### Dynamics Of Beams Undergoing Large Rotations Accounting For Arbitrary Axial Deformation

#### Abstract

It is well-known that flexible beams become stiffer when subjected to high speed rotations. This is due to the

membrane-bending coupling resulting from the large displacements of the beam cross-section. This effect, often

called geometric stiffening, has been largely discussed in the last two decades. Several methodologies have been

proposed in the literature to account for the stiffening effect in the dynamics equations. However, considerable effort

is generally done to derive linear models using steady-state assumptions and membrane-bending decoupling. This

work aims first to present a brief review of the open literature on this subject. Then, a general non-linear model is

formulated using a non-linear strain-displacement relation. This model is used to deeply analyze simplified models

arising in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed and

its consequences are discussed. Thereafter, four finite element models are proposed, one based on non-linear theory

and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing

prescribed high speed large rotations. The analyses show that one must account for the geometric stiffening effect

to obtain realistic results. In addition, it is shown that models disregarding the axial displacement dynamics lead to

erroneous results for the axial stress in the beam, which may be of main importance in structural integrity analysis.

Hence, in the general case, geometric stiffening must be accounted for in association with the inclusion of full axialtransverse

displacements coupling dynamics in the model.

membrane-bending coupling resulting from the large displacements of the beam cross-section. This effect, often

called geometric stiffening, has been largely discussed in the last two decades. Several methodologies have been

proposed in the literature to account for the stiffening effect in the dynamics equations. However, considerable effort

is generally done to derive linear models using steady-state assumptions and membrane-bending decoupling. This

work aims first to present a brief review of the open literature on this subject. Then, a general non-linear model is

formulated using a non-linear strain-displacement relation. This model is used to deeply analyze simplified models

arising in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed and

its consequences are discussed. Thereafter, four finite element models are proposed, one based on non-linear theory

and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing

prescribed high speed large rotations. The analyses show that one must account for the geometric stiffening effect

to obtain realistic results. In addition, it is shown that models disregarding the axial displacement dynamics lead to

erroneous results for the axial stress in the beam, which may be of main importance in structural integrity analysis.

Hence, in the general case, geometric stiffening must be accounted for in association with the inclusion of full axialtransverse

displacements coupling dynamics in the model.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**