TMR4205 Buckling and Ultimate Strength of Marine Structures

Transcription

1 TMR05 Buckling and Ultimate Stength of Maine Stuctues Chapte 5: Buckling of Cylindical Shells by Pofesso Jøgen Amdahl MTS

2 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 CONTENTS 5. BUCKLING OF CYLINDRICAL SHELLS Intoduction Equilibium Equations fo Cylindical Shells Stess Analysis Beam Theoy Lateal Pessue Lateal Pessue-- Solution of the Diffeential Equation Buckling of Cylindes Axial Compession Cuved Panel Bending Extenal Lateal Pessue Tosion Buckling of Impefect Cylindical Shells Geneal Shape Impefections Buckling Coefficients Elasto-Plastic Buckling Combined Loading Buckling of Longitudinally Stiffened Shells Geneal Othotopic Shell Theoy Buckling of Ring Stiffened Shells Lateal Pessue Combined Loading Geneal Buckling Axial Compession and Bending Tosion and Shea Column Buckling Refeences...0

3 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 5. BUCKLING OF CYLINDRICAL SHELLS 5. Intoduction Stiffened and unstiffened cylindical shells ae impotant stuctual elements in offshoe stuctues. They ae vey often subjected to compessive stesses and must be designed against buckling citeia. The buckling behaviou is usually moe violent than it is fo plate and column stuctues. A theoetical load-end shotening cuve epesentative fo cylindical shells subjected to axial compession is shown in Figue 5.. Duing initial loading the stuctue follows the linea pimay equilibium path. At some load level, the pimay path is intesected by an unstable seconday path. The buckling mode in the seconday path is quite diffeent fom the defomations in its stable pimay state of equilibium. The intesection is called a bifucation point, B. σ σ CL Pefect shell B L Impefect shell ε ε CL Figue 5. Equilibium Paths fo Pefect and Impefect Shells. In pactice, it is vey difficult to each the theoetical bifucation loads. The eason fo this is the pesence of initial impefections which causes the shell to fail into a configuation close to the buckled shape of the ideal cylinde at a load significantly smalle than the bifucation load. Theefoe, buckling occus at the limit point L athe than the bifucation load B. Figue 5.a shows the enomous influence of a small axisymmetic initial impefection, δ o, on the buckling load, N, of an axially loaded cylinde. Fo an impefection amplitude of only /0 of the wall thickness, the buckling load is educed to 60% of the theoetical value, N c. The impefection sensitivity is futhe illustated in the plot of expeimental buckling loads in Figue 5.b, whee the wide scatte of the esults is also obseved. Because of this effect, the design of cylindical shells is based on the modification of the theoetical load by an empiical eduction, o a knock-down facto.

4 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 Figue 5. (a) Influence of Axisymmetic Impefections on the Buckling Load of a Cylinde, (b) Expeimental Buckling Loads Of Axially Loaded Cylindes. l θ Longitudinal stiffne L l Ring fame s l x y z Figue 5.3 Geometical Paametes of a Stiffened Cylindical Shell, /5./. The chaacteistic geometic paametes of a stiffened cylindical shell is defined in Figue 5.3. The buckling modes fo stiffened cylindical shell may be categoized as follows:- Shell buckling; Buckling of shell plating between stiffenes and fames. Intefame shell buckling; Involves buckling of the longitudinal stiffene with associated shell plating. Panel ing buckling; Buckling of ings with associated plate flange between longitudinal stiffenes. Geneal buckling; Involves bending of shell plating, longitudinal stiffenes as well as ing fames. Tosional o local buckling of stiffenes and fames. Column buckling of the cylinde. Possible buckling modes fo cylindes with vaious stiffene aangements ae displayed in Table 5-.

5 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 5 of 3 Table 5- Buckling Modes in Cylindes with Vaious Stiffene Aangements, /5./. Buckling Mode Geomety Unstiffened X X SHELL cylinde BUCKLING Unstiffened cuved panel X PANEL Stinge stiffened X BUCKLING cylinde GENERAL BUCKLING OVERALL BUCKLING LOCAL STIFFENER BUCKLING Ring stiffened cylinde Ring / Stinge stiffened cylinde Column Ring Stinge X X X X X X X X X X X X 5. Equilibium Equations fo Cylindical Shells The classical theoy fo buckling of cylindical shells, as suggested by Flügge, may be found in /5.3/. Analytic solutions to the Flügge equations ae known fo seveal load cases. Howeve, fo pactical puposes it is moe convenient to use the equations poposed by Donnel /5./ which leads to vey simple fomulas fo the buckling stess. The simplifications intoduced somewhat limit to thei ange of applicability, but it has been found that they may be used unde the following estictions, /5.5/. The numbe of cicumfeential waves should not be too small, (n ) Stess state fo a cylindical shell. The defomations in the longitudinal and cicumfeential diection ae small as compaed to the adial displacement. Theefoe, the column buckling mode can not be pedicted. Figue 5. shows an infinitesimal element of a shell with its associated stess esultants fom membane and bending actions. Consideing equilibium in the axial, cicumfeential, and adial diections, the following equations ae obtained,

6 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 6 of 3 Membane Bending Lateal load-caying due to N θ Figue 5. Shell Stess Resultants. N x N x + θ 0 x θ (5.) N xθ N + θ 0 x θ (5.) w x x D p N w x N w x N w + + N θ θ θ θ θ (5.3) whee, N N N x xθ θ σ t x Nθ x σ xθ t (5.) σ t θ ( ) x + θ (5.5) The plate stiffness, D, is given by D Et ( ν ) 3 (5.6) The pessue, p, is positive outwads. Note the similaity between Equation (5.3) and the coesponding expession fo plate equilibium. The equations can simply be obtained fom the plane membane and plate equations by substituting

7 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 7 of 3 y θ, y θ (5.7) The only new tem is Nθ which epesents the lateal component of the cicumfeential stess. Thus, unlike plates, cylindical shells can cay lateal loads by pue membane action and no bending. This is a vey efficient popety, but at the same time this makes shells sensitive to buckling. Equations (5.-3) fom a coupled set of thee non-linea equations with fou vaiables-- N x, N xθ, N θ, and w. By intoducing the kinematic and constitutive elationships, and applying the opeato, Equation (5.3) may also be witten as, 8 w + D N w x x N xθ w + x N w Et θ θ θ D w x (5.8) which is the Donnel's equation. 5.3 Stess Analysis 5.3. Beam Theoy A cylindical shell is geneally exposed to the load conditions shown in Figue 5.5. T N Q p Q N T M M Figue 5.5 Load Conditions in a Cylindical Shell. The stess can often be deived by assuming that the wall thickness is much smalle than the adius. The axial stess becomes whee the mean axial stess is and the bending stess is, σ x σ a+ σ b (5.9) N σ a (5.0) πt Mz M sinθ σ b 3 I π t M sinθ π t (5.)

8 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 8 of 3 whee θ denotes the angula position of the stess point elative to the y-axis. If the shell is povided with longitudinal stiffenes, these can be smeaed so as to obtain an equivalent thickness, t e A t + (5.) s whee A is the aea of stiffene without plate flange and s is the stiffene spacing. The total shea stess is given by τ τ T + τ Q (5.3) whee the expessions fo the shea stess due to tosion and the shea stess due to bending ae given, espectively, as and τ T T T (5.) A t π t cyl QS Q t cosθ Q τ Q cosθ (5.5) 3 It π t t π t Stiffenes ae nomally not consideed to influence the shea stesses Lateal Pessue Fo an unstiffened cylinde subjected to lateal pessue, the cicumfeential stess can be obtained by consideing equilibium of a half section as shown in Figue 5.6. σ t p θ p σ > t θ ( Tension if σθ 0) (5.6) Fo a closed cylinde the axial stess becomes πtσ π p x p σ x t σ θ (5.7) The cicumfeential stess may also be detemined diectly fom the equilibium Equation (5.3) because the entie load is caied by the membane action (w is constant aound the peimete so that all the deivates vanish).

9 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 9 of 3 p N θ 0 p σ θ t (5.8) p σ θ t Figue 5.6 Half Section of Cylinde with Lateal Pessue. σ θ t The cicumfeential stesses in a ing stiffened cylinde subjected to lateal pessue and axial stess may be deived by a simple consideation. Assume that the ing has an associated plate flange with effective width, l eo, so that the effective thickness in the ing diection is t e A t + (5.9) eo also, assume fist that the ing is infinitely igid. Due to estained tansvese contaction the axial stess, σ x, causes a cicumfeential stess, νσ x, (tension if σ x > 0). The net extenal foce acting on each half section is accodingly p vσ x t (5.0) When the ing is allowed to defom, a cicumfeential stess, σ θ, is set up in the ing and plate flange. Equilibium yields o, σ t p ν σ t (5.) θ e x p σ θ ν σ x (5.) t A + t eo In the plate flange at the ing the stess due to estained contaction must be added, so that plate p σ θ ν σ x + ν σ x (5.3) t A + t eo If σ x is due to the end pessue alone, and p is constant (σ x p/t), the fomula can be witten as

10 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 0 of 3 plate σ θ p t ν A + t eo ν + (5.) The cicumfeential stess mid-way between the ings depends on the size of the stiffenes and the distance between the ings. Fo an unstiffened cylinde, the cicumfeential stess is given by Equation (5.6). Fo a cylinde with closely spaced ing stiffenes, the stess is given by Equation (5.). It is natual to assume that the cicumfeential stess, in the geneal case, can be obtained by intepolation between these two exteme cases so that, p p σ θ ( ξ) + ξ νσ + ν σ x x (5.5) t t A + eo t whee, ξ 0 fo vey distant stiffnes ξ fo closely spaced stiffnes, Effective width, l eo β l eo / l ξ 0,8 0,6 0, Effective width appoximation. Intepolation paamete, ξ 0, Tansition accoding to DnV 0-0, β Figue 5.7 The Paametes l eo and ζ. Reaanging this gives

11 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 σ θ A p eot p ξ ν σ x t + A t eot (5.6) This fomula can be deived fom the solution to the diffeential equation of equilibium as shown below. The paametes ae defined as follows, sinh β cosβ + cosh β sin β ζ (5.7) sinh β + sin β β Z (5.8) 56. t 3 whee Z is the so called Batdof' paamete. The effective width is given by, eo cosh β cosβ β sinhβ + sinβ (5.9) The paametes ζ and l eo ae plotted in Figue 5.7. It is seen that l eo l fo closely spaced stiffenes and appoaches asymptotically.56(t) fo long cylindes. A good appoximation, which is also plotted in Figue 5.7, is obtained as, { eo min,.56 t} (5.30) Lateal Pessue-- Solution of the Diffeential Equation (A Cylinde unde Lateal Pessue and Axial Stess). The defomation will in this case be axisymmetic so the diffeential equation eads, Dw, + N θ N w, p (5.3) xxxx x xx Fo axisymmetic loading the cicumfeential stain is given by and ε θ w N θ E θ t N Et w x (5.3) ε + ν +ν t σ (5.33) x This gives, Dw, Et tw, w p ν σ x σ + t xxxx x xx p e (5.3)

12 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 whee p e is the effective load. In the following linea theoy is applied, i.e. the lage deflection tem σ x tw, xx is neglected. By intoducing the paamete, k ( ν ) Et D t (5.35) the equation takes the fom, w, + k w p' (5.36) D xxxx The solution to this diffeential equation is given as, w wp+ w h (5.37) whee the paticula and the homogeneous solutions can be witten, espectively, as and, p' p ν σ (5.38) D k t Et w p x -βx ( ) ( ) cos sin cos sin (5.39) x wh e β C kx + C kx + e C 3 kx + C k The integation constants ae detemined fom the bounday conditions. In the following, the case illustated in Figue 5.8 is studied. The coodinate system is located midway between the ings p (<0) w t b x yielding l/ l/ R l Figue 5.8 Ring Stiffened Cylinde Exposed to Lateal Pessue. Symmety yields w(-x) w(x) which implies, C C C 3, C (5.0) kx -kx kx -kx ( ) sin ( ) w w + C coskx e e + C kx e e p Also w, x 0 fo x l/. This yields, w + A coskx cosh kx + A sin kx sinh kx p (5.)

13 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 w, A k sin β cosh β + A k cosβ sinh β + A k cosβ sinh β + A k sin β cosh β 0 (5.) x x whee, ( ν ) k k 3 β β t 3Z (5.3) This gives A A sin β cosh β cosβ sinh β sin βcosh β + cosβsinh β (5.) The condition that the displacement should be equal to the ing fame displacement at the ing yields, sin β cosh β cosβ sinh β wr wp Acosβcosh β + A sin βsinh β sin βcosh β + cosβsinh β A (5.5) sinβ + sinhβ sin βcosh β + cosβsinh β which implies that, ( ) A w w R p ( wr wp) sinh β cosβ + cosh β sin β sinhβ + sinβ ξ (5.6) when Equation (5.7) is employed. Obseving that the tem elated to A vanishes fo x 0, the deflection at mid-way between the ings is, ( ) ( ξ) wr w w + w w x0 p R p w + p ξ ξ (5.7) Thus, the fist tem epesents the contibution fom an unstiffened cylinde, and the second tem the contibution fom a cylinde with closely spaced stiffenes as shown by the physical easoning in Section The cicumfeential stess due to w p is obtained fom Equation (5.38), p p σ θ E w p + ν σ x (5.8) t

14 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 The total foce in the cicumfeential diection between two ings is given by, l l F σ tdx t E w dx φ 0 θ 0 t E w l l l A ( kx kx) dx A ( kx kx) p + dx cosh cos + sinh sin 0 0 Et wl p + A ( β β + β β) + A ( β β β β) sinh cos cosh sin k cosh sin sinh cos k (5.9) Intoducing the expessions fo A and A thee is obtained F φ ( ) Et cosh β cosβ wl p + wr wp k sinhβ + sinβ (5.50) The last tem has a length dimension, and is intepeted as an effective length denoted by l eo coshβ cosβ l coshβ cosβ k sinh β + sinβ β sinh β + sinβ (5.5) Theefoe, Equation (5.50) becomes, F φ Et [ wl p ( wr wp) l eo] [ wr leo wp( l leo ] Et + + ) (5.5) Hence, the total foce in the cicumfeential diection can be consideed to consist two blocks: one with the stess of the ing stiffene acting ove the effective flange, l eo, and the emainde, l - l eo, has a stess equal to an unstiffened cylinde., Hoop stess 0,8 β β Stess in unstiffened cylinde β 3 0,6 Stess at ing stiffene 0, Midway between ings At ing stiffene 0, 0 0 0, 0, 0,3 0, 0,5 Loaction between ing x /(l /) Figue 5.9 Stess Distibution Ove a Half-Ring Fame.

15 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 5 of 3 The exact stess distibution can be obtained by intoducing the displacement function in the stess expession. In Figue 5.9 the stess distibution ove a half-ing fame is sketched fo thee diffeent β-values. Fo β the ings ae closely spaced and the stess mid-way between the ings is almost equal to the stess at the ing stiffene. Fo β3 the stess is appoximately equal to the stess in an unstiffened cylinde fo about 60 % of the fame spacing. Only in the vicinity of the ings is thei effect noticeable. The effective flange fo β,, and 3, is l eo /l 0.9, 0.5, and 0.3, espectively. 5. Buckling of Cylindes 5.. Axial Compession Conside a cylinde subjected to an axial compessive load, P. If the end effects ae neglected, the following assumptions apply, P π N x, N x θ Nθ 0 Intoduction of these values into Equation (5.8) gives (5.53) D w Et w P w x π x (5.5) The solution to this diffeential equation takes the fom m x w n π δ sin sin θ (5.55) l whee m is the numbe of half waves in the longitudinal diection and n is the numbe of entie waves in cicumfeential diection which gives fo the citical stess σ xe π E ( ν ) ( m +n ) t Z l m + π m ( m +n ) (5.56) whee Z is the Batdof paamete, and l Z ( ν ) (5.57) t n nl (5.58) π

16 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 6 of 3 Fo cylindes of intemediate length, a close estimate of the smallest citical load may be obtained by analytical minimization of Equation (5.56) with espect to the quantity m + n m (5.59) Then, the minimum is found fo m + n m 3 Z (5.60) π which gives the following citical load, σ π E ( ν ) t Et Z σ l 3 π (5.6) xe cl This is the classical solution fo axially compessed cylinde. The tem, () CZ 3 π Z (5.6) is intepeted as the buckling coefficient fo an intemediate length cylinde. It is obseved fom Equation (5.60) that seveal buckling modes may coespond to a single bifucation point as illustated in Figue5.0. It should be noted that m and n ae teated as continuous vaiables in the minimization pocess while they ae in effect discete quantities. The eo intoduced by this is, howeve, minimal as seen fom Figue5.0 whee the coefficient is plotted as a function of m and n. Fo shot cylindes the buckling mode will be axisymmetic with m and n0. The following buckling coefficient, is valid fo () CZ Z + π (5.63) π Z < Z () 85. (5.6) 3 An appoximate buckling coefficient which is valid fo shot as well as long shells, may be obtained by applying the elliptical inteaction fomula, () CZ + C (5.65) C Z Z It is seen that C Z appoaches asymptotically the coect values fo small and lage values of the Batdof paamete, Z. Reaanging Equation (5.65) thee comes out

17 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 7 of 3 C Z Z + 8 π (5.66) The fist tem can be intepeted as petaining to buckling of a plane wide plate, the second is the shell contibution. Fo long cylindes, column buckling is a potential collapse mode. The buckling stess fo a shallow shell is expessed by π EI σ E Al π E l (5.67) Thus, the Donnel's theoy is valid fo (3) π 3 Z < Z ( ν ) t (5.68) 0000 Buckling coefficient m,n,,3 &, 3,,,3,3 3,3, &, 3,,, 3,,,,0 Appoximate buckling coefficient Batdof paamete Z Figue5.0 The buckling coefficient fo vaious buckling modes {m,n} 5.. Cuved Panel The buckling load fo a cuved panel of width, s, and length, l, can be found by intoducing the tem k n π (5.69) s into Equation (5.55), whee k is the numbe of half waves acoss the width. Then, by eaanging Equation (5.56), the following expession emeges fo naow panels (k ).

18 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 8 of 3 σ E π E ( ν ) 3Z S t s + π (5.70) whee the Batdof paamete now eads, Z S s t ν (5.7) The last backet in Equation (5.70) is intepeted as the buckling coefficient, C Z, which is valid fo ( ) π Z S < Z S. (5.7) 3 ( ) Figue 5. Buckling Coefficients Fo Axial Compession. An appoximate buckling coefficient valid fo the entie inteval of Z S may be obtained in a manne simila to that fo closed cylindes, (see Equation 5.65) C Z Z + 3 π S (5.73) Again, the fist tem can be consideed petaining to buckling of a plane, long plate, the second tem petains to the cuved shell.

19 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 9 of 3 The vaious buckling coefficients ae shown as a function of the Batdof paamete in Figue Bending It is consideably moe complicated to analyze cylindes subjected to bending because, The initial stess distibution is no longe constant aound the cicumfeence. The pe-buckling defomations of long cylindes is highly non-linea due to ovalization of the coss-section. Howeve, studies caied out in this field indicates that the buckling esistance due to bending may be taken equal to the buckling stess fo axial compession fo all pactical puposes. 5.. Extenal Lateal Pessue In the pe-buckling state the extenal pessue sets up compessive membane stesses in the meidional diection. Retaining only the linea tems in Equation (5.3) gives, N θ p (5.7) Intoduction into Equation (5.8) yields the following stability equation 8 Et w w D w + + p 0 (5.75) x θ The displacement function is of the same fom as fo axial compession. Intoducing Equation (5.55) thee comes out, p π E t σ θ E ( ν ) ( +n ) t Z l n + π n + ( n ) (5.76) whee one axial wave (m) always gives the lowest buckling load. The last tem is intepeted as the buckling coefficient, C θ. The smallest value of C θ may be detemined by tial. If n is assumed lage (>>), analytical minimization of Equation (5.76) gives C () θ 6 3π z (5.77) The appoximate buckling coefficient valid fo small and medium values of Z now eads

20 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 0 of 3 C θ + Z 3π (5.78) The fist tem is identical to the buckling coefficient of a plane, long plate. When l/ appoaches infinity, Equation (5.76) educes to n E t t σ θ E 0.75E( ) (5.79) ( ) ν Long cylindes fail by ovalization fo which n. The value pedicted by Equation (5.79) is, howeve, somewhat too high. A bette appoximation is obtained by substituting n n 3 (5.80) Fo a closed cylinde, the end pessue must be included, whee N x p N θ (5.8) By etaining the tem with N x in Equation (5.8), it can be shown that Equation (5.76) still applies with C ' θ E ( + n ) n + + π Z + n + ( n ) (5.8) Fo plane plates ( ), Z 0 and n 0, theefoe C' θ E (5.83) Fo vey cuved panels (Z >>), n 5>> and Equation (5.8) can be witten as, Z C' θe n + 6 π n (5.8) Minimizing C' θe 6 with espect to n, we obtain n Z Z C' E 6 θ (5.85) π 3π 6 The appoximate, combined expession can now be deived fom,

21 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 which yields, ( Z ) ( π ) C' θe C' θe (5.86) C Z + 8 (5.87) 3 π ' θ E The vaious buckling coefficients ae shown in Figue 5.. The equivalent stess intensity is often used as a paamete in the plasticity coection. When the end pessue is included this becomes, σ σ + σ σ σ 3σ θ (5.88) eq x θ x θ Cuved panels may be analyzed in the same manne as shown in the pevious section, (see Equation (5.73)). Figue 5. The Buckling Coefficient fo Extenal Pessue, /5./ Tosion Fo a cylindical shell subjected to a twisting moment about its longitudinal axis, Equation (5.8) simplifies to

22 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 D w Et w x N w + x 8 θ x 0 θ (5.89) Unde tosional loading, the deflection patten consists of a numbe of waves that spial aound the cylinde fom one end to the othe. Fo a long cylinde this may be epesented by a function of the fom mπ w δ sin x n θ (5.90) Intoduction of Equation (5.90) into Equation (5.89) gives E t ( m n ) m σ x θ,e ( ) ( ) ν _ ν mn ( m + n ) n t (5.9) whee m mπ/l. Fo long cylindes, the shell buckles in two cicumfeential waves, i.e. n. It may also be assumed that m <<. Intoducing this appoximation and minimizing Equation (5.9), thee comes out π E t Z t t σ x θ,e ( ) ν l π ( ν) ( ν) 3 3 (5.9) Donnel's theoy is inaccuate fo small numbe of cicumfeential waves (n). Theefoe, the facto 0.7 in Equation (5.9) should be eplaced by Fo shote cylindes the bounday conditions can no longe be disegaded. A solution can be obtained by use of a deflection function composed of a finite sum of tems. Accoding to Timoshenko and Gee, /5.6/, the following buckling coefficient applies C Z 3 (5.93) θ Fo lage Z, Equation (5.9) can be simplified to, π E t E t 3 σ x θ,e [ Z ] ( ν ) l.. Z (5.9) The buckling coefficient fo cuved panels subjected to shea may be obtained fom Figue 5.3.

23 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 Figue 5.3 The Buckling Coefficient fo Tosion, /5./. 5.5 Buckling of Impefect Cylindical Shells 5.5. Geneal The classical theoy fo buckling of cylindical shells is valid only fo idealized stuctues. Paticulaly, two effects have a detimental effect on the eal buckling load of welded cylindical shells:- mateial impefections, shape impefections. Mateial impefections, like esidual stesses and heteogenities, mainly affect the buckling load in the elasto-plastic ange, and theefoe thei effect is taken into account in the plasticity eduction facto, φ φ( λ ). Shape impefections ae impotant both in the elastic and elasticplastic ange. Hence, thei effect is accounted fo by educing the classical buckling stength. Figue 5. shows, in pinciple, a sketch of expeimental data plotted vesus the educed slendeness atio σ Y λ (5.95) σ E As commented upon, in Section 5., the test esults show a significant scatte, which could be epesented by a distibution function as indicated in Figue 5.. Ideally, a design cuve should be detemined so as to epesent some lowe bound to the test esults, say the 5% pecentile in the distibution function. Of couse, it is not possible to cay out such an analysis accuately, but this is the ultimate goal of the modification of the classical buckling loads due to shape and mateial impefections.

24 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 Figue 5. Pinciple Illustation of Expeimental Data Fo Shell Buckling Shape Impefections The effect of shape impefections is conveniently accounted fo by modifying the classical elastic buckling esistance, σ cl, as deived in Section 5., by an empiical eduction, o knockdown facto ρ. σ ρσ (5.96) E cl Consideing the combined buckling coefficients developed in the peceding sections, e.g. Equations (5.66, 5.73, and 5.78), it is obseved that they can all be epesented by the geneal expession, C + ξ ψ ψ (5.97) whee ψ is the plate buckling coefficient fo a plane plate, and ξ is the contibution fom the cuved shell (asymptote fo lage Batdof paamete (Z) values). In Chapte 7, it was shown that plates ae little sensitive to shape impefections. Hence, the knock-down facto should be applied to the cuved shell contibution. The buckling coefficient is theefoe modified as follows,

25 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 5 of 3 C + ρξ ψ ψ (5.98) The effect of the impefection facto is illustated in Figue 5.5. Vaious fomulas exist fo the knock-down factos. Fo axial compession and bending, DNV /5./ uses ρ 05. t t axial compession - bending (5.99) These impefection factos ae plotted in Figue 5.6 along with ECCS ecommendations. Fo extenal pessue, tosion, and shea, a facto ρ 0.6 is simply used, /5./. Fo othe types of loading the buckling load is much less influenced by impefections. 00 C Z ν t 0 Classical theoy Impefect shell ψ ζ ρζ 0 00 Z 000 Figue 5.5 The Influence of Shape Impefection on Buckling Coefficient. Table 5- Buckling Coefficients Fo Unstiffened Cuved Panels, Mode a) Shell Buckling ψ ξ ρ 05. Axial Stess 0. 70Z S t Shea Stess s l Cicumfencial Compession + s l s 0. l s 3 l Z S 0.6 Z S 0.6

26 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 6 of 3 Table 5-3 Buckling Coefficients Fo Unstiffened Cylindical Shells, Mode a) Shell Buckling ψ ξ ρ 05. Axial Stess 0.70 Z t Bending 0.70 Z t Tosion and Shea foce Z Lateal Pessue 0. Z 0.6 Hydostatic Pessue 0. Z 0.6 Figue 5.6 Impefection Factos fo Cylindes Subjected to Axial Compession and Bending. 5.6 Buckling Coefficients 5.6. Elasto-Plastic Buckling Seveal methods ae available fo modifying the elastic citical stess due to plasticity. As is the case fo plated stuctues, the φ-method is often used in connection with offshoe shell stuctues. The citical load is defined by σ c φ σ (5.00) Y whee φ is a function of the educed slendeness atio, λ 7. A widely used expession is the Mechant- Rankine fomula. This is based on the following inteaction function, σ σ c σ c + E σ Y (5.0)

27 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 7 of 3 It is seen that the citical stess appoaches asymptotically the Eule buckling stess fo slende stuctues and the yield stess fo stocky membes. Solving fo the citical stess we get, σ c Y + λ σ (5.0) In Figue 5.7, Equation (5.0) is compaed with vaious othe φ-elations. Figue 5.7 Vaious Design Cuves Fo Shell Buckling Combined Loading Analytical teatment of buckling unde combined loading is geneally complicated. A convenient technique is to use fomulas of the inteaction type. Fomally, fo N load components this may be witten, N i i σ (5.03) σ i i c γ whee σ i c is the citical load (accounting fo possible plasticity effects) when σ i acts alone. The exponents, γ i, ae patly suppoted by theoetical consideations and patly veified by expeiments. Fo a stuctue subjected to axial compession, bending, extenal pessue, tosion, and shea, the following fomula has been suggested /5./

28 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 8 of 3 σ x σ xc bc σ θ σ xθ + σ θc σ xθc σ b + + σ (5.0) An altenative appoach is to stat with an inteaction fomula fo elastic buckling (substitute σ i c by σ E i in Equation (5.0)) assume popotional loading, calculate the equivalent stess and modify fo plasticity. Assuming a linea inteaction elation (γ i ), the following expession emeges fo the load case consideed above σ x σ xe σ b σ θ σ xθ σ eq (5.05) σ σ σ σ be θe xθe eqe whee the equivalent stess accoding to von-mises is ( ) ( ) eq x+ b x+ b θ + θ + 3 xθ σ σ σ σ σ σ σ σ (5.06) The equivalent educed slendeness atio, λ eq, is obtained fom σ Y σ Y σ σ σ σ λ x b θ x eq σ σ σ σ σ σ eqe The equivalent buckling esistance can then be detemined by Equation (5.0). eq xe be θe θ xθe (5.07) 5.7 Buckling of Longitudinally Stiffened Shells 5.7. Geneal Longitudinally, stiffened shells may be divided into thee categoies /5.80/, see Figue 5.8. Categoy A includes cylindes with few stiffenes. The shell behaves basically like an unstiffened shell. The only effect of the stiffenes is to incease the total coss-sectional aea and the moment of inetia of the cylindes. Categoy B includes cylindes with closely spaced heavy stiffenes. In this case the effect of cuvatue is often neglected and the shell is modelled as an equivalent stiffened plane panel. Categoy C includes cylindes with closely spaced light stiffenes. By using the smeaed stiffene technique, the shell may be assumed to act like an othotopic shell.

29 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 9 of 3 Figue 5.8 Categoies of Longitudinally Stiffened Cylindical Shells.

30 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 30 of 3 The citeion fo subdivision into categoies is set somewhat abitay /5./, (cf. Figue 5.). Fo and, Z S Z S s 9 i.e. > 3 8 apply categoy A t t s 9 i.e. < 3 9 apply categoy B o C t t 5.7. Othotopic Shell Theoy This appoach elates to shells of categoy C. The effect of longitudinal stiffenes can be included by adding the tem EI s lef w x (5.08) in Equation (5.3) whee the tem EI lef /s epesents the bending stiffness of a stiffene, including effective flange, equally distibuted ove the stiffene width. In Equation (5.8) this tem becomes, EI s lef w x (5.09) Axial Compession The solution to the diffeential equation fo axial compession becomes, (cf. Equation (5.56)) ( + n ) π E t ( ) ( ) ( ) m + + Z m σ xe m γ (5.0) + ν l A s t e m π m + n whee A is the atio of the stiffene exclusive of the effective plate flange, and ( ν ) I lef γ (5.) 3 st defines the atio between the stiffness of stiffenes and the shell plating. Most shells will buckle in one wave between the ends. Hence, minimizing Equation (5.0) subject to the constaint m, thee comes out C x 3 γ + A Z + π s t e (5.)

31 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 The last tem epesents the effect of shell plating alone, while the fist tem is the contibution fom the stiffenes. Fo vey shot cylindes, the last tem in Equation (5.0) becomes negligible and the smallest buckling load is obtained with n 0, C x + γ (5.3) A + st This is the buckling coefficient fo an othotopic plate and can also be easily deived fom Equation (7.89). Fo vey lage Z, the effect of stiffenes becomes small. A convenient epesentation of the buckling coefficient is found by applying the asymptotic appoximation, o, ρ + C C x ( 3 π ) C Z (5.) Z C Cx ρ. (5.5) C x The knock-down facto assumed by DNV /5./ is ρ Extenal Lateal Pessue The elastic buckling load fo extenal lateal pessue is expessed by (m ), Ref. Eq p π E t σ θ,e ( ν ) ( n ) t γ Z l n n π n ( + n ) (5.6) The minimum load is found by minimization with espect to n 0. Following the appoach outlined in Section 5.., the buckling coefficient fo a vey shot cylinde (Z 0) is, while fo vey long cylindes ( Cθ + + γ ) (5.7) C θ 6 3π Z (5.8) Thus, the asymptotic expession takes the fom, C θ Z Cθ + ρ 0. Cθ (5.9)

32 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 A knock-down facto, ρ 0.6, is typically used. Buckling coefficients fo panel stiffene buckling ae listed in Table 5-. Table 5- Buckling Coefficients Fo Longitudinally Stiffened Panels ψ ξ ρ + γ S + Ast 0. 70Z 0.5 Axial Stess ( ) Tosion and Shea Foce e γ S s Z Lateal Pessue ( + +γ S ) 0. Z Beam on Elastic Foundation The shell buckling poblem beas consideable similaity with buckling of a beam on an elastic foundation, (see Section in T.H. Søeide, Ultimate load analysis of maine stuctues). Depending on the stiffness of the foundation, the beam may buckle in seveal waves. Hee, a buckling mode with one wave (m ) is consideed. The diffeential equation fo such a beam with a sinusoidal initial impefection, πx δ sin (5.0) w o o is given by, w EI + N x w x w N w 0 + α (5.) x whee the axial load, N, is defined positive in compession. By applying the opeato, / x, the similaity with Equation (5.8) is appaent. The solution is given by π EI l α l N * E + l π EI l (5.) and the citical stess is, σ E N E A πt + st π EI l α l l ( st + A) + π EI (5.3) l Equation (5.3) may be eaanged so that

33 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 33 of 3 σ E π EI l (st + A) 3 3Z st + π ( ν ) I l (5.) Equations (5.3-) ae identical if, α π E ( ν ) 3 t l s (5.5) and Equation (5.) can be used to epesent the behaviou of a stinge stiffened shell. The total deflected shape of the beam is given by w w + w (5.6) tot o The effective deflection is, N * N E w N N * E π δ o sin l x (5.7) whee N E * is given by Equation (5.). The maximum moment becomes, N N E * w N E M max EI δ o (5.8) x N l * N E whee N E is the Eule buckling load fo zeo foundation stiffness. The fist yield occus when N * N N E σ Y + st + A N N σ E * σ E σ x + σ σ x * E * E Nw W o σ x(st + A) w W o (5.9) The solution fo the compessive stess is given by the Pey-Robetsen fomula, whee, σ x σ Y ( ) ( ) + γ + ξ + γ + ξ γ γ (5.30) σ Y σ E ( st + A) w o γ *, ξ * (5.3) σ σ W E E

34 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 3 of 3 in which W is the elastic section modulus of the stinge-shell combination. If a educed effective width, s e, of the shell is assumed, the ultimate load expession eads se t + A σu σ x st + A (5.3) The magnitude of impefection may be pedicted by fomulas simila to those specified fo stiffened plates. It should be noted that stinge stiffened cylindes ae vey pone to inwad buckling, because othewise cicumfeential stetching would occu. This means that the failue is most likely to be plate-induced fo intenal stiffening. As seveal spans may take the same deflection patten, the stinges may expeience consideable otational estaint at the ing fames. 5.8 Buckling of Ring Stiffened Shells 5.8. Lateal Pessue The buckling pessue of a ingstiffened cylinde is conveniently fomulated as the sum of a shell contibution, p c, and a ing-fame tem, p pc p p c + (5.33) A widely used expession is the Byant fomula /5.7/ which is based on the elastic potential enegy t k ( n ) EI pc E + (5.3) 3 l ( k ) n + k n + whee k π/l. The citical numbe of waves is usually in the ange of to 6. Sometimes the shell contibution is neglected and n is selected equal to so that, This epesents ovalization of the ing. p c EI 3 3 (5.35) l The magnitude of the stess pedicted by Equation (5.3) is usually fa above the elastic limit. Vaious methods exist fo taking elasto-plastic effects into account. One method is paallel to the Pey-Robetson method fo column buckling (see efeence /5./). The othe method epesents a genealization of Equation (5.35) p η p + η p (5.36) c c c whee h c and h ae the plasticity coection factos. Vaious fomulas have been suggested. A

35 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 35 of 3 simple definition is EsEt E t η c, η (5.37) E E whee E s and E t ae the secant and tangent modulus espectively. Fom Equation (5.35) the equiement to ing moments of inetia can be deived. The total cicumfeential foce between ings is given by N θ pl. Assuming a sinusoidal impefection with amplitude w o, the bending stess at the top of the fee flange is p w σ b o zt θ I p (5.38) p c whee z t is the distance fom the centoid of the top flange and p c is given by Equation (5.35). A safety facto of is applied in the denominato of the magnification tem. A consevative design citeion is to equie that the total stess at the top flange be less than half the citical stess fo tosional buckling of the ing, σ T. Hence, b σ θ + σ θ σ T (5.39) This yields the following equiement to the moment of inetia of the ing stiffene, 3 p l 3 Ezt wo I + (5.0) 3E (0.5σ σ T θ) If this equiement is not met, a moe accuate calculation of the citical pessue (sketched fo ing buckling below), must be pefomed as shown below. This solution was developed by Bodne in 957. It can be expessed as, σ whee the buckling coefficient is E π E ( v ) t l C (5.) ( ) ( ) Z C + n + γ + n ( + α )(0.5 + n ) + n π (5.) This is identical to the shell buckling coefficient in Equation (5.8), except that the smeaed bending contibution fom the ing-stiffenes ae added ( ν ) e I γ 3 (5.3) t l

36 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 36 of 3 whee l is the ing-stiffene spacing, including the effective shell flange. I e is the moment of inetia of the ing-stiffene An appoximate buckling coefficient may be obtained analogous to the appoach used fo unstiffened cylinde. The expession eads, + γ 8Z γ C + (5.) + α + 3π + γ γ whee, A α (5.5) l t eo is the atio between the ing-stiffene aea and the effective shell flange aea. The effective length of the cylinde is taken as the distance between the end closues, bulkheads o heavy ing fames, as indicated in Figue 5.9. Elasto-plastic buckling is calculated by means of a fist yield citeion as used fo beam columns. It is assumed that the ing stiffenes have an initial out-of-oundness compatible to the buckling mode, w ( θ ) w sin nθ (5.6) o o L 0.H 0.H L a c H H L L H (L +L )/ b d L L Figue 5.9 Definition of the Effective Length. In calculating the bending contibution, it is necessay to take into account that the stess in the ing stiffene and in the shell flange is diffeent due to estained contaction. The total foce in between the ing fames can be expessed as + α Fθ p σ θ tl 05. ν (5.7)

37 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 37 of 3 Hee, it is also taken into account that the stess at the top of the ing, σ θ, is lage than the stess in the plate, so that in calculating the effective bending moment the stess is evaluated at the shell flange. This gives, F θ σ tl t θ + α 05. ν (5.8) whee t is the adius to the top of the ing stiffene. In calculating the moment of inetia of the ing fame, it is assumed that the ing may have some post-buckling capacity beyond the ing failue. The appaent moment of inetia is expessed as, I app I p + p p s I ps p + p s I Cs C (5.9) whee p and p s ae assumed to be the contibution fom the ing and the shell flange to the citical pessue fo ing-stiffene buckling, C s is the buckling coefficient fo unstiffened shell subjected to lateal pessue, and C is given by Equation (5.). The fist yield citeia can now be established as follows, + α t Cs σ θ tl t o σ θ ( 05. ν) C zw + (5.50) σ Y σ θ σ Y I σ whee z t is distance fom the centoid of the stiffene (including effective shell flange) to the top of the ing flange. This equation has the solution, e σ θ σ y ( ) µ λ µ λ λ λ (5.5) whee the equivalent deflection paamete, µ, is defined as t t s µ w z l C o ( i ) l ( ν) e eo C 05. I (i e ) l t( + α ) eo (5.5) (5.53) The solution is also pesented in Figue 5.0 as a function of µ.

38 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 38 of 3, Stess atio σ c / σ Y,0 0,8 0,6 0, , 0,0 0,0 0,5,0,5,0,5 3,0 Reduced slendeness atio λ Figue 5.0 Citical Stess fo Ring-Stiffene Failue Combined Loading When the cylinde is subjected to a combination of axial compession, bending, extenal pessue, tosion, and shea, a linea inteaction fomula is often used. That is, the total ing moment of inetia is taken as the sum of the equiements fo each sepaate load condition. 5.9 Geneal Buckling 5.9. Axial Compession and Bending As fo plane panels, oveall buckling is an undesied event. Hence, fo offshoe stuctues, the ings ae designed so as to ensue that this failue mode is pevented. This is achieved if the geneal buckling esistance is highe than the buckling load of the shell and, if pesent, longitudinal stiffenes. The expessions fo geneal buckling becomes vey complicated fo analytical teatment. Theefoe, consideable simplification is obtained if the plane panel analogy can be applied, neglecting the cuvatue effect of the shell. The following deivation is based on Section 7.5. The elastic buckling esistance fo a longitudinal stiffene with associated plate flange is,

39 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 39 of 3 σ E π D a t + A s i a (5.5) whee it has been assumed that i a >>. The esistance to geneal buckling is expessed by Equation (7.93), π D σ E [ ii a b] (5.55) A b t + s σ > σ yields, The equiement that E E i b b ia b E > I a E a t b o σ + π A s (5.56) If buckling is not elastic, σ E may be substituted by the elasto-plastic buckling esistance, σ c. This is assumed to be consevative because the tansvese gides ae less influenced by plasticity and shape impefections. While a plane panel buckles with one wave in the tansvese diection seveal waves may be geneated in the ing diection of the shell. Howeve, Equation (5.56) may still be applied if b is substituted by p/n, and a by l, I π A t c n + st l σ E (5.57) Equation (5.57) may be used fo both, shell with stinges and shell without stinges. A poblem aises when assessing the numbe of cicumfeential waves. One possibility is to use the values pedicted by elastic theoy. In efeence /5./, the tem n /π is eplaced by 500. When calculating I account should be taken fo educed shell plating. Fo axisymmetic buckling, DNV poposes leff ( min 56. t, l) (5.58) 5.9. Tosion and Shea The equiement is that geneal buckling should be pedicted by buckling in tosion and shea. The elastic geneal buckling esistance can be witten as a genealization of the isotopic buckling expessions, (see Equation (5.93)). whee σ G x θ,e π E ( ν ) t ( γ ) k l + (5.59)

40 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page 0 of 3 k (5.60) Z G L t ( ν ) Z G ( Ast) + + γ (5.6) By intoducing the appoximations, k 0.856Z 3/ G G σ σ, (cf. Equation (5.9)), gives x θ,e> x θ,e and γ >>, the equiement that I 05 > A + st. σ θ,c 35 E 85 5 L Ltl (5.6) 5.0 Column Buckling Vey long cylindical shells may fail by column buckling modes which can not be descibed by linea Donnel s shell theoy. Futhemoe, vey dangeous inteaction between local buckling and column buckling may occu due second ode effects of axial compession. Bending effects may also cause loss of stiffness due to ovalization. Fo pue axial compession of a cylinde with initial out-of-staightness, the Pey-Robetson appoach may be used whee failue is defined when the maximum axial stess eaches the local buckling esistance. A moe compehensive desciption of oveall failue modes can be found in efeence /5./. 5. Refeences /5./ "Buckling Stength Analysis" Classification Note No. 30., Det Noske Veitas, July 98. /5./ Odland, J.: "Buckling Resistance of Unstiffened and Stiffened Cicula Cylindical Shell Stuctues", Nowegian Maitime Reseach, No. 3, Vol. 6, 978. /5.3/ Flügge, W.: "Stesses in Shells", Spinge-Velag, Heidelbeg, N.Y., 973. /5./ Donnel, L.H.: "Stability of Thin-Walled Tubes Unde Tosion", NACA Repot No. 79, 933. /5.5/ Esslinge, M. and Gee, B.: "Postbuckling Behaviou of Stuctues",

41 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 Spinge-Velag, Wien-N.Y., 975 /5.6/ Timoshenko, S.P. and Gee, J.M.: "Theoy of Elastic Stability", McGaw-Hill Kogakusha Ltd., 96. /5.7/ Byant, A.R.: "Hydostatic Pessue Buckling of a Ringstiffened Tube", Naval Constuction Reseach Establishment, Rep. R-306, 95.

42 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 INDEX A asymptotic appoximation 30 axisymmetic buckling 5 axisymmetic loading 0 B Batdof paamete, 5, 7, 8, 3 Batdof' paamete 0 beam on an elastic foundation 3 bifucation point, 5 Bodne 3 Byant fomula 33 buckling coefficient 5, 6, 7, 8, 9,, 3, 30, 3, 35, 36 buckling modes 3, 5, 6 C cicumfeential stain 0 cicumfeential stess 6, 7, 8, 9, classical buckling stength column buckling, 6, 33, 39 Column buckling 3 combined loading 6 constitutive elationships 6 citical load 5, 5, 6 citical stess, 5, 6, 3, 3 cuved panel, 6 cuved shell 7, 3 D Donnel, 6, 6,, 39 E ECCS effective deflection 3 effective load effective width 8, 0, 33 Elasto-Plastic Buckling 5 equivalent educed slendeness atio 7 equivalent stess 0, 7 Eule buckling load 3 Eule buckling stess 6 F Flügge, 39 G geneal buckling 37, 38 Geneal buckling 3 H homogeneous solutions I Intefame shell buckling 3 isotopic buckling 38 K kinematic elationships 6 knock-down facto, 3, 30, 3 L local buckling 3, 39 M mateial impefections Mechant- Rankine fomula 5 N naow panels 6 O Othotopic Shell Theoy 9 othotopic shell. 7 ovalization 8, 9, 33, 39 P Panel ing buckling 3 paticula solution Pey-Robetsen fomula 3 Pey-Robetson 33, 39 plate stiffness 5 pimay path

43 TMR05 Buckling and Ultimate Stength of Maine Stuctues 5. Buckling of Cylindical Shells Page of 3 R educed slendeness atio 5 estained contaction 8, 35 S seconday path shallow shell 6 shape impefections, 3, 38 Shell buckling 3 T to Timoshenko and Gee Tosional buckling 3 V von-mises 7