On the free vibrations of grid-stiffened composite cylindrical shells

Transcription

1 Act Mech 5, DOI 0.007/s M. Hemmtnezhd G. H. himi. Ansri On the free virtions of grid-stiffened composite cylindricl shells eceived: 7 Mrch 03 / evised: 0 July 03 / Pulished online: 0 Septemer 03 Springer-Verlg Wien 03 Astrct A unified nlyticl pproch is pplied for investigting the virtionl ehvior of grid-stiffened composite cylindricl shells considering the fleurl ehvior of the ris. A smered method is employed to superimpose the stiffness contriution of the stiffeners with those of the shell in order to otin the equivlent stiffness prmeters of the whole pnel. The stiffeners re modeled s em nd considered to support sher lods nd ending moments in ddition to the il lods. Therefore, the corresponding stiffness terms re tken into considertion while otining the stiffness mtrices due to the stiffeners. Theoreticl formultions re sed on first-order sher deformtion shell theory, which includes the effects of trnsverse sher deformtion nd rotry inerti. The modl forms re ssumed to hve the il dependency in the form of Fourier series whose derivtives re legitimized using Stokes trnsformtion. In order to vlidte the otined results, 3-D finite element model is lso uilt using ABAQUS CAE softwre. esults otined from two types of nlyses re compred with ech other, nd good greement hs een chieved. Furthermore, the influence of vritions in the shell thickness nd chnges of the oundry conditions on the shell frequencies is studied. The results otined re novel nd cn e used s enchmrk for further studies. Introduction The virtion nlysis of cylindricl shells e.g., frequencies, mode shpes, nd modl forces is well-estlished rnch of reserch in structurl dynmics. Cylindricl shells due to their high strength s well s light weight hve gined mny pplictions in civil, mechnicl, nd erospce engineering e.g., lunch vehicles, reentry vehicles, ircrft fuselges, spcecrfts, etc. in prticulr. The lrge numer of pulictions which hs een epnded rpidly in the pst decdes testifies to this fct. An ecellent collection of reserch in this re ws crried out y eiss. There re lso some good reviews on virtion of composite shells using eperimentl,3 nd nlyticl methods 4 0, nd numericl techniques 5. Bsed upon the first-order sher deformtion theory, Ansri nd Drvizeh 6 presented generl nlyticl pproch for investigting the virtionl ehvior of functionlly grded cylindricl shells. ecently, Hemmtnezhd et l. 7 investigted the virtionl ehvior of composite cylindricl shells using the sme nlyticl pproch s in 6 nd sed on different shell theories. Grid-stiffened cylinders re cylinders with stiffening structures either on the inner, outer, or oth sides of the shell. These stiffeners significntly increse the lod resistnce of cylinder without much increse in weight. The optimum kind of stiffener configurtion is chosen sed on the type of ppliction, the loding condition, M. Hemmtnezhd G. H. himi B Deprtment of Mechnicl Engineering, Trit Modres University, P.O. Bo 45-43, Tehrn, Irn E-mil: rhimi_gh@modres.c.ir. Ansri Deprtment of Mechnicl Engineering, University of Guiln, P.O. Bo 3756, sht, Irn

2 60 M. Hemmtnezhd et l. cost, nd other fctors. The promising future of stiffened cylinders hs led to wide rnge of reserch work 8 5. The numer of pulictions deling with the mechnicl ehvior of composite cylinders with cross stiffeners is scrce, nd most of the relted reserches re ssocited with stiffened cylinders with longitudinl nd circumferentil stiffeners. Kidne nd his ssocites 6,7 derived the uckling lods of generlly cross nd horizontl grid-stiffened composite cylinder y developing smered method for determintion of the equivlent stiffness prmeters of grid-stiffened composite cylindricl shell. Afterwrd, Yzdni 8 nd Yzdni nd himi 9 performed eperimentl investigtions on the uckling ehvior of composite cylindricl shells with cross stiffeners. They lso studied the effects of the numer of helicl ris nd chnges in grid types on the uckling lod of these structures 30,3. ecently, himi nd his co-ssocites studied the effect of the stiffener cross-section profile on the uckling strength of composite stiffened cylindricl shells y implementing the finite element method 3. The virtion nlysis of grid-stiffened cylindricl shells hs not een seen in the literture to the est of the uthors knowledge. In the present work, clcultion of the overll response of composite cylindricl shells with cross stiffeners is presented using n ect nlyticl pproch nd sed upon first-order sher deformtion theory. A smered method is employed to superimpose the stiffness contriution of the stiffeners with those of the shell in order to otin the equivlent stiffness prmeters of the whole pnel 6,7. In the smered method proposed in the previous studies 6,7, it is ssumed tht the trnsverse modulus of the unidirectionl stiffeners is much lower thn the longitudinl modulus. Therefore, the stiffeners re ssumed to e two-force memers supporting il lods only, nd the cpility of supporting sher lods nd ending moments is not considered for them. The virtionl nlysis of the grid composite cylinders considering the fleurl ehvior of the ris hs not een crried out efore. Since sher lods re effective forces in composite structures nd ending moments re more importnt in the uckling nd virtionl ehviors, ignoring theses effects my give nlyticl results with higher errors. The present work dels with free virtion nlysis of composite stiffened cylindricl shells under ritrry oundry conditions. The im is to find precise nd effective mthemticl model cple of predicting the virtionl chrcteristics of stiffened cylinders to void performing multiple eperiments. A smered method is employed to superimpose the stiffness contriution of the stiffeners with those of the shell in order to otin the equivlent stiffness prmeters of the whole pnel. In ddition to the il lod, the stiffeners re lso ssumed to support sher lods nd ending moments. Then, generl nlyticl pproch sed on the firstorder sher deformtion theory 6 is used to find the nturl frequencies of virtion. The modl forms re ssumed to hve the il dependency in the form of Fourier series whose derivtives re legitimized using Stokes trnsformtion 0,7. Furthermore, the influence of vritions in the shell thickness nd chnges of the oundry conditions on the shell frequencies is studied. The results otined re novel nd cn e used s enchmrk for further studies. Equivlent stiffnesses Consider cylindricl shell reinforced with lozenge-type stiffener structure s shown in Fig.. First of ll, it is required to determine the equivlent stiffness prmeters of the overll structure in order to clculte the virtion frequencies of composite cylinder with inner stiffening structure. The nlyticl tool employed for this, so-clled the smered stiffener pproch, uses mthemticl model to smer the stiffeners into n equivlent lminte nd determine the equivlent stiffness of the lminte. The procedure is similr to tht used in 6,7 ut sed on the following refined ssumptions:. The sher stresses in the cross stiffeners re not to e ignored.. The cross-section dimensions of the stiffeners re very smll compred to the length. 3. A uniform stress distriution is ssumed cross the cross-sectionl re of the stiffeners. 4. The stiffeners re modeled s em nd considered to support sher lods nd ending moments further to the il lods. 5. The lod on the stiffener/shell is trnsferred through sher forces etween the stiffeners nd shell. 6. In the loding condition, the cross sections of the stiffeners do not go under torsion.. Force nlysis Bsed on the lminted plte theory, the strins on the interfce of the stiffener nd the shell re given y

3 Composite cylindricl shells 6 Fig. Unit cell nd coordinte system for stiffened cylindricl shell t ε = ε 0 + κ, t ε = ε 0 + κ, t ε = ε 0 + κ, ε z = εz 0, ε z = εz 0, which cn e considered s the strin components in the cross-section of the stiffeners. In the ove reltion, t is the shell thickness nd ε 0,ε0,ε0,ε0 z,ε0 z,κ,κ,κ descrie the mid-plne strins nd curvtures of the shell. To resolve these strin components long the stiffener direction l nd norml to it t, one cn use the following trnsformtion mtri: ε l c s 0 0 cs ε ε t s ε tz ε = c 0 0 cs ε 0 0 c s 0 ε z zl 0 0 s c 0 ε, z ε lt cs cs 0 0 c s ε where c = cos φ,s = sin φ, ndφ re the stiffener orienttion ngle. Figure shows the forces on the unit cell of the stiffener structure. Sustituting the pproprite ngle into Eq. nd pplying ssumptions nd3, one cn rech the il nd sher forces in the stiffener direction s F l = A l E l ε l = A l E l c ε + s ε + csε, F l = A l E l ε l = A l E l c ε + s ε csε, F lt = A l G lt ε lt = A l G lt csε csε + c s ε, F lt = A l G lt ε lt = A l G lt csε + csε + c s ε, 3 where E l nd G lt re the longitudinl nd sher modulus of the stiffeners, respectively. esolving the il nd sher forces in the nd directions nd then summing up the il nd circumferentil forces pplying on the sides of the unit cell, one rrives t F = F l + F l c + F lt + F lt s, F = F l + F l s F lt + F lt c. 4

4 6 M. Hemmtnezhd et l. Fl F lt F lt F l φ F l F lt F lt F l Fig. Force distriution on the unit cell The sher force is otined y summing the forces long ny sides of the unit cell s F = F l F l c + F lt F lt s. 5 Sustitution for strin components from Eq. into Eq. 3, using Eqs. 4 nd5 nd dividing the force epressions y the corresponding edge width of the unit cell, gives the forces per unit length s N st = A l E l c 3 + A l E l s ct N st = A l E l c s + A l E l s 3 t = 4A lg lt cs N st A lg lt cs t ε 0 + A l E l s c ε 0 + A lg lt c s s κ + A lg lt c s st κ, ε 0 + A l E l s 3 ε 0 + A lg lt c s c κ + A lg lt c s ct κ, ε 0 4A lg lt cs κ + A l E l c st κ, ε 0 + A l E l c s ε 0 + A l E l c 3 t κ ε 0 + A l E l c st κ ε 0 A lg lt cs t κ 6 where the superscript st stnds for the stiffeners. These forces should e trnsferred to the mid-plne of the shell. While trnsferring, ending moments re produced due to the distnce mong the points of ppliction of the rection forces in the stiffeners nd the mid-plne of the shell. These ending moments will e otined in the following section.. Moment nlysis As pointed efore, the rection moments due to the stiffeners re produced y the sher forces on the interfce of the stiffener nd the shell. The moment cused y these forces on the mid-plne of the shell equls the forces multiplied y one hlf the shell thickness. Figure 3 shows the moment free ody digrm of unit cell. M nd M re the moments resulting from forces F nd F, respectively. Following the sme mnner s ove for the force nlysis on unit cell, the resultnt moments on the horizontl nd verticl sides cn e otined s

5 Composite cylindricl shells 63 M M φ M M Fig. 3 Moments due to the stiffeners M st M st = A l E l c 3 t + A l E l s ct = A l E l c st + A l E l s 3 t = A lg lt cs t M st A lg lt cs t ε 0 + A l E l s ct ε 0 + A lg lt c s st κ + A lg lt c s st κ, ε 0 + A l E l s 3 t ε 0 + A lg lt c s ct κ + A lg lt c s ct κ, ε 0 A lg lt cs t κ + A l E l c st κ. ε 0 + A l E l c st ε 0 + A l E l c 3 t κ ε 0 + A l E l c st κ ε 0 A lg lt cs t κ 7.3 Sher force nlysis In plne with norml vector z, the strin component ε lz cn e written s The sher forces resulting from sher strins re given s ε lz = sε z + cε z. 8 F lz = A l G lz sεz 0 + cε0 z, F lz = A l G lz sεz 0 + cε0 z. 9 esolving these forces in the nd directions nd then summing up the forces on the upper nd lower sides of the unit cell, we rrive t q st = F lz + F lz c, q st = F 0 lz + F lz s, whichintermsofeq.9 cn e rewritten s q st = A lg lz c ε 0 z, qst = A lg lz csε 0 z. The resultnt sher forces per unit length cn e otined y dividing the ove forces y the corresponding length s

6 64 M. Hemmtnezhd et l. Q st = A lg lz c εz 0, which cn e rrnged in mtri form s Q st Q st = = A lg lz cs εz 0, Qst Al G lz c 0 ε 0 z A l G lz cs 0 εz Stiffness mtrices Equtions 6nd7 denote the force nd moment contriutions of the stiffener, respectively. These equtions re summrized in mtri form elow: A N st l E l c 3 A l E l s c A l G lt c s s A l E l c 3 t A l E l s ct A l G lt c s st ε 0 N st A l E l c s A l E l s 3 A l G lt c s c A l E l c st A l E l s 3 t A l G lt c s ct ε 0 N st 4A l G lt cs M st = 4A l G lt cs A l E l c s A l G lt cs t A l G lt cs t A l E l c st ε 0 A l E l c 3 t A l E l s ct A l G lt c s st A l E l c 3 t A l E l s ct A l G lt c s st. κ M st A l E l c st A l E l s 3 t A l G lt c s ct A l E l c st A l E l s 3 t A l G lt c s ct κ M st κ A l G lt cs t A l G lt cs t A l E l c st A l G lt cs t A l G lt cs t A l E l c st Also, the sher forces due to the stiffeners re given y Eq. 3. The resultnt force nd moments due to the shell in terms of the strin components of the mid-plne surfce of the shell cn e written s N sh N sh N sh M sh M sh M sh Q sh Q sh A A 0 B B A A 0 B B A B B B 0 D D = B B 0 D D B D A A 55 ε 0 ε 0 ε 0 κ κ κ ε 0 z ε 0 z 4, 5 in which A 44 = KA 44, A 55 = KA 55, K is known s correction fctor nd sh superscript stnds for the shell. Moreover, the stiffnesses of the shell re given s t Aij, B ij, D ij = Q ij, z, z dz, 6 t where Q ij re the plne stress-reduced stiffnesses, nd t is the uniform thickness of the shell with the reference middle surfce. The totl force nd moment on the pnel re the superposition of the forces nd moments due to the stiffeners nd the shell ccording to their volume frctions s N Vst N st + V sh N sh = M V st M st + V sh M sh. 7

7 Composite cylindricl shells 65 Fig. 4 A 3-D model of the grid-stiffened cylinder used in the present work Also, the superposition of the sher forces gives Q = Q Vst Q st + V sh Q sh V st Q st + V sh Q sh where V st nd V sh re the volume frctions of the stiffener nd the shell, respectively., 8 3 Cylindricl shell equtions A stiffened composite cylindricl shell, whose schemtic view is shown in Fig. 4, is considered here. The governing shell equtions in terms of resultnt forces nd moments including the trnsverse sher nd rotry inerti terms re 33 Q N where I, I, I 3 re the inerti terms otined s N + N N = I ü + I ψ, + + Q = I v + I ψ, + Q N ˆN w + q n = I ẅ, M + M Q = I ü + I 3 ψ, M + M Q = I v + I 3 ψ, I, I, I 3 = t 9 ρz, z, z dz. 0 t The resultnt forces nd moments in Eq. 9 re the resultnt forces nd moments of the shell-stiffener structure nd should e sustituted from Eqs. 7nd8. 4 Anlyticl procedure Bsed on the first-order sher deformtion theory, the strins nd curvtures on the mid-plne surfce of the shell cn e written s

8 66 M. Hemmtnezhd et l. ε 0 = u, ε0 = v ε 0 = u + v, ε 0 z = ψ + w, κ = ψ, + w, ε0 z = ψ + w v, κ = ψ, κ = ψ + ψ + v u. Utilizing Eq., Eq. 9 cn e epressed in terms of displcement field nd its corresponding derivtives s follows: A u, + A v, + w, + A 6 u, + u, + A 6 v u,, + w, + A 66 + v, + B ψ, + B ψ, + v, + B 6 ψ, + ψ, + v, + B 6 ψ, + v, ψ, + B 66 + ψ, + v, = I ü + I ψ, A u, + A v u,, + w, + A 6 u, + A 6 + u u,,w, + A 66 + v, + KA 44 ψ + w, v + KA 45 ψ, + w, + B ψ, + B ψ, + v, ψ, + B 6 ψ, + B 6 + ψ, v, + B 66 ψ, + ψ, + v, = I v + I ψ, A u, A v, + w A 6 + KA 45 ψ, + w, v, w, ψ, + v, KA 45 ψ + w, v B B 6 + B 6 u, + w, + B 66 u, + ψ, + v, ψ, + ψ, + v, w, + KA 44 ψ, v, + KA 55 W, + ψ, B KA 55 ψ + W, + B u, + B u, + D 6 ψ, + ψ, + V, + U, + D 6 = I ü + I 3 ψ, KA 44 ψ + W, v + B 6 u, + v u,, + W, + B 66 + u, + D 6 ψ, + ψ, + v, + D 66 ψ, = I ẅ, c u, + w, + B 6 u, + v, + D ψ, + D ψ, + v ψ, + v, + D 66 ψ, + ψ, + U, KA 45 ψ + W, + B u, + B + D ψ, + D ψ, + ψ, + v, u, + W, + B 6 u, d ψ, + v, + D 6 ψ, = I v + I 3 ψ. e For circulr cylindricl shell, the displcement field is ssumed to e function of the circumferentil wve numer n nd the il wve numer m. A generl epression for the displcement field my e written s

9 Composite cylindricl shells 67 u,,t = u cosnsinωt, v,,t = v sinnsinωt, w,,t = w cosnsinωt, ψ,,t = ψ cosnsinωt, ψ,,t = ψ sinnsinωt, 3 where ω is the nturl frequency of the shell, nd u, v, w, ψ,nd ψ re the il modl functions. The crucil prt of the present nlysis involves choosing pproprite series forms for these modl functions. A convenient set of Fourier series which stisfies the oundry conditions of shell with simply-supported ends with no il constrint SNA term-y-term cn e given s 6 u = A 0n + v = w = mπ A mn cos mπ B mn sin mπ C mn sin ψ = D 0n + ψ =,,, mπ D mn cos mπ E mn sin It is cler tht the sine series give zero vlues t the ends unless one specifies the following ffected terms:., 4 v 0 = v 0, v = v, w 0 = w 0, w = w, 5 ψ 0 = ψ 0, ψ = ψ. Sustitution of the set of displcement functions nd their derivtives into Eqs. e leds to n eigenvlue prolem whose eigenvlues give the nturl frequencies of SNA SNA stiffened shells. Since the im is to consider generl cses nd not necessrily considering ny prticulr type of oundry conditions, shell with freely supported ends with no tngentil constrint FSNT, which hs the end conditions u = N = Q = ψ = M = 0 = 0,, 6 is chosen s se prolem for the set of displcement functions given in Eq. 3. None of the ten oundry conditions given y Eq. 4 re stisfied y the set given in Eq. 4, on term-y-term sis. Therefore, Stokes trnsformtion is used to enforce constrints to stisfy the oundry conditions. While differentiting the displcement functions using Stokes trnsformtion, the end vlues given y Eq. 5 re required see the Appendi. The derivtive formuls for modl displcement functions re riefly given in the Appendi. Sustituting the set of displcement functions nd their derivtives into Eqs. e, the results cn e simplified into two distinct mtri equtions in which the Fourier series coefficients re coupled s λ λ λ 3 λ 4 λ 5 A mn F λ λ 3 λ 4 λ 5 B mn F nd λ 33 λ 34 λ 35 Symm. λ 44 λ 45 λ 55 λ0 λ 0 A0n = λ 03 D 0n λ 0 C mn D mn E mn F6 = F 3 F 4 F 5 7 F 7, 8

10 68 M. Hemmtnezhd et l. where λ ij i, j =,...,5, λ 0,λ 0,ndλ 03 re coefficients tht depend upon the mteril properties, the shell frequency, geometricl prmeters, nd the circumferentil nd il wve numers. The vlues of F to F 7 re in terms of unspecified end vlues, N 0, N, M0, M,v 0,v,w 0,w,ψ 0,ψ, due to pplying the Stokes trnsformtion. Using Eqs. 7 nd8, Fourier coefficients cn e epressed eplicitly in terms of the ten unspecified oundry quntities. Now, it is necessry to enforce the oundry conditions ssocited with FSNT shells which re oth geometricl nd nturl types. The geometricl oundry conditions which must e imposed re relted to u nd ψ, while those of nturl type re N = 0, Q = 0, nd M = 0t oth ends. For further detils, the reder is referred to efs. 5,6. This results in generl eigenvlue prolem which cn e used for ny possile comintion of oundry conditions. After imposing the constrint conditions due to the geometric nd nturl oundry conditions, one rrives t the following homogeneous mtri eqution: eij N 0 N M 0 M v 0 v w 0 w ψ 0 ψ T = 0, i, j =,,...,0. 9 A nontrivil solution for this homogeneous liner system eists if the determinnt of the coefficient mtri vnishes. This leds to e ij = 0, i, j =,,...,0, 30 resulting in chrcteristic eqution whose eigenvlues re the nturl frequencies of the FSNT shell. To derive the pproprite chrcteristic eqution of specified oundry condition, its ssocited end conditions must e imposed. The chrcteristic eqution required for ny type of oundry condition cn e etrcted y tiloring Eq. 9 in n pproprite wy 5. Therefore, the frequency determinnt ssocited with ny specified oundry condition cn e derived s sumtri of the generl ten y ten mtri in Eq esults nd discussion The stiffener cylinder structure shown in Fig. 4 is considered to e mde of Hs-Grphite/epoy with mteril properties listed in Tle. The cylindricl shell is ssumed to e four-lyered with 30/ 30 s stcking sequence, while in the stiffener structure, fiers re considered to e oriented in the ris directions. The geometricl prmeters ssocited with the present model re tken s those reported in Tle. Tle 3 illustrtes comprison of the nturl frequencies otined vi the present nlyticl pproch nd those reported y ABAQUS for SNA SNA grid-stiffened shell, nd good greement hs een chieved. esults re given for four circumferentil wve numers nd three different shell thicknesses. As cn e seen, the trends of the frequency response otined from the two nlyses gree well. Since ABAQUS considers the structure with its rel geometry, it seems tht the numericl results given y ABAQUS re closer to the rel vlues of the frequencies. However, from the comprison etween the two nlyses, it cn e concluded tht the present nlyticl scheme is good s well s cple one for predicting the virtionl ehvior of grid-stiffened cylindricl shells. Tle 4 gives comprison etween the nturl frequencies of stiffened Tle Mteril properties of Hs-Grphite/epoy Hs-Grphite/epoy Young s Modulus GP E, E, E , 0.34, 0.34 Sher Modulus Gp G, G 3, G 3 7., 7., 7. Poisson s rtio ν,ν 3,ν 3 0.8, 0.8, 0.8 Density kg/m 3 ρ,389.3 Tle Geometricl prmeters of the present model Shell height 54 mm Shell inner dimeter 40 mm Shell thickness 0.4 mm Unit cell height 7 mm Unit cell circumferentil length 73.3 mm Stiffener orienttion ±60 Stiffener cross-section 6 6mm

11 Composite cylindricl shells 69 Tle 3 Comprison of the nturl frequencies reported y two types of nlyses for SNA SNA stiffened cylindricl shell n t = 0.4 mm t = 0.8mm t = mm ABAQUS Anlytic ABAQUS Anlytic ABAQUS Anlytic,057,,3,908,36,786 3,654,4,66,457,663,48 4,37,006,445,978,46,959 5,947,955 3,074,87 3,8,807 Tle 4 Comprison of the nturl frequencies for cylindricl shell with nd without stiffeners n C C SNA SNA C F Unstiffened Stiffened Unstiffened Stiffened Unstiffened Stiffened 4, ,85.5 3, ,675.3, ,94.44,96.9 5,997.6,45.44,.4,45.53, , , , , , , , , ,9.6 3, , , ,8.9 4, , , , ,707.76,0.3 5,47.9, , Unstiffened Stiffened n=6 Nturl frequency Hz n=5 n= Shell thickness m 0 3 Fig. 5 Vrition of the nturl frequency with the shell thickness for SNA SNA cylindricl shell nd unstiffened cylindricl shells for seven vlues of circumferentil mode numers nd different oundry conditions. The oundry conditions considered here re SNA SNA, clmped clmped C C, nd clmped free C F. As would e oserved, the vlues of the frequencies for the stiffened shell re higher thn tht of unstiffened shell. This is minly ecuse of the fct tht lthough the grid structure results into n increse in the mss of the structure, ut it increses the stiffness s well nd tht is why the presence of the grid structure increses the nturl frequencies of the shell. As epected efore, the fully clmped composite shell hs the highest nturl frequencies mong the selected oundry conditions. In ddition, for oth stiffened nd unstiffened cses, frequency curves converge for circumferentil wve numers greter thn si. Figure 5 ehiits the frequency vrition over the shell thickness of SNA SNA grid-stiffened cylindricl shell. esults re lso compred to those of n unstiffened shell. As cn e seen from this figure, the influence of shell thickness vrition on the frequency curve is more significnt for the unstiffened shell, nd the nturl frequencies for this cse increses y n increment in the shell thickness sed upon nerly liner trend. This is ecuse of the fct tht in the cse of the unstiffened shell, n increse in the shell thickness increses the stiffness fster thn the mss, wheres the trend is different in the cse of the grid-stiffened shell. Also, it cn e pointed out tht for shell thicknesses higher thn.5 mm, the frequency curves for oth stiffened nd unstiffened shells converge, nd this mens tht for shell thicknesses greter thn specific vlue, the presence of the grid structure does not provide ny effect on the nturl frequency vlues. Figure 6 depicts the mode shpes of virtion of grid-stiffened cylindricl shell ssocited with different oundry conditions.

12 60 M. Hemmtnezhd et l. n = n = n = n = 3 n = 3 n = 3 n = 4 n = 4 n = 4 Fig. 6 Mode shpes ssocited with SNA SNA, C C, nd c C F grid-stiffened cylindricl shell c 6 Conclusions A unified ect nlysis sed on Fourier series is employed to investigte the dynmic ehvior of grid composite cylindricl shells with cross stiffeners. Theoreticl formultions re sed on first-order sher deformtion shell theory which includes the effects of trnsverse sher deformtion nd rotry inerti. A smered method is employed to superimpose the stiffness contriution of the stiffeners with those of shell in order to otin the equivlent stiffness prmeters of the whole pnel. In the smered method, the fleurl ehvior of the ris is lso tken into considertion which provides us with more precise model of the stiffeners. These equivlent stiffnesses will then e entered into the nlyticl procedure in order to otin the nturl frequencies of virtion. To vlidte the correctness of the otined results, 3-D finite element model is uilt using the ABAQUS CAE softwre. The results given re novel nd cn e used s enchmrk for further studies. The otined results clrify tht the vlues of the frequencies for the stiffened shell re higher thn for the unstiffened shell. This is minly ecuse of the fct tht lthough the grid structure results in n increse in the mss of the structure, ut it increses the stiffness fster nd tht is why the presence of the grid structure increses the nturl frequencies of the shell. Also, the shell thickness hs significnt effect on the virtion frequencies of grid shells, nd this effect is more pronounced for the unstiffened shell cse rther thn the stiffened one. Furthermore, for shell thicknesses greter thn specific vlue, the presence of the grid structure does not provide ny effect on the nturl frequency vlues, nd the frequency curves ssocited with oth stiffened nd unstiffened shells converge.

13 Composite cylindricl shells 6 Appendi: Stokes trnsformtion When differentiting Fourier series, cre must e tken with respect to the end vlues. For emple, the end vlues of the functions represented y sine series re forced to e zero, ut using Stokes trnsformtion, the end vlues of the sine series re relesed y eing defined seprtely. Consider function f represented y Fourier sine series in the open rnge 0 < < ndyvlues f 0 nd f t the end points s f = n sin nπ, 0 < <, n= f 0 = f 0, f = f, Since we re not sure out the derivtive f to e represented y term-y-term differentition of the sine series, the derivtive is insted represented y n independent cosine series of the following form: f = 0 + n= n cos nπ. Now, Stokes trnsformtion consists of integrting y prts in order to otin the reltionship etween n nd n s follows: n = = 0 f cos nπ d f cos nπ + nπ 0 0 f sin nπ d = n nπ f f 0 + n. A similr mnner s ove must e tken when finding the correct sine series corresponding to f. Therefore, the complete set of derivtive formuls for the sine series cn e written s { f = n= n sin nπ, 0 < <, f 0 = f 0, f = f, f = f f 0 { f0 n } π f n n cos nπ, 0, n= { f = π n= n { f 0 n f } π n n sin nπ, 0 < <, f 0 = f 0, f = f Similr trnsformtion formuls must e used to otin the correct form of the successive derivtives of the cosine series. These formule re used for the derivtives of displcement functions in the solution procedure of the present nlysis. Some of the derivtives of displcement functions re given in the following: u,, t = A 0n + π u, = π u, 0, = π u, = A mn cos mπ ma mn sin mπ ū 0 cos n,u,,= ū 0 +ū + cos n sin ωt, 0, cos n sin ωt, 0 < <, π ū cos n, {ū0 +ū m m A mn } cos mπ. cos n sin ωt, 0,

14 6 M. Hemmtnezhd et l. v,,t = B mn sinmπ / sin n sin ωt, 0 < <, v0, = π v, = π v 0 sin n,v,= v 0 + v π v, = π w, = w 0 + w π w, = π v sin n, { + v0 + v m } mπ + mb mn cos sin n sin ωt, 0, { v0 m + v m m + m } mπ B mn sin sin n sin ωt, 0 < <, { + w0 + w m } mπ + mc mn cos cos n sin ωt, 0, { w0 m + w m m + m } mπ C mn sin cos n sin ωt, 0 < <, mπ ψ,,t = D 0n + D mn cos cosnsinωt, 0, π ψ, = md mn sin mπ π ψ, 0, = ψ 0 cos n, π ψ 0 + ψ ψ, = + ψ,,t = cos n sin ωt, 0 < <, π ψ,,= ψ cos n, { ψ 0 + ψ m m } mπ D mn cos E mn sinmπ / sin n sin ωt, 0 < <, ψ 0, = π ψ, = π π ψ 0 sin n,v,= ψ 0 + ψ π ψ, = + ψ sin n, } {ψ 0 + ψ m + me mn cos mπ {ψ 0 m + ψ m m + m E mn } sin mπ cos n sin ωt, 0, sin n sin ωt, 0, sin n sin ωt, 0 < <, The successive derivtives with respect to nd t re simply chieved. For emple, the successive derivtives of u,,t with respect to re s follows: u, = n A 0n + A mn cos mπ sin n sin ωt, u, = n A 0n + A mn cos mπ cos n sin ωt. eferences. eiss, A.W.: Virtion of shells, NASA SP-88 US Govt Printing Office 973. Egle, D.M., Bry, F.M.: An eperimentl study of free virtion of cylindricl shells with discrete longitudinl stiffening. School of Aero-Spce nd Mechnicl Engineering, University of Oklhom, NSF Grnt, Finl eport, GK-490, Novemer 968

15 Composite cylindricl shells Shrm, C.B.: Clcultion of nturl frequencies of fied-free circulr cylindricl shells. J. Sound Vi. 35, Shrm, C.B., Drvizeh, M.: Free virtion chrcteristics of lminted, orthogrphic clmped-free cylindricl shells, developments in mechnics. In: Proceedings of the 9th Midwestern Mechnics Conference, Deprtment of Engineering Mechnics, The Ohio Stte University, Columus, Ohio, 9 Septemer Drvizeh, M.: Free virtion chrcteristics of orthotropic thin circulr cylindricl shell. Ph.D. Disserttion, UMIST Shrm, C.B., Drvizeh, M., Drvizeh, A.: Free virtion response of multilyered orthotropic fluid-filled circulr cylindricl shells. Compos. Struct. 34, Birmn, V.: Ect solution of isymmetric prolems of lminted cylindricl shells with ritrry oundry conditions: higher-order theory. Mech. es. Commun. 9, m, K.Y., oy, C.T.: Influence of oundry conditions nd fier orienttion nd the nturl frequencies of thin orthotropic lminted cylindricl shells. Compos. Struct. 3, m, K.Y., oy, C.T.: Influence of oundry conditions for thin lminted rotting cylindricl shell. Compos. Struct. 4, Drvizeh, M., Hftchenri, H., Drvizeh, A., Ansri,., Shrm, C.B.: The effect of oundry conditions on the dynmic stility of orthotropic cylinders using modified ect nlysis. Compos. Struct. 74, Bert, C.W., Mlik, M.: Differentil qudrture method: powerful new technique for nlysis of composite structures. Compos. Struct. 39, Shrm, C.B., Drvizeh, M., Drvizeh, A.: Free virtion ehvior of heliclly wound cylindricl shells. Compos. Struct. 44, Hftchenri, H., Drizeh, M., Drizeh, A., Ansri,., Shrm, C.B.: Dynmic nlysis of composite shells using differentil qudrture method DQM. Compos. Struct. 78, Gnesn, N., Kdoli,.: Buckling nd dynmic nlysis of piezothermoelstic composite cylindricl shell. Compos. Struct. 59, Kdoli,., Gnesn, N.: Free virtion nd uckling nlysis of composite cylindricl shells conveying hot fluid. Compos. Struct. 60, Ansri,., Drvizeh, M.: Prediction of dynmic ehvior of FGM shells under ritrry oundry conditions. Compos. Struct. 85, Hemmtnezhd, M., Ansri,., Drvizeh, M.: Prediction of virtionl ehvior of composite cylindricl shells under vrious oundry conditions. Appl. Compos. Mter. 7, Junky, N., Knight, N.F., Amur, D..: Optiml design of generl stiffened composite circulr cylinders for glol uckling with strength constrints. Compos. Struct. 4, Helms, J.E., i, G., Smith, B.H.: Anlysis of grid stiffened cylinders. In: Proceedings of the Engineering Technology Conference on Energy ASME/ETCE, Houston Blck, S.: A grid stiffened lterntive to cored lmintes. High Perform. Compos. 0, Junky, N., Knight, N.F., Amur, D..: Formultion of n improved smered stiffener theory of uckling nlysis of gridstiffened composite pnels. NASA technicl Memorndum 06, June 995. Phillips, J.., Gurdl, Z.: Structurl nlysis nd optimum design of geodesiclly stiffened composite pnels. NASA eport CCMS-90-05, July Gerdon, G., Gurdl, Z.: Optiml design of geodesiclly stiffened composite cylindricl shells. AIAA J. 3, Junky, N., Knight, N.F., Amur, D..: Optiml design of grid stiffened composite pnels using glol nd locl uckling nlysis. J. Aircrft 35, Wng, J.T.S., Hsu, T.M.: Discrete nlysis of stiffened composite cylindricl shells. AIAA J. 3, Kidne, S., i, G., Helms, J., Png, S., Woldesenet, E.: Buckling lod nlysis of grid stiffened composite cylinders. Compos. B 34, Wodesenet, E., Kidne, S., Png, S.: Optimiztion for uckling lods of grid stiffened composite pnels. Compos. Struct. 60, Yzdni, M.: Anlyticl nd eperimentl uckling nlysis of grid stiffened composite shells under il loding. Ph.D. Disserttion. Trit Modres University, Irn Yzdni, M., himi, G.H., Afghi Khtii, A., Hmzeh, S.: An eperimentl investigtion into the uckling of GFP stiffened shells under il loding. Sci. es. Essy 4, Yzdni, M., himi, G.H.: The effects of helicl ris numer nd grid types on the uckling of thin-wlled GFP-stiffened shells under il loding. J. einf. Plst. Compos. 9, Yzdni, M., himi, G.H.: The ehvior of GFP-stiffened nd -unstiffened shells under cyclic il loding nd unloding. J. einf. Plst. Compos. 30, himi, G.H., Zndi, M., souli, S.F.: Anlysis of the effect of stiffener profile on uckling strength in composite isogrid stiffened shell under il loding. Aerosp. Sci. Technol. 4, Toorni, M.H., kis, A.A.: Generl equtions of nisotropic pltes nd shells including trnsverse sher deformtions, rotry inerti nd initil curvture effects. J. Sound Vi. 37,