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# Computational buckling of a three lobed crosection cylindrical shell with variable thickness under combined compression and bending loads-7

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### Computational buckling of a three lobed crosection cylindrical shell with variable thickness under combined compression and bending loads-7

3. 3. International Journal of Mechanical Engineering Research and Development (IJMERD) ISSN 2248-9347 (Print), ISSN 2248-9355 (Online) Volume 3, Number 1, Jan-March (2013)  2 (1− ζ ) cosθ + ζ 2 − 4(1 − ζ ) 2 sin2 θ , 0 ≤ θ ≤ θ1      cosec(θ + 300 ) , ρ = 0, θ1 ≤ θ ≤ 1200 − θ1  f (θ ) = a1  , (1) 2 (1 − ζ ) cos(θ −1200 ) + ζ 2 − 4(1 − ζ ) 2 sin2 (θ −1200 ) , 1200 − θ1 ≤ θ ≤ 1200 + θ1      − secθ , ρ = 0, 1200 + θ1 ≤ θ ≤ 1800   a1 = A1 / a , ρ = a / R1 , ζ = R1 / A1 , θ1 = tan −1 { } 3 / (4 / ζ − 3 ) .L1 and L2 are the axial and circumferential lengths of the middle surface of the shell, and thethickness H (θ) varying continuously in the circumferential direction. The cylindricalcoordinates ( x, s, z ) are taken to define the position of a point on the middle surface of theshell, as shown in Figure (1.1) and Figure (1.2) shows the three-lobed cross-section profile ofthe middle surface, with the apothem denoted by A1, and the radius of curvature at the lobedcorners by R1. While u , υ and w are the deflection displacements of the middle surface of theshell in the longitudinal, circumferential and transverse directions, respectively. We supposethat the shell thickness H at any point along the circumference is small and depends on thecoordinate θ and takes the following form: H( θ )= h0 ϕ (θ ) (2)where h0 is a small parameter, chosen to be the average thickness of the shell over the length L2.For the cylindrical shell which cross-section is obtained by the cutaway the circle of the radius r0from the circle of the radius R0 (see Figure (1.3) function ϕ (θ ) have theform: ϕ (θ ) = 1 + δ ( 1 − cos θ ), where δ is the amplitude of thickness variation, δ = d /h0, andd is the distance between the circles centers. In general case h0=H( θ = 0 ) is the minimum valueof ϕ (θ ) while hm= H( θ = π ) is the maximum value of ϕ (θ ) , and in case of d = 0 the shell hasconstant thickness h0. The dependence of the shell thickness ratio η = hm/h0 on δ has theform η = 1 + 2 δ .2.2. GOVERNING EQUATIONS For a general circular cylindrical shell subjected to a non-uniform circumferentiallycompressive load p (θ ) , the static equilibrium equations of forces, based on the Goldenveizer-Novozhilov theory [29-30] in 1961 and 1964 can be shown to be of the forms: • N x + N sx − P(θ ) u′′ = 0 , ′ N xs + N s• + Qs / R − P(θ )υ ′′ = 0 , ′ Qx + Qs• − N s / R − P(θ ) w′′ = 0 , ′ • M ′ + M sx − Qx = 0 , x (3) M ′ + M s• − Qs = 0 , xs ′ S s − Qs − M sx = 0 , N xs − N sx − M sx / R = 0 ,where N x , N s and Qx , Qs are the normal and transverse shearing forces in the x and sdirections, respectively, N sx and N xs are the in-plane shearing forces, M x , M s and M xs , M sx arethe bending moment and the twisting moment, respectively, S s is the equivalent ( Kelvin-Kirchoff ) shearing force, R is the radius of curvature of the middle surface, ≡ ∂ / ∂x , and 13