Analysis Of Large Post-buckling Deformation Of Elastic Rings

It is important for the designer of undersea vessels to understand thebuckling and postbuckling behaviour of circular, elastic, andinextensional rings under uniform compression. This paper isconcerned with large post-buckling deflections of such a ring.This paper presents analytical approximations to large post-bucklingdeformation of a circular, elastic, and inextensional ring under uniformcompression. By combining the linearization of the governingequation with the method of harmonic balance, we establish analyticalapproximations to the post-buckling deformation of the ring. Unlikethe classical method of harmonic balance, linearization is performedprior to proceeding with harmonic balancing thus resulting in a set oflinear algebraic equations for unknown coefficients instead ofnon-linear algebraic equations. We are hence able to establishanalytical approximate formulas for the solution. These formulas showexcellent agreement with the exact solutions, and are valid for small aswell as large deformation.Considering the dimensionless governing equation of a circular,elastic, and inextensional ring under uniform compression:where (' ')denotes the second derivative with respect to s , v (s) isthe difference of the curvature of the undeformed and the deformedring, s ∈[0 , 2π] is the independent variable (arc length), p is theloading parameter, c is the integral constant andμ = 32?c, β = c+p?12. (2)Closeness of the ring requires one consider periodic solution to Eq. (1)only.And the period of v (s) is π , with the boundary conditionsv ′( 0)=v′(π 2) =0 (3)and the supplementary condition2()0∫0 =πv sds. (4)Here, v (s) has the following Fourier series expression:()cos(2)1vshnsnn∑∞== . (5)We will establish the analytical approximate solution to Eq. (1) interms of the initial valuev ( 0)=A. (6)A initial approximation satisfying conditions in Eq. (6) can be taken asv 0 (s,A)= Acos2s. (7)Substituting Eqs. (7) into Eq. (1), and setting the resulting constant term and thecoefficient of term cos 2s equal to zeros give? β +3 A 2 4=0,(μ ?4) A+3A38=0. (8)We can obtain the first analytical approximate values of μ and β :μ 0 ( A) = 4?3A28,β0(A)=3A24. (9)Next, we express v (s), μ in the forms of v 0 ( s)+ ?v0(s),μ 0 ( A) + ?μ0(A) respectively. Making linearization of Eq. (1) withrespect to the correction ?v 0 (s)and ?μ 0 (A) leads to3320.32202000000003020000?+?=″+?+++?″+?+?+vvvvv μ vβvvvμvvμ (10)and?v 0 (0)=0, ?v 0 ′(0)=0. (11)Now, ?v 0 (s) which satisfies Eq.(11) is taken of the form? v 0 ( s)=x0(cos2s?cos4s) . (12)Substituting Eqs. (7), (12) into Eq. (10), and setting the resulting constant itemand the coefficients of the items cos 2s and cos 4s equal to zeros,respectively, yield? β +3 A2 4+(? 3A2+3A28)x 0=0, (13a)8 ?μ 0 A +3A3?32A+8Aμ0?x0(3 2?8μ0+12A?9A2) =0, (13b)3 A 2 4? (? 16+μ 0?3A2+3A24)x 0=0. (13c)Solving Eqs.(13a)-(13c) to obtain β , ?μ 0 and x 0. Finally, weget the second approximation to μ , β and v .μ1 (A)= μ 0+ ? μ0, β 1 (A),v1 ( s,A)= v0+?v0. (14)To obtain the third approximation to the exact solution, we expressv (s), μ in the forms of v1 ( s)+ ?v1(s), μ1 ( A) + ?μ1(A). Makinglinearization of Eq. (1) leads to3320. (15)32212111111113121111?+?=″+?+++?″+?+?+vvvvv μvβvvvμvvμand? v1 ( 0)=0,v1′(0)=0. (16)Setting ? v1 (s) which satisfies Eq.(16) in the form of? v1 ( s)=x1(cos2s?cos4s)+x2(cos4s?cos6s). (17)Substituting the expressions of v1 (s) and ?v 1 (s) into Eq. (15), and settingthe resulting constant item and the coefficients of the items cos 2s, cos4s andcos 6s equal to zeros, respectively, yieldf 11 ( β ,?μ 1,x1,x2)=0, (18a)f 12 ( β ,? μ1,x1,x2)=0, (18b)f 13 ( β ,?μ 1,x1,x2)=0, (18c)f 14 ( β ,? μ1,x1,x2)=0, (18d)Solving Eq. (18a)-(18d) to achieve β , ?μ 1 ,x1,x2. Finally,we canobtain the third approximation to β , μ and v .μ 2 (A)= μ1 + ? μ1, β 2 (A),v 2 ( s,A)= v1+?v1. (19)It should be clear how the procedure works for constructing furtherapproximate solutions.Keywords: Elastic ring;Post-buckling;Large deformation;Linearization;Harmonic balance method...