Generalised path-following for well-behaved nonlinear structures

Groh, Rainer; Avitabile, Daniele; Pirrera, Alberto

doi:10.1016/j.cma.2017.12.001

Cited by 109 publications

(81 citation statements)

References 95 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the following, we assume a link system comprising 20 links of total horizontal length NL 0 = 1 m. The linear and quadratic spring stiffness are constant at k 1 = 1 kN/m and k 2 = 0 N/m 2 , whereas the cubic stiffness k 3 is varied to modulate the behavior from stiffening (positive k 3 ) to softening (negative k 3 ). The finite-element model is solved using numerical continuation [17] that allows path following of equilibrium manifolds, pinpointing of bifurcations, and branch switching.…”

Section: Representative Modelmentioning

confidence: 99%

Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Groh

Pirrera

2019

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

Shell buckling is known for its extreme sensitivity to initial imperfections. It is generally understood that this sensitivity is caused by subcritical (unstable) buckling, whereby initial geometric imperfections rapidly erode the idealized buckling load of the perfect shell. However, it is less appreciated that subcriticality also creates a strong proclivity for spatially localized buckling modes. The spatial multiplicity of localizations implies a large set of possible trajectories to instability-also known as spatial chaos-with each trajectory affine to a particular imperfection. Using a toy model, namely a link system on a softening elastic foundation, we show that spatial chaos leads to a large spread in buckling loads even for seemingly indistinguishable random imperfections of equal amplitude. By imposing a dominant imperfection, the strong sensitivity to random imperfections is ameliorated. The ability to control the equilibrium trajectory to buckling via dominant imperfections or elastic tailoring creates interesting possibilities for designing imperfection-insensitive shells.

show abstract

Section: Representative Modelmentioning

confidence: 99%

Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Groh

Pirrera

2019

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

show abstract

“…These techniques are, however, not classically used for structural mechanics (statics) applications, where specialized arc-length techniques restricted to a single parameter-the applied load-are predominantly used [15,16]. The application of broader numerical continuation methods within structural finite element solvers often go by the name of "generalized path-following" [17][18][19], and it is such a framework that we use herein to analyze the stability of the axially compressed cylinder.…”

Section: Theorymentioning

confidence: 99%

“…Additionally, we compute bifurcation-tracking curves that trace the evolution of a critical point with respect to two parameters. For implementation-specific details on the modeling framework the interested reader is directed to a previous publication [19].…”

Section: Theorymentioning

confidence: 99%

Localised post-buckling states of axially compressed cylinders and their energy barriers

Groh

Pirrera

2019

AIAA Scitech 2019 Forum

View full text Add to dashboard Cite

The buckling of axially compressed cylinders remains an interesting problem in the engineering mechanics literature. Here we revisit the problem by considering fully localized buckling modes in the form of a single dimple. By applying a combination of nonlinear finite element methods and numerical continuation algorithms, we study the evolution of a single dimple into a ring of circumferential diamond-shaped waves and further into the Yoshimura post-buckling pattern. This post-buckling sequence is established by cellular buckling (homoclinic snaking) as the axial compression is increased; i.e. a sequence of de-and restabilizations causes the single dimple to multiply circumferentially. The initial single-dimple seed corresponds to a saddle in the energy landscape, and the energy barrier provided by the single-dimple saddle around the stable pre-buckled path is quantified. The low magnitude of this energy barrier suggests that the pre-buckling equilibrium is metastable once the single dimple exists as a postbuckling equilibrium. We parameterize the onset of metastability with respect to geometric parameters and assess the suitability of this onset as a lower-bound design load. As NASA's well-known lower-bound curve is overly conservative for shells manufactured using modern, tightly-toleranced manufacturing techniques, the onset of metastability could provide a viable and less conservative lower-bound design load.

show abstract

“…Utilizing that only the signs of eigenvaluesκ i are interesting for the stability conclusions, and that this sign spectrum is not affected by the contents of the mass matrix M, as long as it is positive definite, the treatment below will also set the mass matrix in Eqs. (23) and (26) to M = I N u . This corresponds to the implicit assumption when the static stability criterion is a demand for a positive definite tangent stiffness matrix K.…”

Section: Constrained Free Vibrationsmentioning

confidence: 99%

“…With several parameters in x , any kind of auxiliary equations can be used to select specific parameterized equilibrium states [16,23]. Addition of further parameters in x and related functions in g x are then used for transformations between parameters in the simulation.…”

Section: Convenience Transformationmentioning

confidence: 99%

Constrained stability of conservative static equilibrium

Eriksson

Nordmark

2019

Comput Mech

View full text Add to dashboard Cite

Modelling of static structural stability problems is considered. Focus is set on problems where passive physical constraints affect the response to applied forces, and where more than one free parameter describes the setting. The existence of vibration frequencies at equilibrium states is investigated, as an indication of stability. The relevant Jacobian matrix is developed, with an emphasis on the necessity to formulate the constraint equations from an energy form in a conservative problem. The corresponding mass matrix is introduced, with zero mass contribution from constraint equations. Three different forms of the relevant Jacobians are considered, and alternative methods for the eigenvalue extraction given. Stability is discussed in a context of generalized equilibrium problems, where auxiliary parameters and equations can be included in a continuation setting. Examples show the formulation, implementation and interpretation of stability.

show abstract

Generalised path-following for well-behaved nonlinear structures

Cited by 109 publications

References 95 publications

Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Spatial chaos as a governing factor for imperfection sensitivity in shell buckling

Localised post-buckling states of axially compressed cylinders and their energy barriers

Constrained stability of conservative static equilibrium

Contact Info

Product

Resources

About