Elastic Buckling Analysis of Ring and Stringer-stiffened Cylindrical Shells under General Pressure and Axial Compression via the Ritz Method

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1 Journal of Solid Mechanics Vol., No. 4 () pp Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells under General Pressure and Axial Compression via the Ritz Method A. Ghorbanpour Arani *, A. Loghman, A.A. Mosallaie Barzoi, R. Kolahchi Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Islamic Republic of Iran Received 5 September ; accepted November ABSTRACT Elastic stability of ring and stringer-stiffened cylindrical shells under axial, internal and external pressures is studied using Ritz method. The stiffeners are rings, stringers and their different arrangements at the inner and outer surfaces of the shell. Critical bucling loads are obtained using Ritz method. It has been found that the cylindrical shells with outside rings are more stable than those with inside rings under axial compressive loading. The critical bucling load for inside rings is reducing by increasing the eccentricity of the rings, while for outside ring stiffeners the magnitude of eccentricity does not affect the critical bucling load. It has also been found that the shells with inside stringers are more stable than those with outside one. Moreover, the stability of cylindrical shells under internal and external pressures is almost the same for inside and outside arrangements of stringers. The results are verified by comparing with the results of Singer at the same loading and boundary conditions. IAU, Ara Branch. All rights reserved. Keywords: Bucling analysis; Stiffened cylindrical shells; Ritz method. INTRODUCTION E LASTIC bucling analysis of stiffened cylindrical shells is more difficult than unstiffened one. The difficulty is more considerable when the shell is reinforced by irregular stiffeners with different material properties. Despite numerical analysis, it is not always possible to find out an exact solution. Although a stiffened shell may be modeled with an equivalent orthotropic shell, the results are only accurate if the stiffeners are distributed evenly and closely spaced. One of the best methods for deriving eigen-values is the Ritz method. In this method, the assumed displacement functions may satisfy any combination of boundary conditions. The power of polynomial equations assumed for the displacement functions can be adusted according to the boundary conditions in Ritz method. The general instability of a simply supported cylindrical shell under hydrostatic pressure is analyzed by Singer and Baruch []. They showed that the critical loads for inside-stiffened rings are about -5 percent greater than outside ones. Ghorbanpour et al. [] studied elastic stability of cylindrical shell with an elastic core under axial compression by energy method. Their results show that using an elastic core increases elastic stability and significantly decreases the weight of the cylindrical shells. The stiffeners are attached to the shell as a fully welded ring. As Oalvo and Newman [3] have pointed out, an effective line of attachment must be assumed in these cases. Thermoelastic bucling of thin cylindrical shell based on improved stability equations is analyzed by Eslami et al. [4].They showed the magnitude of thermoelastic bucling of thin cylindrical shells under different thermal loading and their results * Corresponding author. address: aghorban@ashanu.ac.ir; a_ghorbanpour@yahoo.com (A. Ghorbanpour Arani). IAU, Ara Branch. All rights reserved.

2 A. Ghorbanpour Arani et al. 333 were extended to short and long thin cylindrical shells. Singer et al. [5] studied the effect of eccentricity on the critical bucling load of cylindrical shells under axial compression. Their results showed that the effect of eccentricity is very similar to clamped and simply supported shells. Bucling behavior of long steel cylindrical shells subected to external pressure is studied by Hubner et al. [6]. Their numerical findings are verified by a set of experimental tests with a series of cylindrical shells subected to external pressure. The tests provide the fundamental basis for the proposed improved assessment procedure for cylindrical shells. Buermann et al. [7] considered a semi- analytical model for local post-bucling analysis of stringer and frame-stiffened cylindrical panels. They captured the effects of local sin bucling and stiffener tripping. For the stiffeners, a uni- axial membrane stress state is assumed while plane stress state is considered for the sin. Strength and stability of double cylindrical shell structure subected to hydrostatic external pressure is investigated by Din [8]. This research aims at the bucling of shell section with some curvature and cut out of a shell by two radial planes through the axis. Andrianov [9] proposed bucling analysis of discretely stringer-stiffened cylindrical shells. They show that a discrete character of reinforcement in the prebucling state can be taen into account in an analytical way using the homogenisation approach. This research aims at the bucling of stiffened cylindrical shells using Ritz method. The critical bucling load of stiffened cylindrical shells under different loading and stiffener arrangements are presented in this paper. The new specific wor done in this paper is showing the significant effect of stringer and ring in inner and outer surface of cylindrical shell as reinforcer, loading condition include axial, lateral and hydrostatic pressure and eccentricity of reinforcer on the bucling load parameter of, for stiffened cylindrical shell. DERIVATION OF THE ENERGY EQUATION Consider a thin, isotropic, stiffened, cylindrical elastic shell with uniform thicness, radius, length, Young modulus, Poisson s ratio and shear modulus represented by h, R, L, E, ν and G respectively. The shell is reinforced by N number of rings and M number of stringers which may have different arrangements as shown in Fig.. The elastic strain energy for cylindrical shell due to planar stretching and bending of the shell in the coordinate systems is shown in Fig. using Timosheno relations and considering strain-displacement relations of Sanders theory yields [9]. U Shell = L æ u ö æ v ö w ç + x R - ç è ø çè ø æ öæ ö æ ö - ç è øèç ø çè ø ú û 3 Eh L æ w ö æ w ö w 4 - ò ò ç x çè ø R è ç ø æ w öæ w ö æ w v u ö - ç x ç R ç 4 4 è øè ø è ø ú úû Eh ( ) - ò ò v v v v u + - w + + Rd R x x R ( ) ( ) w R d R x x R () where u, v and w are the displacements in the longitudinal, tangential and radial directions, and x and are the longitudinal and circumferential coordinates, respectively. Only the strain energy due to the normal strain in the direction of the stiffeners axis and shear strain due to twisting about this axis are considered for the stiffeners Fig.. The normal strain includes the effects of extension and bending of the stiffener about two axes. Thus -U s, the total strain energy of the sth stringer- located at, is [3]. Læ ö ì L JG s s w d Es u w v s= s s s ò + í + - R x x x x çè ò ø= ú s î û U x A e c s IAU, Ara Branch

3 334 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells 3 3 v w w w zs s yz s x R x x R x ú ú û û v 3 w w 3 ü w ú yzs s s x R x x R x úý ú û ú û þ = s + I + e + I + c + I + e + c () where the sth stringer has an axial rigidity of EsA s, torsional rigidity of GJ s s, flexural rigidities about z and y axes of, EsI zsand EI s ys, respectively. I yz is the product of inertia about y and z axes, e s is the radial distance between the centroid of the cross-section of the sth stringer and its attachment line at mid-surface. While c s is the circumferential distance between the centroid of the cross-sections of the sth and (s+)th stringers. Note that a positive value of eccentricity e s implies external stringer and a negative value of eccentricity represents an internal stringer. Since the depth of the stiffener rings are much smaller than the shell radius, it is permissible to use Euler- Bernoulli theory for evaluating the strain-energy of the rings. Thus, strain energy of the th ring is [3, ] U EI æ x w ö w 3 d ( R + e ) ò ç çè ø E IzR ì æ 3 w u ö e æ w öü + w ò í + x R + + R x ý çè ø çè î ø þ E A( R + e) ìæ v ö e æ v í w w öü + ò ç - + ç è + ý d ø R ç î è ø þ G JR æ u w ö + d ò - ç R x çè ø ú û = + d (3) The th ring-stiffener is located at rigidity of GJ, and flexural rigidities about z and x axis of EI z and EI x between the centroid of the th ring-stiffener cross-section and shell mid-surface Fig.. Lie stringers a positive value of eccentricity e implies external ring while its negative value represents an internal one. The total potential energy of shell with a symmetric lateral pressure q(x), and a uniform axial pressure p at its end, expressed as [3]: L measured from the end of the shell with axial rigidity of EA, torsional L ì qx ü ( ) w p L æ dd wö í ý ú 4 ç x è û ø î þ V=- + w w x - R d, respectively while e is the distance ò ò ò ò (4) where is a scalar indicator which taes the value of or, depending on whether there is or there is not an end axial pressure. In this analysis, the effect of shell imperfection in bucling is neglected when there is an end axial load. Thus, the total potential energy is given by: = U + U + U + V = U + U + U + V Shell Stringer Ring Loads Shell s Loads s= = Eh L u v v v v v u = æ ö æ ö æ öæ ö - æ ö w w Rd ( - ) ò ò çè x ø R çè ø R çè x øè ç ø çè x R ø úû 3 L Eh æ wö æ w ö + 4 w + + 4( - ) ò ò çè x ø R è ç ø M å N å IAU, Ara Branch

4 A. Ghorbanpour Arani et al. 335 æ wöæ w ö ( - ) æ w 3 v uö + w Rd R x R x 4 x 4R ú è ç øè ç ø è ç ø úû M ì L JG æ L s s w ö E ì s u w v + åí + ía s + es -c s S ò çr x ò è ø x x x ú = = î s î û 3 3 v w w w + I zs + e s + I yz + c s x R x ú x R x ú û û 3 3 v w w w ü ü + I yzs + e d s + c ú s x x R x ú ý ý û x R x ú ûþ = s þ r N r r ì ì EI z æ w u ö EI æ x r w ö + ò í + + w åí 3 + î R+ e = ç è x R+ e ø ( R+ e ) è ç ø î r r r EA æ ü ü v ö r GJ æ w uö + -w + d - + ý ý R+ e ç è ø R+ e è ç x R+ e ø þ þ L qx ( ) ì w ü p L - æ wö ò ò í + ww dd x Rd ý - ú 4 ò ò çè x î û þ ø (5) In this wor, only one term of the double Fourier series is taen into account. It is noted that one term has a good accuracy. For computational simplicity and accuracy of the total potential energy, variables x and can be separated using the following trigonometric functions. u= u( x, ) = u( x)sinn (6a) v= v( x, n) = v( x)cosn (6b) w= w( x, ) = w( x)sinn where n is the circumferential wave number. For generality and convenience, also, the following non-dimensional terms are considered: (6c) w u v x w=, u=, v=, x= R h h L J es cs E I s zs J =, es =, Cs =, Es =, Izs = 3 3 Rh h h E Rh E e I I A E =, e =, I =, I =, A = E h Rh Rh h Iys Iyzs Iys =, I 3 yzs =, A 3 s = As, J s s = J Rh Rh h Rh3 = R, = h, ( x) = q( x) pr( ), = - L R pl Eh z x z x 3 3 (7) By substituting Eqs. (6a), (6b), (6c) and (7) into Eq. (5) and integration with respect to, the non-dimensional total potential energy function may be expressed as: M N ( - ) = = Uc+ Us+ U + V RLEh å å (8) s= = The non-dimensional strain energy function of cylindrical shell is: IAU, Ara Branch

5 336 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells ì du du Uc æ ö ( nv w) ( n w) æ ö = ò í ç ç + è ø è ø î - æ dv ö 4 d w + nu æ ö w (-n ) èç ø çè ø æ d wö æ dw 3 dv n ö ü + w( - n ) + ( - ) n + - u ý ç è ø çè 4 4 ø ú ú ûþ (9) and the non-dimensional strain energy function of stringers-stiffened: M where M ì U ( - ) = E í U + U + U + U + U s î å å () s s s s s3 s4 s5 s= = ü úý û þ æ cos d ö + ò ç çè x ø Jsn Us = n w ( ) d ì æ duö æ d wö Us = As sin n d x + es h sin n í ò èç ø ò èçd x ø î æ duö æd wö æ d vö +eh s sin n d x+ c cos d s h n ò x ç è ç ç ò øè ø çè ø æ d - sin n cos n csh u öæd v ö æd d d + w öæ v ö ü x e d scsh x ò ç ç ý è çç ò øè ø èç øè ç ø ú ú ûþ æd vö æ d wö æd vöæd wö Us3 Izsh cos n d x + e d + d s n x e sn x ç ç ç ç è ø è ø è øè ø ú û (a) (b) = ò ò ò (c) æd wö s4 ys s ç U = I h sin nθ+ c nξ cosnθ û ò çèd x ø (d) d v d w d w Us5 Iyzsh cos n( sin n+c sn cos n) æ öæ ö æ ö d x + e d sn x çè øè ç ø è ç ø ú û = ò ò (e) The non-dimensional strain energy functional of ring-stiffened is: N å = - U E U U U U ( ) = e úû () where d z ( ) U = I + e -n e -n u d x ú û (3a) U = I x ( n ) w - úû (3b) ( + e ) U3 = A n( + e) v + ( + n e) w úû (3c) IAU, Ara Branch

6 A. Ghorbanpour Arani et al. 337 B B Ring detail Fig. Geometry and coordinate system of a stiffened cylindrical shell. Stringer detail J dw ú û U4 = - nu + n (+ ) d x (3d) The non-dimensional potential energy function with axial and lateral pressure is written as: V Vl Va dw æ ö ( xw ) ( n) - + ç çè ø úû = + =- ò (4) where V l and V a are the non-dimensional potential energy of the lateral and the axial pressures, respectively. 3 DERIVING EIGEN-VALUE EQUATION In continuous members lie cylindrical shells, infinite coordinates are needed to investigate the stability of the deformed shape. The critical load is determined by using polynomial function of the Ritz method.this method have a good agreement with the critical load which is determined in the previous studies. The polynomial functions in Ritz method must satisfy all inematic boundary conditions. In simply supported condition ( w, xx = w= v= ), moreover, the following Ritz functions for cylindrical shells may be adopted as []: u = c x = cu v = d x úx - x= d v w= e x úx - x = ew û m m i- i i i ú i= û i= m m i- i i i ú û û i= û i= m m i- i i i ú û û i= i= å å (5a) å å (5b) å å (5c) where m is the number of polynomial terms of displacement function. Applying the Ritz method on the nondimensional total potential energy function leads to: = i =,,..., m c i = i =,,..., m d i (6a) (6b) IAU, Ara Branch

7 338 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells = i =,,..., m e i (6c) where Substituting displacements of Eqs. (5a), (5b) and (5c) in Eq.(8) and then in Eqs.(6a), (6b) and (6c) yields: { } { } ( K G û- û ) C = (7) { C} ì ü c d e {} c { d} {} e ìc ü ìd ü ìe ü = í ý, {} c = í ý, {} d = í ý, {} e = í ý î þ c n d n e î þ î þ î n þ (8) And the elements of the stiffness matrix are: M N c s å å (9) K = K + K + K û û û û s= = The elements of stiffness K û and geometric stiffness G û matrices are given in Appendix A. To determine the bucling load parameter, which is defined in non-dimensional form as = P r \(- )/( Eh), it is necessary to obtain the nontrivial solution of Eq. (7). Therefore, the determinant of the coefficient matrix in Eq. (7) is set equal to zero. By solving the linear eigenvalue problem the smallest eigenvalue which is minimum bucling load parameter ( min ) and the corresponding eigen-vector { C }, can be found. It is noted that the bucling load is computed by multiplying the initial load with this smallest eigenvalue. The important point in solution of the eigenvalue equation for obtaining minimum bucling critical load is selection of appropriate values for m and n parameters. Appropriate value for n may be chosen by trial which leads to the smallest eigen-value cr. 4 NUMERICAL RESULTS AND DISCUSSION The numerical results of the critical bucling load of stiffened cylindrical shells under different loading, boundary conditions and stiffener arrangements are presented in this paper. These arrangements included inner and outer rings and stringers stiffener with variable distances. The results were considered for boundary conditions of free and fixed ends and loading conditions of axial, lateral and hydrostatic pressure or combinations of them. The results obtained by solving the eigen-value equation of Ritz method are in good agreement with those reported by other researchers. 4. Stiffened cylindrical shell with rings under axial pressure The influence of the ring-stiffened shell geometry on axial pressure bucling parameter = PR / D, are shown for both inside and outside rings in Figs. and 3 respectively. The numerical results of the bucling parameter are then compared with those obtained from Singer et al. [5] in Table which indicate that cylindrical shells with outside rings have more stability than those with inside rings under axial pressure. The numerical results are in good agreement with those reported by Singer et al. [5]. The eccentricity effect for cylindrical shell with stiffened rings under axial pressure on bucling load parameter is shown in Fig.. With inside rings, in general, the bucling load is reduced by decreasing the eccentricity of the rings, whereas with outside rings the magnitude of the eccentricity does not affect the bucling load. The curves of bucling load parameter variation in cylindrical shell with 4 evenly spaced external ring-stiffeners under axial, lateral and hydrostatic pressures verses length variations are drawn in Fig. 3. These curves show that for small ratio of R/L, there is little difference between the values of shell bucling load parameter under hydrostatic and lateral pressures, but as R/L increases, this difference increases IAU, Ara Branch

8 A. Ghorbanpour Arani et al. 339 too, and the stability of ring-stiffened cylindrical shell under axial compression becomes greater than hydrostatic and lateral pressures. Fig. Eccentricity effect on axial pressure bucling parameter, for ring stiffened cylindrical shell. Fig. 3 Comparison of the bucling load parameter, for ring stiffened cylindrical shell under hydrostatic, lateral and axial pressures. Table Comparison studies of the bucling axial pressure parameter, for ring stiffened cylindrical shell ( L / R=.5, =.3 ) Present study Singer,Baruch,Harari[5] Rh er h 3 Ar bh Ibh Outside(+) Inside(-) Outside(+) Inside(-) A n A n A n A n * * ** 46 5 ** * n means axisymmetric bucling. ** Results for outside and inside rings are the same because of e r. IAU, Ara Branch

9 34 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells 4. Stiffened cylindrical shell with stringers under axial pressure The influence of the stringer-stiffened shell geometry on axial pressure bucling parameter = PR / D is shown in Figs. 4 and 5. To evaluate the accuracy of the Ritz method in this study, these results are compared in Table with those obtained by Singer et al. [5]. The bucling load parameter for stiffened cylindrical shell with 6 stringers located either inside or outside, under axial load, is shown in Fig. 4. The curves of bucling load parameter are also plotted in this figure to indicate the influence of the stringers on the bucling load in order to compare with the case of an unstiffened equivalently thicened cylinder. It is noted that the bucling load diagram of the unstiffened shell is dependent on the non-dimensional parameter R/L, but variation of this case become low compared with other cases. This shows the shell length variation more specifically. This point was also demonstrated for bucled shells under axial compression in Timosheno relation [6]: N cr cr = = h R Eh 3( ) - () However, the difference between the critical axial load for inside and outside stringers decreases with increase in the shell length. The difference between two curves increases with increase in R/L, and bucling load parameter for shells with outside stringer is more than those with inside ones. In contrary, the shells with inside stringers are more stable than those with outside ones for high ratio of R/L, i.e. shell with short lengths. The effect of rings and stringers on bucling load parameter for shells under axial pressure is compared in Fig. 5 which data points associated to this figure has been shown in Table 3. These curves and table indicate that under axial load, outside rings have more stability than inside ones. It can also be seen that for low R/L ratios, there is no significant variations for both rings and stringers. For R/L ratios of higher than., the stability of the stringers improves considerably compared with that of the rings, as increases rapidly. It is observed from Table 3 that bucling load parameter,, in case of equivalent shell is validated by ref [4]. Fig. 4 Axial pressure bucling parameter, for stringer stiffened cylindrical shell. Fig. 5 Comparison of ring and stringer effects on bucling load parameter, for stiffened cylindrical shell under axial pressure. IAU, Ara Branch

10 A. Ghorbanpour Arani et al. 34 Table Effect of geometry parameters on the axial pressure bucling parameter, for stringer stiffened cylindrical shell Present study Singer et al. [5] LR Rh As bh Izs 3 bh Outside(+) Inside(-) Outside(+) Inside(-) A n A n A n A n es h Table 3 Effect of rings and stringers on bucling load parameter versus R/L for shells under axial pressure (Data point associated to Fig. 5) Stiffeners Shell Equivalent (Ref. Ring:Inside Ring:Outside Stringer:Inside Stringer:Outside Log ( R / L ) [4]) Stiffened cylindrical shell with stringers under lateral pressure and hydrostatic pressure In this section the influence of stringers on bucling load parameter under lateral and hydrostatic pressures is investigated. For better understanding, stringers and rings are compared and shown in separate curves. The geometrical dimensions of the proposed stiffeners and shell are as follow: L R R H I bh A bh e h 3 / = 3.98, / = 338, / =.3, / = 9.79, / = 5.83 The influence of rings and stringers on bucling load parameter under lateral pressure and hydrostatic pressure are shown in Figs. 6 and 7, respectively. The curve of unstiffened equivalent thicened shell is also plotted in these figures. As shown in Figs. 6 and 7, the stability of stiffened cylindrical shell by ring under lateral or hydrostatic pressures is more than the one stiffened by stringers. Bucling load of shell with inside and outside rings are almost the same, thus short shell length does not change the value of bucling load of rings stiffened shells under hydrostatic pressure. Similar to the curves shown in Figs. 6 and 7, stiffened shells with outside stringers under lateral and hydrostatic pressures are more stable than those with inside stringers. The stability of unstiffened long shells is more than stringer-stiffened shells. The bucling load parameter of stringer-stiffened cylindrical shell under hydrostatic, lateral and axial pressures is compared in Fig. 8. As shown in Fig. 8, the stability of stringers against axial loads are much more than lateral and hydrostatic pressures, and the stability of stringer stiffened cylindrical shell against lateral and hydrostatic loads are the same, and with increase in R/L, this stability increases too in the same manner. IAU, Ara Branch

11 34 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells Fig. 6 Comparison of ring and stringers effect on bucling load parameter, for stiffened cylindrical shell under lateral pressure. Fig. 7 Comparison of ring and stringers effect on bucling load parameter, for stiffened cylindrical shell under hydrostatic pressure. Fig. 8 Comparison of the bucling load parameter, for stringer stiffened cylindrical shell under hydrostatic, lateral and axial pressures. 4.4 Stiffened cylindrical shell with combination of stringers and rings In this section, the influence of stringers and rings combination on bucling load parameter are investigated. A simply supported cylindrical shell with 6 stringers and 4 evenly spaced rings with the following dimensions are analyzed here: L R R H I bh A bh e h 3 / 3.98, / 338, /.3, / 9.79, / 5.83 In Figs. 9-4, variation of bucling load versus R/L under axial, lateral and hydrostatic pressure is shown. Fig. 9 illustrates the best stability position of stiffened shell under axial pressure when stringers and rings are outside the shell surface. However Fig. shows that for very small values of R/L, and very long stiffened cylinder shells under axial pressure, outside stringers and inside rings represent maximum stability. The critical bucling load parameter of stiffened cylindrical shell, under lateral and axial pressures, for small R/L (Figs. and 4) have very little IAU, Ara Branch

12 A. Ghorbanpour Arani et al difference (about ). This shows that stability behavior of long shell under lateral and hydrostatic pressures are very similar. The number of circumferential waves is also the same during bucling. In stiffened shell under lateral and hydrostatic pressure, with high R/L ratio (Figs. and 3), the shell stability is more when rings and stringers are outside the cylindrical shell. Fig. 9 Axial pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R/ L 5). Fig. Axial pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R/ L.). Fig. Lateral pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R / L 5). Fig. Lateral pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R/ L.). IAU, Ara Branch

13 344 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells Fig. 3 Hydrostatic pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R / L 5). Fig. 4 Hydrostatic pressure bucling parameter, for stringers and rings stiffened cylindrical shell ( R/ L.). The bucling load parameter curve of unstiffened shell of Fig. 3 shows that, with high R/L ratio, the stability of unstiffened shell is more than the stability of stiffened cylindrical shell with stringers and rings. 5 CONCLUDING REMARKS According to our numerical results and comparisons, the following concluding remars are itemized: - In stability of stiffened cylindrical shells under axial pressure loading, the cylinders reinforced with outer rings are more stable than the inner one. In this case, the stability is independent of the length of the cylinders. The critical bucling load decreases with increasing eccentricity of the reinforced inner rings. For the outer ring cases, however, the eccentricity has almost no effect on the critical bucling load. - In cylindrical shells stiffened with stringers, the stability of cylinders with outer stringers is higher than the inner one. Also, the critical bucling load increases with increasing the eccentricity of the stringers. - The stability of cylinders stiffened by stringers under axial loading is much higher than the hydrostatic lateral loadings - In general, stability of the stiffened cylindrical shells under hydrostatic lateral loading, the ring stiffeners are more stable than the stringers one. Also, the stability behavior of the stiffened long cylinders under hydrostatic lateral loading is very close to each other. APPENDIX A The shell partitioned stiffness matrix c K û is summarized as K pp K pq K úû úû prúû ú û ú ú ú û û û T T K pr K qr K rr úû úû ûú û c T K = K pq K qq K qr ú (A.) The elements of the shell stiffness submatrices are given by IAU, Ara Branch

14 A. Ghorbanpour Arani et al. 345 du K d x n û ú ç uu du pp = + - æ ç + ö i 48 çè ø i ò ( ) ç ò (A.) æ d ö v du i K pq n ( ) ui d x v úû = - ç çè ø úû 4 dw dui K pr ú =-n (-) ui d x - w ò ò (A.3) û ò ò (A.4) d dv vi K n vv d x ( ) úû ç 6 3 qq = i + - æ ç + ö çè ø ò ò (A.5) d dw vi K = n vw d x + n (- ) 4 3 qr ú i û ò ò (A.6) æ ö d w K ( n ) û ç ww 4 d wi rr = ç ò i + 6 ò çè ø æ d d w w ö i + (- n ) w d d x wi x 6 + ç ò ò çè ø d dw w + i n (- ) 3 ò (A.7) The stringer partitioned stiffness matrix û is summarized as s K = s K spp K spq K úû úû spr úû s ( ) T K - E s K spq K sqq K û = = sqr s úû úû úû (A.8) T T K spr K ú sqr ú K û û srr ûú û = s The elements of the stringer stiffness submatrices are given by K K K úû spp úû spq 3 du du i = α Assin n ò (A.9) d v =- ò x (A.) 4 3 dui csh Assin n cos n d d w = ò x (A.) dui spr úû esh α Assin n d æ 4 3 d v d v ö i K sqq ( cs h Ascos n+ αξh Izscos nθ) ûú ç çè ø = ò (A.) 4 K sqr úû = - A ssin n cos + αξh Izscos nθ escs h n esn æ d d v w ö i αξ h Iyzscos nθ ( sin nθ+csnξ cos nθ) + ú û ç ò çè ø (A.3) IAU, Ara Branch

15 346 Elastic Bucling Analysis of Ring and Stringer-stiffened Cylindrical Shells dw K = x+ e h n + e n n J sαξn dwi srr û d { s Assin s Izsh cos cos nθ ( +ν) ò d w + I ysh sin n csn cos n + û + esn h Iyzscos n (sin n+c n cos n) d d wi s } ò (A.4) The ring partitioned stiffness matrix K x = L û is summarized as K pp K pr úû úû - ν K E x L 3 K qq K û = = qr ( + e ) ú ú û û T T K pr K ú qr ú K û û rr ûú û x= L (A.5) The elements of the ring stiffness submatrices are given by J K ( L ) ( L pp n æ Izn ui u ) úû + ç ( + ) çè ø (A.6) dw ( L ) J K ( ) ( L pr n Iz e n e ç u i ) úû ç è ( + ) øèç ø (A.7) 4 K ( ) ( L ) ( L qq i ) úû = n A + e v v (A.8) 3 K ( )( ) ( L ) ( L qr i ) úû = na + e n + e v w (A.9) nj dw K rr æ Iz ( e n e ) ö û = èç ( + ) ø ( L i ) æ A ö + I ( n ) ( e n ) ( e ) w w ç è ø ( L ) ( L ) ( L x i ) dw (A.) and G û= (A.) G rr ûúû where dw dw i G n ww ( ) rr û = - ò i - ò (A.) where i,,, 3. IAU, Ara Branch

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