Buckling of Cylindrical Shells

5 5.1 Buckling of Cylindrical Shells Governing Equation for Buckling of Cylindrical Shells The starting point of the analysis is the strain-displacement relation for plates: 1 1 (uα ,β +uβ ,α ) + w,α w,β 2 2 καβ = − w,αβ ε◦αβ = (366) Consider a flat plate (x, y) and a segment of a cylinder (x, aθ), where a is the radius of a cylinder and θ is the hoop coordinate. a x x y aθ Can the strain-displacement relation for a cylinder be derived from similar re lation for a flat plate? x→x ; dx → dx (367) y → aθ ; ∂ 1 ∂ → a ∂θ ∂y (368) Consider component by component. Use the notation: u1 = ux → u u2 = uy → v u3 → w (369) Then, one gets: 1 ε◦xx = u,x + (w,x )2 (370) 2 u,θ 1 ³ w,θ ´2 w ε◦θθ = + + (371) a 2 a a ´ 1 1 ³ u,θ w,θ ε◦xθ = + u,x + w,x (372) 2 a 2 a There is a new term w/a in the expression for the hoop strain. The physical meaning of this new term becomes clear if we consider axisymmetric deformation with v = 0 and w independent of θ. Define the hoof strain as a relative change in the length of circumference when the original circle has a radius of "a" before deformation and a new circle has a radius of "a + w" after deformation: ε◦θθ = 2π (a + w) − 2πa w = 2πa a (373) a θ w Mathematically, Eq. (370)-(372) can be derived from its counterparts by trans forming the rectangular coordinate system into the curvilinear coordinate system. x = r sin θ (374) y = r cos θ The step-by-step derivation can be found, for example, in the book by Y.C. Fung, "First Course in the Continuum Mechanics." The expression for curvature are transformed in a similar way: κxx = w,xx w,θθ κθθ = 2 a w,xθ κxθ = a (375) Using the variational approach explained in details for the plate problem, one can see that the only new term in the expression for δΠ = 0 is Z δw dS (376) Nθθ a S Therefore, the new term should be added in the equation for out-of-plane equilib rium: µ ¶ 1 2 1 4 D∇ w + Nθθ − Nxx w,xx + Nxθ w,xθ + 2 Nθθ w,θθ = q (377) a a a where ∇4 w = w,xxxx + 2 1 w,xxθθ + 4 w,θθθθ 2 a a (378) The above equations are the nonlinear equilibrium equations for quasi-shallow cylin drical shells. The linear equilibrium equations are obtained by omission the non linear terms, i.e. terms in the parenthesis. The resulting equations are: aNxx ,x +Nxθ ,θ = 0 (379) aNθθ ,x +Nθθ ,θ = 0 1 D∇4 w + Nθθ = q a with ³ w´ Nxx = C u,x +ν (380) a ´ ³w 0 Nθθ = C + ν u,x a ¡ ¢ where C is the axial rigidity, C = Eh/ 1 − ν 2 . The pre-buckling solution should satisfy the system Eq. (379) and (380). These solutions will be denoted by Nαβ = ◦ . −Nαβ 5.1.1 Special Case I: Cylinder under Axial Load P , q = 0 The membrane forces in the pre-buckling state are: ¯ ¯ P ¯¯ 1 0 ¯¯ ◦ Nαβ = 2πa ¯ 0 0 ¯ (381) and the corresponding displacement field reads: P x 2πahE ν Pl w= E 2πah u (x) = − (382) u ( l) l w x aθ It is easy to prove that the above solution satisfies all field equation. 5.1.2 Special Case II: Cylinder under Lateral Pressure ¯ ¯ ¯0 0¯ ◦ Nαβ = Nθθ ¯¯ ¯¯ 01 (383) where from the constitutive equations Eq. (380): Nθθ = Eh w a (384) Substituting Eq. (384) into Eq. (379) leads the following linear forth order inho mogeneous ordinary differential equation for w (x): Dw0000 + Ehw =q a2 (385) or in a dimensionless form: d4 w q + 4β 4 w = (386) 4 D dx where ¡ ¢ 3 1 − ν2 Eh 4 β = 2 = (387) a D a2 h4 The dimension of β is [L−1 M 0 T 0 ], so βx is dimensionless. There are four boundary conditions for a simply supported cylinder: Mxx = w = 0 at x=0 Mxx = w = 0 at x=l (388) where one gets from the moment-curvature relation: Mxx = −D d2 w dx2 (389) Mθθ = 0 The general solutions of the above boundary value problem is: w (x) = e−βx [c1 sin βx + c2 cos βx] + eβx [c3 sin βx + c4 cos βx] − q D4 β 4 (390) The four integration constants can be found from the boundary conditions. typical term of the solution is a rapidly decaying function of x. A It can be conducted that the curvature and bending is confined to a narrow boundary zone of the width xb = π/ (2β). The remainder of the shell undergoes a uniform radial contraction: q (391) w0 = − D 4β 4 For the sake of simplicity, this localized bending can be neglected (D → 0). Then, from Eq. (379), the hoop membrane force is related to the lateral pressure by Nθθ = qa and the pre-buckling solution is: ¯ ¯ ¯0 0¯ ◦ (392) Nαβ = qa ¯¯ ¯¯ 01 5.1.3 Special Case III: Hydrostatic Pressure For a cylinder subjected to the hydrostatic pressure, the total axial compressive force is: P = qπa2 (393) The pre-buckling solution is: ◦ Nαβ ¯ ¯ ¯ 1/2 0 ¯ ¯ = qa ¯¯ 0 1¯ which is a classical membrane stress state in a thin cylinder. (394) 5.1.4 Special Case IV: Torsion of a Cylinder The pre-buckling stress in a cylinder subjected to the total torque of the magnitude T is: ¯ ¯ M ¯¯ 0 1 ¯¯ ◦ = (395) Nαβ 2πa ¯ 1 0 ¯ x aθ N θx M 5.2 Derivation of the Linearized Buckling Equation We are now in the position to linearize the nonlinear buckling equation Eq. (377). It is assumed that the state of membrane forces does not change at the point of buckling from the pre-buckling value. Thus, ◦ Nαβ = −Nαβ (396) ∙ ¸ 1 2 ◦ 1 ◦ ◦ D∇ w + Nθθ + Nxx w,xx + Nxθ w,xθ + 2 Nθθ w,θθ = q a a a (397) and Eq. (377) becomes: 4 where the hoop membrane force Nθθ , the second term in Eq. (397), depends linearly on three component of the displacement vector (u, v, w): µ ¶ 1 w v,θ + + νu,x (398) Nθθ = C a a Therefore, the out-of-plane equilibrium equation is coupled with the in-plane dis placement (u, v) through the presence of the term Nθθ . Note that in the plate buckling problem the in-plane and out-of-plane response was uncoupled. It is possible to eliminate the terms involving in-plane components using the full set of equilibrium and constitutive equations in the in-plane direction. By doing this, the order of the governing equation has to be raised by four to give eight: ∙ ¸ 1 − ν2 2 ◦ 1 ◦ 8 4 ◦ Cw,xxxx +∇ Nxx w,xx + Nxθ w,xθ + 2 Nθθ w,θθ = 0 (399) D∇ w + a2 a a The above equation is called the Donnell stability equation in the uncoupled form. Not that w (x, θ) in the above equation represents additional lateral deflection over and above those produced by the pre-buckling solution. The total deflection is a sum of the two. 5.3 5.3.1 Buckling under Axial Compression Formulation for Buckling Stress and Buckling Mode We are now in a position to develop solutions to the buckling equations, Eq. (399) for four different loading cases discussed in the previous section. Consider first Case I of a simply-supported cylindrical shell in which the pre-buckling solution is given by Eq. (381). In this case, Eq. (399) reduces to: D∇8 w + 1 − ν2 P 4 Cw,xxxx + ∇ w,xx = 0 a2 2πa The buckling deflection of the shell is assumed in the following form: ³ mπx ´ sin (nθ) w (x, θ) = c1 sin l (400) (401) where c1 is the magnitude, and the integer numbers (m, n) denote the number of half-waves respectively in the axial and circumferential direction. The above deformation satisfies both simply-supported boundary conditions at the ends, x = 0 and x = L, and periodicity conditions along the circumference. Here, the half-length of the buckling wave is defined: λ= l m It is convenient to introduce a dimensionless buckling number, m̄: mπx ³ mπa ´ x x = = m̄ l l a a mπa ⇒ m̄ = l (402) (403) Using the dimensionless buckling number, substituting the solution Eq. (401) into the governing equation Eq. (400) leads: ¸ ∙ ³ x´ ¢ ¡ ¢ ¢ D¡ 2 P ¡ 2 4 2 4 2 2 2 2 c1 m̄ + n + m̄ 1 − ν C − n m̄ sin m̄ sin (nθ) = 0 m̄ + a2 2πa a6 a (404) By setting the coefficient in the square bracket to zero, the critical buckling mem brane force per unit length becomes: Pcr 2πa ¡ ¢2 ¡ ¢ D m̄2 + n2 m̄2 2 = 2 + 1 − ν C a m̄2 (m̄2 + n2 )2 Ncr = (405) Here, by introducing the dimensionless parameter χ: ¡ 2 ¢2 m̄ + n2 χ= (406) m̄2 Eq. (405) reads: ¡ ¢ 1 D Ncr = 2 χ + 1 − ν 2 C (407) a χ The dependence of the buckling force on the parameter χ is shown in the figure below. Treating χ as a continuous variable, one can find an analytical minimum: ¢ 1 dNcr D ¡ = 2 − 1 − ν2 C 2 = 0 dχ a χ from which the optimum value of the parameter χ is: r (1 − ν 2 ) Ca2 χopt = D ap = 12 (1 − ν 2 ) h a ' 3.3 h (408) (409) Introducing the expression for the optimum parameter χ into Eq. (407) and using definitions of bending and axial rigidity, one gets: E h2 (Ncr )min = p 3 (1 − ν 2 ) a (410) The buckling stress is obtained by dividing the critical membrane fore by the shell thickness h: (Ncr )min h E h =p 2 3 (1 − ν ) a σ cr = σ cr ' 0.605 E h a (411) (412) This is the classical solution for the buckling stress of a cylindrical shell subjected to axial compression. While the buckling load is unique and does not depend on (m̄, n), the buckling mode is not unique as: χopt ¡ 2 ¢2 m̄ + n2 a = = 3.3 m̄2 h There are infinity of combinations of m̄ and n that give the same expression. Example 8 Let a/h = 134, then ¡ 2 ¢2 m̄ + n2 = 3.3 × 134 ' 442 χopt = m̄2 m̄2 + n2 ⇒ ' 21 m̄ (413) n= 5.3.2 p 21m̄ − m̄2 Buckling Coefficient and Batdorf Parameter Let us introduce the dimensionless buckling stress or the buckling coefficient kc : kc = σ cr hl2 π2 D (414) Additionally, the Batdorf parameter Z is defined: Z= p l2 1 − ν2 ah (415) Using Eq. (411), the buckling coefficient becomes: h E hl2 kc = p × 2 3 (1 − ν 2 ) a π D 12 p l2 =√ 1 − ν2 ah 3π 2 12 =√ Z 3π 2 kc ' 0.702 Z (416) (417) This relation between kc and Z are shown in the figure below together with two limiting cases of very short and very long cylindrical shells. These two limiting cases are discussed below. Limiting Cases: Short Cylinders, a À l Consider a case of the following conditions: aÀl a ⇒ →∞ (418) l Then, it is natural to assume that the number of half-waves in the axial direction is unity: m=1 Consequently, one gets: a →∞ l Since m̄ is much larger than n, one also gets: m̄ = π (419) m̄2 + n2 ' m̄2 (420) Now, from Eq. (405), the membrane force reads: P 2πa ¡ ¢ 1 D = 2 m̄2 + 1 − ν 2 C 2 a m̄ 2 m̄ D = 2 a π2 D = 2 l Ncr = (421) The buckling stress reads: Ncr π2D = h hl2 Now, from the definition of the buckling coefficient, Eq. (414), the buckling coeffi cient for the very short cylindrical shells reads: σ cr = kc = 1 This solution indicated in previous figure as the lower bound cut-off value. upper bound cut-off values is given by the Euler buckling load. The Limiting Cases: Very Long Cylinders, l À a The cylindrical shell is becom ing the Euler column. The buckling load of the column is: Pcr = where I = πa3 h for cylinder sections. π 2 EI l2 (422) The buckling stress reads: Pcr 2πah π 2 ³ a ´2 = E 2 l σ cr = (423) For this very long cylindrical shells, the buckling coefficient can be written: ¡ ¢ ³ a ´2 kc = 6 1 − ν 2 h kc = 6 µ a l ¶4 Z2 (424) (425) From Eq. (411) and (423), the transition between the local shell buckling and global thin-walled column buckling occurs when (σ cr )shell = (σ cr )column which gives: 5.4 E h π 2 ³ a ´2 p = E 2 l 3 (1 − ν 2 ) a p π 2 3 (1 − ν 2 ) hl2 ' 8.15 ⇒ 3 = a 2 (426) (427) (428) Buckling under Lateral Pressure From the pre-buckling solution, Eq. (392), the governing equation Eq. (399) be comes: 1 − ν2 q Cw,xxxx + ∇4 w,θθ = 0 (429) D∇8 w + 2 a a Assuming the double sine buckling deflection function, similar to the case of axial compression, the governing equation becomes: ∙ ¸ ³ x´ ¢ ¡ ¢ ¡ 2 ¢ D¡ 2 2 4 4 2 2 2 2 c1 m̄ 1 − ν C − qa n m̄ + n + m̄ + n sin m̄ sin (nθ) = 0 a2 a6 a (430) By setting the coefficient in the square bracket to zero, the equation for the buckling pressure becomes: ¡ ¢2 ¡ ¢ m̄4 D m̄2 + n2 2 C + 1 − ν qa = 2 a n2 n2 (m̄2 + n2 )2 (431) It can be shown that the smallest eigenvalue is obtained when m = 1 or m̄ = πa/l. With this observation, the solution becomes a function of the parameter n, or dimensionless parameter n̄: nl (432) n̄ = πa Additionally, we define the dimensionless buckling pressure, q̄: q̄ = q l2 a π2 D (433) Now, substituting m̄, n̄ and p̄ into Eq. (431) leads: ¡ ¢2 ¡ ¢ µ ¶ 1 + n̄2 1 − ν2 C l 2 1 q̄ = + n̄2 a2 a n̄2 (1 + n̄2 )2 (434) By introducing the Batdorf parameter Z, one gets: ¡ ¢2 1 + n̄2 1 12 2 q̄ = + 2 π Z 2 2 n̄2 n̄ (1 + n̄ ) (435) For any value of the geometrical parameter Z, there exists a preferred n̄ which minimize the buckling pressure. Treating n̄ as a continuous variable, the optimum n̄ can be found analytically from d¯ p/d¯ n = 0. Substituting this back into Eq. (432), there will be a unique relation between the buckling pressure and the Batdorf pa rameter. The solution is shown graphically in the figure below. Limiting Cases: Very Long Cylinders, l À a In the limiting case of a long tube (l À a), one gets: a m̄ = mπ → 0 (436) l In this case, the last term in Eq. (431) vanishes and the buckling pressure becomes: q= n2 D a3 (437) Imagine a long cylinder consisting of a change of ring, each of the height b. The moment of inertia of the ring along the axial axis reads: I= bh3 12 (438) From Eq. (437) and (438), the intensity of the line load Q = qb can be written: Q = qb n2 Eh3 = 3 b a 12 (1 − ν 2 ) n2 EI = 3 a (1 − ν 2 ) (439) The above approximation is due to Donnell. The smallest integer value is n = 1 which gives the following buckling mode. The Donnell solution should be compared with more exact solution of the ring buckling problem which take into account a more complex incremental displacement field with both w (θ) and v (θ). Here, a distinction should be made between the centrally directed pressure (as in all preceding analysis) and the field -pressure loading where pressure is always directed normal to the deformed surface. In the later case, the ring buckling occurs at: ¡ ¢ EI Qc = n2 − 1 3 a (440) where the smallest integer n = 2 so that Q = 3EI/a3 . The buckling mode has now eight nodal points rather than four. The solution of centrally directed pressure loading case is: ¡ 2 ¢2 n − 1 EI Qc = n2 − 2 a3 (441) where again the smallest n = 2. Thus the smallest buckling load intensity is: Qc = 4.5 EI a3 (442) It is seen that for a realistic value n=2, the present solution, Eq. (439) gives the buckling pressure between the two cases of field-pressure loading and centrally directed loading, 3 < 4 < 4.5. 5.5 Buckling under Hydrostatic Pressure Special case of the combined loading in which the total axial load P is: P = πa2 q (443) The pre-buckling solution is given by Eq. (394). The solution of the buckling equa tion still can be sought through the sinusoidal function, Eq. (401). The optimum solution can be found by a trial and error method varying parameters m̄ and n̄. A graphical representation of the solution is shown in figure below. 5.6 Buckling under Torsion A twist applied to one end of a cylindrical shell process a twisting force Nxθ : ◦ Nxθ = M 2πa (444) ◦ = N◦ = The other two components of the pre-buckling membrane forces vanish Nxx θθ 0. Moreover, the force is constant. Under these conditions, the governing equation reduces to: D∇8 w + 1 − ν2 2 ◦ C w,xxxx + Nxθ ∇4 w,xθ = 0 2 a a (445) In view of the presence of odd-ordered derivatives in the above equation, the sep arable form of the solution for w (x, θ), assumed previously, does not satisfy the equation. Under torsional loading, the buckling deformation consists of circumferential waves that spiral around the cylindrical shell from one end to the other. Such waves can be represented by a deflection function of the form: ³ x ´ (446) w (x, θ) = C sin m̄ − nθ a with mπa (447) L where m and n are integers. The alone displacement field satisfy the differential equation and the periodical condition is the circumferential direction, but does not satisfy any commonly used boundary at the cylinder ends. Consequently, this simple examples can be used only for long cylinders. For such cylinders, introduction of the additional displacement field into the governing equation, yields: ¡ 2 ¢2 ¡ ¢ m̄ + n2 D m̄3 ◦ + 1 − v2 C (448) Nxθ = 2 2 2 m̄ n a 2 (m̄2 + n2 ) n m̄ = For sufficiently long cylinders, the shell buckles in two circumferential waves, n = 2. Also, the term m̄2 is small compared with 4. Then, the approximate expression is: ◦ Nxθ = ¢ 4 D m̄3 ¡ + 1 − v2 C 2 m̄ a 64 (449) An analytical minimization of the alone experiment with repeat to m̄ gives: µ ¶2 64 h 4 (450) m̄ = 2 9 (1 − ν ) a Upon substitution, the final expression for the buckling force, or better, critical shear strain causing buckling is: µ ¶3/2 N◦ 0.272 E h τ cr = xθ = (451) 3/4 2 h a (1 − ν ) The above solution was given by Donnell. As noted, the above solution is invalid for short shells due to the difficulties in satisfying boundary condition. A more rigorous analytical-numerical solution is shown in the figure below. As the radius of the shell approaches infinity, the critical stress coefficient for simply supported and clamped edge approaches respectively the value 5.35 and 8.98 corresponding to plates under the shear loading. 5.7 Influence of Imperfection and Comparison with Experiments Because of the presence of unavoidable imperfection in real shells, the experimen tally measured buckling load are much smaller than the ones found theoretically. Reduction in the buckling strength due to imperfection Comparison of theoretical and experimental results for four different type of loads: • Axial compression • Torsion • Lateral pressure • Hydrostatic pressure are shown in the subsequent two pages. Note that the graphs were presented in log − log scale. Replotting the results for axially loaded plate yields the graph shown below. 1.1 Theory �cr = 0.605 Eh/a �exp 1.0 0.8 0.6 A design recommendation 0.4 0.2 0 0 500 1,000 1,500 a 2,000 2,500 3,000 3,500 h Distribution of test data for cylinders subjected to axial compression. Figure by MIT OCW. The differences are shown to be very large. Design curves for cylindrical and other shells are based on the theoretical solution modified by empirical prediction factors called "know-down factors". For example, the solid curve shown in the receding page is a "90 percent probability curve." For a/h = 50, the reduction factor predicted by this probability curve is 0.24. Thus, the theoretical solution σ cr = 0.605 Eh/a should be multiplied by 0.24 for the design stress σ = 0.15 Eh/a. Cylindrical shells loaded in different way are been sensitive to imperfections and the resulting knock-down factors are smaller. Most industrial organization establish their own design criteria. They are fre quently loose-leaf and are continuously updated on the volume of the experimented evidence increase. The situation can be converted to the firmly established manual of steel constructions for the design of columns and beam. The reason is that col umn are not sensitive to imperfection as far as the ultimate strength is concerned.