VNU 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 IN E A R ST A B IL IT Y O F S T IF F E N E D L A M IN A T E D C O M P O S IT E PLATES K h u c Van P h u Military Technical Academ y A b s tra c t This paper deals with the non-linear stability of th e stiffeiK'd lai.ilt( composite plate su b jected to biaxial loads Numerical results are presented for illutr ii theoritical analysis of stiffened and unstiffened lam inated co m p o site plates Key words Stiffened lam inated com p osite plate, Shape m em ory alloys (SMA), stbity I n t r o d u c t i o n Stiffened laminated 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 attention [3, 4, 5] sjedl] tor muilysis of noil-linear stiffened laminated composite plates and shells subjoti compress l)i-axial loads This problem is studied in the present paper 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 la te s Let’s consider a rectangular stiffened laminated composite plate, ill which call y ( is a unidirectional composite m aterial This plate is subjected to H uniform coin)(^i( o i l Vcich edge, with result ants p , and Py respectively (Figure ), where p x and 'y 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 hi kx = d2w d IV ky ~ ỡy2 ’ Ỡ2 W k Xy — —2 where a, V w are the midplane displacements along the X, y and z axes respectively S2 Zl X f Y ỌI N Fz # 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 £y 4- { E \ A \ / S i ) z \ k x 4- p ^ / s , N y = ( A 2‘ 4- E A / S2 )Sy + A \ EX + ( E 2A 2/ S 2) z 2ky -f P y / s o, ^1*1/ — ^4f) (Slxy, (3) M e — (£>11 + f ii /i/5 i) f c « 4- D i2ky 4- ( E i A i / $ i ) z i £ x i , M y = ( D 2 + E I / so)ky 4- D i o k x + ( E A / S2)z2Sy, Mxy — Dị-t(\h’Xy i where - Dij ( i j = ,2 and 6) are e x te n d in g and b e n d in g stiffn esses of th e plate without stiffeners, - Ei: E are the Young modulus of the longitudinal and transversal stiffeners respectively, N o n - lin e a r s ta b ility o f s t i f f e n e d la m i n a te d c o m p o s ite p la te s 35 - A\ A o cire the section areas of the longitudinal and transversal stiffeners respec tively - 1 In are the inertial moments of cross-section of the longitudinal and transversal St iffoners, respectively, - ,sI , s2 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 the longitudinal and transversal stiffeners, respectively, - IP r ' , lP'\ are m thee recovery rensiie tensile iorce force 111 ill rile the OIVI/V SMA wires VVIIt'b The equilibrium equations of a plate according to [2] are ONr , d N xy Ox Oy ^ OX o 2M r 0~I\I,I ỊI d 2M y 2w - —-7— - - —- H T—7T 4- iV;r- —r -f ỡư:2 Oxỡy ỡy2 dx2 - (4) ay n d 2w d 2w d 2w 2m XỊJ — + iv y — 5- + :r — Ộ + i yTTT — ởxỡy dy2 Ox dy2 Substituting (2) and (3) into (4) after some operations we obtain the equilibrium equations of the lam inated plate (An + + 1^ 12 12 + A ° ^ ddxx dd yy ~ /Sl w + , 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 G>:r y.T2 dy d x d y u x ay- + (Ao2 - E o A - ỉ / so ) ^ ^ + + (^12 + ^60)77-77 - (^2^2/^2)2:277-^ + v ỡy Ỡ;/ J Oxay OIJ „ xd w d 2w tA A , d w d 2w , d w d 2w + ( A 22 4- E Ả / Sn) — 77 Õ" + (^12 + ^OGyTT- o + ^66 o o — Uv J dy y ' dx dxdy ay dx- ( D u + E i / i / S i ) ^ ^ + ( Ơ + 2D(ifi) - ^ Q^ + (£*12 + £ '2/ / S2) ỹ ^ ' _ ^ - ( p v + p y i ^ h f ~ ( E i A 1/ s 1) z 1^ - - ( E 2A 2/ s2) z ị ^ ~ Oiu d :i tu , , d w d 3w „ - ( E l A l / s ) z 1i^ ^ - (E 2A 2/ s )z2^ ^ - ụAn + , D2w f d i v \ Q2 w ( d w \ 2w ( d w \ d 2w ( d w \ “T o ( ) - ^ 12 T T h r - ( ^ 2 + E 2A s ) ^ - y ( I ỡy2 Vy r) Ớ.7;2 V d y J dy Vd y ) A Ow du) w , Vd u d 2w On d 2w d u d~w Aq6 —— 7-—— —— — (-All + E \ A \ / s 1) — — 2" —2 ^4(56 n 0 ^12 fj o o ÔX d y d x d y Ox d x dy d xd y ỜX UIJ - - ^ d v d 2w 12 dy Ox dv d w " 66 Ỡ£ d x d i r-, A / X9 v d W (Aoo -f- E o A o / s o ) —— _ — 0- ỠU d i r 36 K h u c Van P h u For a plate simply supported on all edges, the following boundary conditions are imposed -f At edges X = and X = a w = 0, V — 0, M x = 0; (6) u = 0, M y = 0; (7) + At edges y = and y = b w = 0, The boundary conditions discussed here can be satisfied if the buckling mode shape is represented by _ nny rriTTX u = u mn COS — a V = sin —- - T rriTTx V m n s ill — TITTV COS a Jir ■ m7TX w = w rnn s ill — , b a , (8 ) bv1 nny sin —-— , b 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 ( ) and applying the Galerkin procedi ice yield the set of three algebraic equations with respect to the amplitudes u,nn, Vnnii w mn, where th e first two eq u ation s o f th is s y s t e m are linear algebraic equations foi umm V-mn (l\ U tmi + O2 V 11U1 — (l,\W mil 4- (1 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 • a 9W^lw + a i 0W?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 — (All + E \ A \ / S \ ) —■ — h A 66 a 0*2 = &5 = ( A \2 -f A ẽo)m n, „2 _ rra b ' _ / El , \ m 3nb — {E\A\/S\)z\ a2 16 ( ^ n +£7 v41/ S l) ( — )) 2-— - (A 12 - A m ) ~ a //* ỈĨ7T /77Ĩ \ (I l)7T 7n 2b (lQ — (A 22 4- E 0A 2/ S2) - J - + L66 , a “4 = - y ^ „ _ / , n A / \ n a A m b Ò2 16 08 2(^22 + E 2A 2/ s 2) ( j ) - ^ \b / rriTT - ( A n - A (M) ~ 'an N o n - lin e a r s ta b ility o f s t i f f e n e d la m i n a te d c o m p o site p la te s a9 ( - A l l + jE’i A i / s i ) — 3- 4- ( ^ A \ + 2^ 66) “— ~b~ + ( ^ 22 37 + ^ 2^ 2/ 52 ) - p / / i ( g 6a4 - Q2fl8) -í- H (fi-1^8 - ^ 5^ 4) a i a — «2^5 [(jElA l/Sl)2l( ? ) 3^ flio -f + ( ^ A 2/ s 2)z2( ^ ) £ ^ j + H \ (a.ịO.ịị — Ci2a ĩ ) + / / ‘2 (a i a — a 3a ĩ>) + H^(ãQã4 — a i a e — a2ữ5 Try^ an 4- / / 4( ^ 1^,« — (1 (1 ) f77777Ì ^ ( D l l + ^ l A / s i ) — — I" ( L >12 + D q q ) - h ( - D 22 + _ _Tỏ(I E I I S ) —g f + / / 3(a3a6 - a 2a 7) + #4(ai&7 - a 3a 5) aựiG — a2«5 n 2a / m 2b A Va7T2 + a )■ 16 Ỵ - ) - ^ ( i 1i + J i ^ i/ i ) + (i412 + 2A66) | ^ tm ~9~ A a / n7T:i 16 1G r r n \ a / A n A / \ i A n A \ 771 ' H = - n ( u ) ~^3 ^ ( ^ 22 + E 2A 2/S 2) H~ (^12 + 2^4fiei) — —9 LVb / 77Ì7T3 a7T;i J Hi = ,„ , , m 36 # — —- ( £ i > W s i ) z i , tt 7T „ , n 3a H = —- ( E A / s o ) z ~7Tf I r 7Ĩ • From (10) we can express compression load with respect to H^mn as follows p* = V(Wmn) ( 11) The lower buckling load of the plate can be analysed by the minimum of