 2

“ 4 = - y 2 ( ^ n +£71v41/ Sl) ( — ) — - (A 12 - A m ) ~

(lQ — (A 22 4- E 0 A 2 / S 2 ) - J - + ,

)

2-a /* /77Ĩ

7n 2b

L66 a

Ò2

16

08 9 2(^22 + E 2A 2/ s 2) ( \ b / j ) 2 - ^ - ( A n - A (M) ~ rriTT ' a n

N o n - l i n e a r s t a b i l i t y o f s t i f f e n e d l a m i n a t e d c o m p o s ite p la te s 37

a 9

flio

-f

( - A l l + jE’i A i / s i ) — 3- 4- 2( ^ A \ 2 + 2^ 66) “— ~ b ~ + ( ^ 22 + ^ 2^ 2/52) - p

/ / i ( g 6a4 - Q2fl8) -í- H 2(fi-1^8 - ^5^ 4)

a i a 6 — «2^5

9 [(jElA l/Sl)2l( ? ) 3^ + ( ^ A 2/ s 2)z2( ^ ) £ ^ j +

H \ (a.ịO.ịị — Ci2a ĩ ) + / / ‘2(a i a 7 — a 3a ĩ>) + H^(ãQã4 — 4- / /4( ^ 1^,« — ( 1 5 ( 1 4 )

a i a e — a2ữ5

a n ( D l l + ^ l A / s i ) —Try ^ 3 — I" 2 ( L >12 + 2 D q q ) - - h f777 77 Ì ^ ( - D 22 + E 2 I 2 I S 2 ) —g f _ _ Tỏ (I+

/ / 3(a3a6 - a 2a 7) + #4(ai&7 - a 3a 5)

aựiG — a2«5

A 1 / m 2b

4 V a7T2 + a

n 2a

)■

H i =

H 2 =

-16

~9~

16

Ỵ - ) - ^ 3 ( i 4 1i + J 5 i ^ i / 5 i ) + (i412 + 2A66) | ^

1 G r r n \ 2 a / A n A / \ i A n A \ 771 '

n ( u ) ~^ 3 ^ ( ^ 22 + E 2 A 2 / S 2 ) H~ (^12 + 2^4fiei) ——-

1 , „ , , m 36

# 3 — —- ( £ i > W s i ) z i 2 ,

1 „ , n 3a

H 4 = — - ( E 2A 2 / s o ) z2 ~ 7 Tf •

From (10) we can express compression load with respect to H^mn as follows

The lower buckling load of the plate can be analysed by the minimum of <p(W)

it means that:

Ỡ R

a w mn

Tho value of W m n corresponding to the lower buckling load is found from the equation (12) and then su b stitu tin g into equation (1 1 ) we obtain lower buckling load P x

We can determ ine the minimum critical buckling force P x min by the way of varying

777,, 7Ĩ and P y m in — O l P x m ill*

3 N u m e r i c a l e x a m p l e s

Let’s consider a simply supported stifferned rectangular symmetrical composite

plate: a = 0.8 m; Ò = 0.5 m;

The materials of the plate have Thornel 300 graphite fibers and Narmco 5205 T h er­ mosetting Epoxy resin [5], the properties of these materials are

3 K h u c V a n P h u

El = 127.4 GPa; Ẽ 2 = 13GPa; Ơ12 — 6.4 GPa; Z^12 — 0.38;

- The plate has six layers: [45/ - 4 5 /9 0 /9 0 / - 45/ 45];

- Thickness of each layer: t = 0.5 mm;

- T h e la m in a t e p la te is reinforced by lo n g itu d in a l a n d tr a in s versa] stiffen ers, w hich

na.de of CPS, SMA and combined materials In combined case: longitudinal and taisverse stiffeners are SMA and CPS material respectively;

Tablcl Effect of thickness of the plate (W ith a = 1 )

h/b

Critical buckling loads Px (.N / m )

Stiffeners StiffenersSMA

9,6.1 0 -3 3.3622e+004 4.4277e+004 6.1802e+008 2 2 0 0 0 e + 0 0 9

1 5 , 0 1 0 - :i 1.2826C+005 1.5338e+005 6.1813e+008 2.2001O+009

The stiffeners have the same sizes, as follows: b g X h g = 4 mill x6 mm:

Diameter of SMA wire is: d = 1.2 X 10~ 3 m;

Spacing OÍ longitudinal and transverse stiffeners are: Si = .S'2 = 0.1 m

Table 2 Effect of orientations of the plate (W ith a = 1 , CPS Stiffeners'

The stacking Sequence

30/-45/90/90/-45/30

0 /9 0 /0 /0 /9 0 /0 45/-45/0/0-45/45 60/-45/30/30/-45/60 45/-45/90/90/-45/45

0 /-4 5 /9 0 /9 0 /-4 5 /0

Critical buckling loads Px (N / r n )

0.93695e+004 0.93939e+004 1.1554e+004 1.2680e+004 1.3027e+004 1.3599c+ 004

Table 3 Effect o f tr a n sv e r se stiffeners oil critical b u ckling load s

(W ith a = I)

N o n - l in e a r s t a b i l i t y o f s t i f f e n e d l a m i n a t e d c o m p o s ite p la te s

C PS Stiffeners CPS + SMA Stiffencrs SMA Stiffcner:

4 C o n c lu s io n s

When considering non-linear geometry of laminated composite plate reinforcedb stiffeners, we obtain:

+ Critical force of CPS lam inate plate not reinforced by stiffeners:

Px min = 8.2084e 4- 003JV/m;

4- Critical force of CPS laminate plate reinforced by CPS stiffeners:

p ,:min = 1.3027e + 004N/m;

+ Critical force of CPS lam inate plate reinforced by SMA and CPS stiffeners:

Pj-I nil = 6.1799c + 00 SN/ m]

■f Critical force of C PS lam inate plate reinforced by SMA stiffeners:

P * min = 2,2002e + 009JV/m;

Critical force of the plate being reinforced by S M A stiffeners is higher th an th it t CPS stiffeners

The plates, which are reinforced by SMA stiffeners work more stably than tioe

of CPS stiffeners (Fig 2) when geometrical parameters of plate are varied In th e ae

of plates under biaxial compression, the siffeners will influence strongly oil the C11K.1 force(Fig 3)

^0 K h u c V a n P h u

Fig 2 Effect o f dimension OI 1 critical loads (with a = 1)

•V • if •

t

big 3 Effect of longitudinal stifteners on critical buckling loads (w ith 0 = 1 ) Depending oil arranged layers of the plate, we can receive different critical buckling

01C( Ill th is e x a m p le, w<‘ received m in im u m cr itica l b u ck lin g force co rc ssp on clinc to the

,a>e [3 0 /-4 5 /9 0 /9 0 /-4 5 /3 0 J

V £ k n o w l 6 d g 6 m 6 n t s Ỉ he a n th er w ould like t o tbrink P ro fesso r D ho H u y Bicli for h elping

lim to complete this work This publication is partly supported by the National Council

01 Natural Sciences

References

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Ed Education, (1994) (in Vietnamese)

2 S P Timoshenko, J M Gere, Theory of Elastic Stability Science and Technical

Publisher, (1976) (ill Vietnamese)

ỉ M w Ilyer, Stress analysis o f fiber Reinforced Composite materials McGraw-Hill

International Editions, (1998)

L M Kolli and K Chanclrashckhara, Nonlinear static and dynamic analysis of stiff-

rnod laminated plates Int J Non-linear Mechanics, Vol.32, Nol(1997) pp

89-5 Victor Birman I heory anil comparision of the effect of composite and shape mem­ ory alloy St iffrners Nability of composite shells and plates, Int ,J Mech Sri

Vol.39 XnlO pp,M39 1149