Research Papers

Constrained Buckling of Spatial Elastica: Application of Optimal Control Method

[+] Author and Article Information
Anna Liakou

Department of Civil, Environmental,
and Geo-Engineering,
University of Minnesota,
Minneapolis, MN 55454-0116
e-mail: liako005@umn.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 9, 2018; final manuscript received April 25, 2018; published online May 28, 2018. Assoc. Editor: Pedro Reis.

J. Appl. Mech 85(8), 081005 (May 28, 2018) (11 pages) Paper No: JAM-18-1139; doi: 10.1115/1.4040118 History: Received March 09, 2018; Revised April 25, 2018

A post-buckling analysis of a constant or variable length spatial elastica constrained by a cylindrical wall is performed for a first time by adopting an optimal control methodology. Its application in a constrained buckling analysis is shown to be superior when compared to other numerical techniques, as the inclusion of the unilateral constraints is feasible without the need of any special treatment or approximation. Furthermore, the formulation is simple and the optimal configurations of the spatial elastica can be also obtained by considering the minimization condition of the Hamiltonian. We first present the optimal control formulation for the constrained buckling problem of a constant length spatial elastica, including its associated necessary optimality conditions that constitute the Pontryagin's minimum principle. This fundamental constrained buckling problem is used to validate the proposed methodology. The general buckling problem of a variable length spatial elastica is then analyzed that consists of two parts; (1) the solution of the optimal control problem that involves the inserted elastica inside the conduit and (2) the derivation of the buckling load by taking into account the generation of the configurational or Eshelby-like force at the insertion point of the sliding sleeve. A variety of examples are accordingly presented, where the effects of factors, such as the presence of uniform pressure, the clearance of the wall, and the torsional rigidity, on the buckling response of the spatial elastica, are investigated.

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Grahic Jump Location
Fig. 1

A spatial elastica constrained by a cylindrical wall with clearance C located with respect to the centerline with the left end fixed: (a) fundamental problem: the right end is constrained to move along the x-axis and (b) insertion problem: the elastica is gradually inserted through a fixed frictionless sleeve at the right end

Grahic Jump Location
Fig. 2

Bifurcation diagram for a constrained clamped–clamped elastica of constant length with clearance c = 0.1 and torsional rigidity α = 5/7. The black line represents the evolution of the applied load Q/π2 (points 1, 2, …, 12) with respect to the end shortening δ.

Grahic Jump Location
Fig. 3

Bifurcation diagram for a constrained clamped–clamped elastica of variable length with clearance c = 0.1 and torsional rigidities α = 5/7, α = 0.2; (1–2) planar one point (P), (3–4) spatial one point, (4–5) one line contact (L), (5–6) two discrete contacts (P–P), (6–7) two line contacts (L–L), (7–8) one line contact (L), and (8–9) line–line–line (L–L–L) (same holds for the corresponding points denoted by ′). The black and gray lines represent the evolution of the internal force Rx/π2 (points 1, 2,…, 9) and the applied load Q/π2 (points 1′,2′,…,9′) with respect to the change of the inserted length δ¯ for α = 5/7 and α = 0.2, respectively.

Grahic Jump Location
Fig. 4

Bifurcation diagram for a constrained clamped–clamped elastica of variable length with clearances c = 0.2, c = 0.3 and torsional rigidity α = 5/7: (1–2) unconstrained, (2–3) spatial one point (P), (3–4) one line contact (L), (4–5) two line contacts (L–L), and (5–6) three line contacts (L–L–L). The black and gray lines represent the evolution of the internal force Rx/π2 (points 1, 2,…, 6) and the applied load Q/π2 (points 1′,2′,…,6′) with respect to the change of the inserted length δ¯ for c = 0.2 and c = 0.3, respectively.

Grahic Jump Location
Fig. 5

Bifurcation diagram for a constrained clamped–clamped elastica of variable length with uniform pressures p = 100, p = 200, clearance c = 0.1, and torsional rigidity α = 5/7: (1–2) planar continuous contact, (2–3) spatial continuous contact (L), (3–4) two line contacts (L–L), (4–5) three line contacts (L–L–L), and (5–6) three point contacts (P–P–P). The black and dashed gray lines represent the evolution of the internal force Rx/π2 (points 1, 2,…, 6) with respect to the end shortening δ for p = 100 and p = 200, respectively. The case of p = 0 is also shown using a gray line (see also validation example).

Grahic Jump Location
Fig. 6

Method 3 – 2 – 1 of Euler angles



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