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1 -!" %&''(* +,+ -&(. /( ( * &+78 (4 ', 9:4' ;< =4 * )(>*/? A* ) B 9: ' 8 4 '5- *6 &7,- &. % / + % & % '*!" #$ C )* &%E.@ ; D BC &' 8 ; & % - ;% ;< &= > &=.@ ' +. ; %. '(. 7 / %.@ ; A H I(J. F ;% #; G.. '( K ;* &7 #;.@A " #CM. ;* &7 #;. *6 &7,- &. % /.7 I M>: %. O% &7/ E6 *6,-.7 I %N % &7RS,- &. % &' I(J A H.@ ( M>: '.. -) 7/ & % - $ K.; $ &UN VF (angle-ply). ' &7RS & % I(J &AH TL $. (cross-ply) *. ' &7RS & % I(J A H %.@ ; & WAM:. 4 C" &7 F &7 %!" #$ ' ; D &F &7!" #$ ;* #; BC &' 8 ; /. '( I '5- RS :)8)=D'E Free Vibration of Delaminated Composite Timoshenko Beam Considering Poisson s Effect H. Biglari Assistant Professor, University of Tabriz, Faculty of Mechanical Engineering M. Sattari Sarebangholi B.Sc., Student, University of Tabriz, Faculty of Mechanical Engineering Abstract In this paper, exact solution of free vibration analysis has been established to study the effect of delamination on natural frequencies of thick laminated composite beams. Timoshenko beam theory with various boundary conditions has been employed. The Poisson s effect is considered in the formulation. Both free and constrained modes have been considered. The influence of the size and location of the delamination on the natural frequencies and mode shapes of a beam are investigated. Studying of mode shapes clarifies the reason of difference of free and constrained modes results, well. The natural frequency shows a high sensitivity for the long and close-to-the-midplane delaminations, but low sensitivity for the others. Considering the Poisson s effect has almost not influence the cross-ply laminate natural frequencies. But, ignoring it in angle-ply laminates may cause large error. Obtained results are compared with the analytical and experimental data reported in the literature to verify the validity of the present analysis. By considering Poisson s effect for angle-ply laminates new data has been released. Keywords: Thick laminated composite beam, Free and constraint delamination mode shapes, Various boundary conditions, Exact solution. hbiglari@tabrizu.ac.ir : *

2 I<7 &7/ % &7@'5- #CM & %. F 994 E.@ A 9: >C &% s F ). %. ;< &= n % [] _ <7 [] ; 995 E. T &. %. *U.- % /. % t. 9 M [.] ;. R.- - u4. '( &7 '. T %.- > I<7 &" % *6 &4- / F. $ ;< &= 8 u4 '5- RS &=.A " %!" #$ 8 8 ; & % K.; A H I(J %. '(. / %.; D BC &'.. '( / &7 I<7 %.@ ; % 8 ; & % / '5- I J + $ &7&.( '!" #$ 8 BC &'.;%.&)*H+(- IJ (4', 9:4' -- RS % -$ ;* &= I<7 %-C. &=.; I% '5-.;% &*% 9: % RS 9M>: v-. ; H w: ;% &. I. ' &= &=.A H _<N 8 _NS %. ; E<L F )% #; G ;< #; G ' *% M>: % <L xun # <7 N )* % -$ *.4 M>: % <L ;< &=!6 / I.% I '5- F ' 9: % &M 9;* ' H w: %. w φ x w kg A + =bi xy x x t φ w EI x -kg A φ + =bi x xy φ x x x t () I, I \U zu. J A, b / U% w ;% zm{ V k.. UN &7C<F M E$ M>: % E I. φ. M>: N ;% E.. E* <N > G xy, EI t<7.@ y.;% '5- RS E* A- V- 9L I( &'. T> % F % % & % I<7 X S Y N # %.I( ' & F % Y N T.. J S TM C<F ' ZCU #[ J \: '5- A. % % % '( 9;* #CM. I. %-.N. 7 \: & % $ 6 4 '5- RS &7.;% ] 7. <L &7%- / '5- &7RS % N K Z. & % &7% % & F #- I; 4S TL. R. I; F 9L. ( C$ > _7- `L% /. % N a '..; I( *6,- _7-. &7 *6,- &. % / + % ' T N & F #%" 9*U '5-. &C _7.bJ.! M ' - &. '.@ ;,- _7- (.% T 987 E [] _ <7..- %_J / U. % F #%" *6 98 E.@; %N <7 % K % %_J <N-- icj- * % [] _.. i. % &" & % V %.% % CCM E % K % K M>:. - E6 % / J. R% R. - - l (.; F 988 E,5.(' -< N% 7 &= n % & CCM &= [4] I.. R% _%. - - l (. D %-C. % [5] &A. ;. n< 7 % <7 / J / A;'% 4 " " E ' > n % CCM E / #M A; 4%. '% %. D UN &7 ' > % ;< &= &C '( 9;* [7] _ <7. &>*F <7.< % ;< &= & % / V. b b. \% & % &F CCM< ). _ <7. A- qj % I(J V + N.@ A D b AR ). <7 & % [9.8]. 979 E [] _ <7. ' I - & % [6] A7..- %_J ;* &7 #; % / '5- &F )*. '*.<...( &' $% #!" Classical Laminated Plate Theory Delamination

3 +,-(./,( + n< X % 4%. 7A 8 ; % F %,5 ). J. R% '. % ; A H 8 ; % F %.; E<L &A' 8%. ( '(. *6,-. A #$ b. 4 &A'.; / ;* &7 #; ', +--- / J. R% '. - ; 7 ' # 4. ; < 7 % )* 9 T<.@ '% <7 / 7 9L %.- I ' 4 ' U% 6 % '* #$ & % x;. x, x 4 x 7A 8 x, x x J 8 ; ' &A' &7. E*. V; N J 8 ; '( F %.@ "% / &. % <N. ;% ' 9{% / & % 7 J x; #; % w =w =w φ = φ = φ x x x <N.A. ;% &. E* 8 ; V =V +V / % &7 J. 7-K :@ (5) :@ ' " % #; % F % / & % H-H H-H M =M +M +P -P 7' #<*,L. #<L &7.A. 7.-K (6) % ω &. ',- % qj '* φ iωt iωt x = Φ ( x) e, w =W ( x) e :; l ' 9: ',J. () 9R* () U% & WF % E$ () #4 > * &' & T. *. I- :( W(x) d 4 W d W p +q -rw= dx 4 dx EI I p=ei, q= + bi ω, GA I κ ω bi r= - ω bi kga & ' <* () :@ ' 9: % () #4 > * <L #$ 4 W x =c sinλx+c cosλx+c sinhµx+c coshµx -q+ q +4pr µ=,λ= p q+ q +4pr p &' 8 ; ' (4) #$ F &7@% 6+ (4', 9:4' -- / & '5- &'4 M _% S % / #M ' * E$.( ; ' 7 & % (4 #; % H & %.; % #;) ; # S- ' U% ) I( qj. () ;< / 7 &7 J.; x=x % & % 9{ #;!%U.@ ; % &. x=x % / & x=x % / & % / E6 % l % % #- E6.@ ; x=x 4 l 4. l % % V % 4. ' E6. (l I<7) l % % V %. ' X>. H #- X>.;% ' 7 & % % / #CM & %.@ H. H.; ; (4 U% * qj Free mode

4 Q 7R * N R &<; k &J b I( - &R ' zu X>. T k &R > z k.;% [N ' T k A +--- }J. R% '. % 4S 7 % v- )* 9 T< (. ) /. =W ) - I 7 %.. & % * W.- #<L I L % :A S5 ' 9{%. ' 8 ; < W(x) V4$ % (4) U% }qj % F %.; 4'% W(x) V4$ % % J :@ ' 9{% <N. )% &. 8%. ;< w V =kg A + x xy φx x (7) φx M xx =EI ( 8%.) % -$ 9R* ' :; #:$ ' K ( wɺɺ, ɺ φ ) x =...( &' $% #!" d W d W p +p + q +q - r +r W i = 4 dx dx ( i) ( i EI I ) ( i) q i = + bi ( i) ( i) ω kga I ( i ω bi ) ( i) ( i r ) i = - ω bi ( i), p i=ei i=, kga i i / &. % J 8 ; I L %. '. - 9.> %.@ '( <7 V; N J 8 ; W.- #<L ' :; ' 9{% / &. % I<. ;% &. x=x : W =W, W =W EIW = EI +EI W EA EA h EIW + W x -W 4 x EA +EA 4l x=x : W 4=W, W 4 =W EI4W 4 = EI +EI W = EI +EI W EA EA h EI4W 4 + W x -W 4 x EA +EA 4l = EI +EI W % C:$ b. 4 '* % % F % () V, I I " >: % % %.A *6,- +,+ (L' HM' -- A H %F I % F % 7 &= L _..; >: (ε yy ) L _- ; Constrain mode 4 () w M xx =-EI w φ x = x x w dm x Vx = = EI dx x (8) :[6]@ "% 7' &M &. & % ' 8%. t<7 P =P +P = Pl Pl H - = ( W W4 ) EA EA x x (9). (6.5) 8%. ( & WF. (9.8) 8%. ' > % # ' " % J 8 ; &' & T :; x=x : W =W =W, W =W =W EIW =EIW +EIW EA EA H EIW + ( W x -W ) 4 x EA +EA 4l =EIW +EIW x=x : W 4=W =W, W 4 =W =W EI4W 4 =EIW +EIW EA EA H EI4W 4 + ( W x -W ) 4 x EA +EA 4l =EIW +EIW <N >. (EA) E* - > () U% % % '5- RS (EI) E* N ( i) ( k EA =A =b Q ) (z -z ) i k+ k k= N (i) b ( k) EI i=d = Q (zk+-z k ) k= () () 4

5 +,-(./,( + #$ b. &7 % &.$ N < I L %.;% W T i T, - &7> I<7 (4) U% :@ ' 9: % #; <; ' x =T sinλ x+t cosλ x+t sinhµ x+t coshµ x 4 I< (6) * A4 &' 8 ; E<L & % (6) U% 9L %.; l % % (7) I( &F %. ; ; % A,T, &4 T< ' U T + T + T T 6= T :; & WF ' * # <7 * % <7 * V % 7 #;. 9R,. ; [ T ] = [ A] - (9) N&- )'O "-- % [ / 9] s IS % A - < % 7 mm / 7 / 6 *% %. ' & 9{ #CM. '( / & % ;< &= n E =4 GPa,E =. GPa,G =G =5 GPa ν =., ρ=48 kg/m :; % I J % F >: (σ yy ) &7. &C &7 I J.@ WA+.@ S (F 9. w. ') * RS & %.@ l<~ #%" ~. ' &7RS. F. <L ).. I J &AH &7_ &.A.. (% ). :E. ). >: % % RS &N 8%. RS RS - $ & % )..; " > )..@ >. (B=) ;% I ' U% ' () U% &% ; v: E* <N * EI= b D * * * D =D /D, D =DD +DD +D6D D =DD66-D6D6 n D =D6D6 -DD 66, D ij= Q zk+-zk D k= =DD6 -DD6 :[4] ; M ( k ) ij.;% RS <N > D ij R% 8%..;% C- E. ). ). :T. ). (4).@ &A% #%" I &7RS * ' R 7 &N 8%. (τ xy. σ yy ) L & ). - > W.; " >: % % RS Q4Q4 -QQ σ 44 xx = Q +Q QQ44-Q4 :[5] ; v: ' 9{% x & QQ 4-QQ 4 * +Q4 ε x =Qε x QQ44-Q 4 / &A " &7@*".@ 8. / ; % F. & % J,- K E.F.@ A & WAM:. 4 [] \F F K ' Q > &F % ().; > - * * Q * I( ; v: :4' )=+ K-4- ;* &7 #; *6,- ',J b. &7 % % 7 #; & %.A " F,- & % ;* 9R* ' T % <7 * *6 [ ] [ ] [ ] [ ] [ ] [ ] A 6 6 T = 6 6 (free mode) A T = (constrain mode) (6) & R% F. / &A Angle-ply 5

6 ( F. /) / '5- (Hz) J,--HE [] ;. R 8 / 88 8 / / 6 66 / 4 55 / 67 '( 8 / 88 8 / / 6 66 / 55 / 67 (;<) $ - 8 / 9 8 / 75 / 67 / 57 / '( 8 / 9 8 / 75 / 67 / 57 / / E6 5 / 4 5 / 8 76 / / 6...( &' $% #!" T. T. *6,- / E6 +-5K % [ / 9 / 9 / ] s IS % A - < 6 cm / 67 / 54 / 778 *% %. ' & 9{ :; #CM F. / 6 cm E6 % / & E =4.4 GPa, E =.7 GPa,G =G =4.4 GPa ν =.7, ρ=9.5 kg/m K % E. S *6,- E.F () 8%..@ A 4 [7] \F ' #:$ %.A Hw: &M 9;* ' - ; I% 6 9;* + ; I [8] \F E.F.@ S / }J &7 K % C:$ K!%U - ; 7 4 wvn R% &7..@ L J &7.% _ < / E. S *6,-- HE % UN /6 /4 -/ -5/7 $ K <; 4,- / E6 _ % E.F K % F K % % t<7.% _7- *6 *.@ I4 K - ; 7. '(.;< M / F. / - $ % (4. F.) / #M. / E6 + 4 #; A 4 [] \F F K % J,- &. 4% % + #;!%U.@ % ) F. / F.. *6,- F. *6,- `L% ( M>: I W.@ A (FN zu ) 4 F. / aun M>: % &7R / FN zu F. I( F. ' R% &7 *6,- _7- % / + *6,- % / E6 5 #;.@ ; \F %. M A4 '5- T. T.!%U. " F. /.@ ; 4 [6.8] 4% R% &7,- / %,- _7- % ' / & % W.@ ;.A " F R% *6 &7,- 4 &7#;. E.F ; D K 4 % #-!%U $ - ' #:$ K - ; 7 5..;% \F % %N (4)(RS-&(.(LQ4--. %-C. &= ' #:$ K 9.> I I & % J,- % (L/H) &~R V + ;<. 7 9{.@ ; I E.F mm / E6.;% < E.F.@ A " F..@ G E6. (H= / 6 mm) ( ) J,- % / E6. #M +-4K 6

7 +,-(./,( + J % R. / '5- (Hz) J,--4 HE '( 9 / 77 5 / 68 / 59 9 / 55 8 / 4 F. '(. &7 & % ;< &= 9 / 77 5 / 7 / 7 9 / 5 / 9 '( 9 / 77 6 / 79 6 / / 9 8 / 67 % F. 9 / 77 6 / 79 6 / / 9 8 / 67 / E6 5 / 4 5 / 8 76 / / 6 E.F K!%U % K 9.> RS - ' A C:..% / A% &7E6 9.>.; %. '( & <N ' / ;* G..% &7R % % #; _% X.;% '( % %.; +,+ (L'-4- I J V _%.A " % 9M IS % / '5- A - < :% H ' & 9{ % [θ/-θ/θ/-θ] S G = E =44.8 GPa,E =9.65 GPa,G = 4.4 GPa *%% / E6 ν =., ρ=89. kg/m (L/H=5) &~R V. " 8. 6 H ' /. ( L = L /L ) &=. K wvn L/H _7- % - ; 7 + I- M _.;% ;< &= F ;% #; G..,- 6+4 % / G + J+V T:U+(LQ4-- 6 #;.@ ; I < &7 & % / J *6 % #;!%U.@ I4. '(. 7 K / + A4 '( % _7- \"..; <-,- _7-. _7- > - A / % >.@ % S+J ( E =4 %&'(-- & %. / J *6,- 4 E.F ' /.@ ; % < E.F!%U. " 8. 6 H _7- *6,- / E6 _ % R FN % / % t<7.@.@ ; <- *6,- &. / + RS (Hz) J,- % (L/H) &~R V +- HE % wvn 6 / 4 5 / 4 / / 7 < / '5- ;< 77 / 7 / 6 89 / 68 / 5 %-C. 77 / / 7 97 / 89 7 / L/H E6 / J,- % I J V + -7K (' < '5- BC &" &7@*" & % A 4 / J,--6K (C 5 / 8 / E6) BC 6

8 )* / #M. E6. 9 #; ( '() / A4 [ / 9] s '5- E. ) F. / #M 9 #;.@ ; I.;% 4 F. / #M #;. ( M>: " F. / #;!%U.( ) ;< '% / 7 / BC &7,- I% I4 % C N.;% F. '( E6. " 4 F. / 7 #;!%U _ %.@ / S- / #; 4 %.@ '( 9{% / )* / E6 I( &A ". / E6 _ - ;. 9 ' / )* G `L% R% F..; '( ; *%% *6,- % I J + 7 #; ' #:$ K % BC &" E6 ( ω ω ω ω w J,- ω int wo w )/ int.@ A 4 [8] \F J,- ω wo. I J % / J,- E<L & % [8] \F.@ I J I.% / ). ' $ - K. (-) _% E. ). ' I J S- &.' I J.@ A > T. ; 7 - <7 F 45 w. '..@ l<~ #%". S w.- (%46) $ &UN I J A.% _7- / E6 _ % I J t<7 &.' T.. E. ). wvn 7 I K 4 x% UN ' &.; % O% w -. [8] \F % / )* #$ ). wvn %.;% $ 5 E.F T.. E. ). wvn!" 4 & %.@ A M < & % / J,- 4 F H ' 5 / 4 mm / E6. % wvn - ; H$v.@ ; \".. N [- $ % F 45 w &.' )....( &' $% #!" 9HM / A4 [ / 9] s )* M-8K (4 F.. C / 6 / E6) BC > % T. ). 8 / 56 / 79 5 / 46 < / J,- wvn -5 HE ( <) T.. E. ). ' E. ). 8 / 4 59 / 8 7 / 48 SR [/9] s [/-] s [45/-45] s & % / A4 E. ;* #;-9K ( F. /) / BC &7E6 :4' )=+ K-5- - F. I / % 4% T %. '% v:. ; '% uv- / 7 '( &7 % & %.- )*.A " % / ;* '5- E. ;* #; 8 #; /.@ ; I ( <) / A4 #; % F %.;% C / 6 E6 %. 4 F. / 7 & <N _ % N% # % T7.; '% 4% 7. F. / A; '% I J R %,- ' >C J,- 9L %. "% I% # % - ;%. '(.@4 7,- ' A % &' 8

9 +,-(./,( + Institute, Society of Plastics Industry, Section -E, pp. 5, 979. [] Lee B. T., Sun C. T., and Lin D., "An assessment of damping measurement in evaluation of integrity of composite beams", J. Reinf. Plastics. Compos. vol. 6, pp. 4-5, 987. [] Wang J. T. S., Liu Y. Y., and Gibby J. A., "Vibration of split beams", J. Sound Vib., vol. 84(4), pp. 49 5, 98. [4] Mujumdar P. M., and Suryanarayan S., "Flexural vibrations of beams with delaminations", J. Sound Vib. vol. 5(), pp. 44 6, 98. [5] Shen M. H. H., and Grady J. E., "Free vibrations of delaminated beams", AIAA J., vol. (5), pp. 6 7, 99. [6] Luo H., and Hanagud S., "Dynamics of delaminated beams", Int. J. Solids Struct, vol. 7, pp. 5-9,. [7] Jafari-Talookolaei R. A., and Abedi M., "Analytical solution for the free vibration analysis of delaminated Timoshenko beams", The Scientific World Journal, vol. 4, A. ID 856, pages, 4. [8] Kargarnovin M. H., Ahmadian M. T., Jafari- Talookolaei R. A., and Abedi M., "Semianalytical solution for the free vibration analysis of generally laminated composite Timoshenko beams with single delamination", Composites: part B, vol. 45(), pp ,. [9] Kargarnovin M. H., Jafari-Talookolaei R. A., and Ahmadian M. T., "Vibration analysis of delaminated Timoshenko beams under the motion of a constant amplitude point force traveling with uniform velocity", Int. J. Mechanical. Sci., vol. 7, pp. 9 49,. [] Ju F., Lee H. P., and Lee K. H., "Freevibration analysis of composite beams with multiple delaminations", Compos. Eng., vol. 4(7), pp. 75, 994. [] Shu D., "Vibration of sandwich beams with double delaminations", Compos. Sci. Tech., vol. 54(), pp. 9, 995. [] Della C. N., and Shu D., "Vibration of beams with double delaminations", J. Sound Vib., vol. 8(-5), pp. 99 5, 5. [] Della C. N., and Shu D., "Vibration of beams with two overlapping delaminations in prebuckled states", Compos: Part B, vol. 8, pp. 9 8, 7. [4] Reddy J. N., "Mechanics of Laminated Composite Plates and Shells-Theory and )(-4 *6,- &. / + $ *U '* % x% 9R* %.; % '5- / ;* 9R*,5. ;% #; G.. 9R*.(.A M E6 7A 8 ; C<F ' BC &7 J + &. w IS. / / / '5- )* #;. *6,-. '(. ' / %.@A " % & %. ' K.A > ;% A " FN F. /. ' / E6 -.@ 7 % / *6,- % I(J V + t<7 % < `L% I(J ' l<~.; %.A. ' &7RS %5.$ *6,- *6,- _7- / + A 7.% _ R% &7,- & % / A4 E. ;* #;-K (4 F. /) / BC &7E6 Y'( [] Ramkumar R. L., Kulkarni S. V., and Pipes R. B., "Free vibration frequencies of a delaminated beam", 4th Annual Technical Conference Proceedings, Reinforced/Composite 9

10 Analysis", Second edition, CRC Inc., New York, 4. [5] Aldraihem O. J., and Khdeir A. A., "Exact deflection solutions of beams with shear piezoelectric actuators", Int. J. Solids Structures, vol. 4, pp.,. [6] Radu A.G., and Chattopadhyay A., "Dynamic stability analysis of composite plate including delaminations using a higher order theory and transformation matrix approach", Int. J. Solids Struct, vol. 9: pp ,. [7] Okafor A., Chandrashekhara K., and Jiang Y. P., "Delamination prediction in composite beams with built-in piezoelectric devices using modal analysis and neural network", Smart Mater Struct, vol. 5, pp. 8 47, ( &' $% #!"

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