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Reut and Beck columns: effects of end gravity force, translational and rotational inertias. Columnas de Beck y Reut: efectos de una fuerza de gravedad en el borde libre, inercias traslacionales y rotacionales. Luis G. Arboleda Monsalve 1 , David G.

Room A Evanston, IL United States of America. The stability of Reut and Beck columns subjected to any combination of gravity and non-conservative fixed-line or follower compressive axial forces is presented using the dynamic formulation. The proposed method is general capturing the static or divergence buckling as well as the dynamic flutter instability of cantilever columns.

Special attention is given to the effects of the end gravity force, translational and rotational inertias along the member. Analytical results are intended to capture the limit on the range of applicability of the static or Euler's method in the stability analysis of slender cantilever columns, and to define the transition from static instability with zero frequency to dynamic instability ''flutter''.

Finally, the comparison between the characteristic stability equations of slender Reut and Beck columns is presented. Keywords: Columns, buckling, dynamic stability, static stability, flutter, non-conservative loads, Beck and Reut columns. Los efectos de la fuerza de gravedad y las inercias traslacionales y rotacionales a lo largo del elemento se analizan cuidadosamente.

The static and dynamic stabilities of slender beam-columns subjected to non-conservative end loads like those produced by jet engines or rockets and cantilevered pipes conveying fluid are of great importance in mechanical, aeronautical, structural and aerospace engineering.

The problem of follower forces on slender columns has been the main subject of many textbooks such as those by []. It has also been presented in several state-of-art review papers such as those by []. The stability problem has also been verified experimentally by [7, 8] while numerical verifications using the finite element program LS-DYNA were presented by [9], This topic has been extensively studied by numerous researchers from different points of view, but due to space limitations just a few of them are presented herein.

For instance, [10] studied the stability of a clamped-elastically restrained column subjected to a partially follower force using the Timoshenko approach. The post-buckling behavior of the Beck column was presented by [15] showing the effect of a tip mass with rotatory inertia.

Recently, [20] studied the influence of an attached end mass on the static and dynamic stability of an elastically restrained Beck column. The main objective of this publication is to present the closed-form eigenvalue equation for the dynamic stability of both Reut and Beck columns including the effects of an axial gravity load applied at the top end, and the translational and rotatory inertias of the column itself. A sensitivity study is carried out showing the transition from static instability with zero frequency to dynamic instability and the interactions between the seven input parameters.

Consider a prismatic element that connects points A perfectly clamped end and B free end , see figure 1. It is assumed that: 1 the beam-column AB is made of a homogenous linear elastic material with modulus E ; 2 its centroidal axis is a straight line; 3 is subj ected to a combination of a gravity axial force P 0 , and a non-conservative axial force P f , applied at the free end B; 4 its transverse cross section is doubly symmetric with a total area A. The transverse and bending equilibrium equations of the differential element shown in figure 1 c are:.

Assuming exponential variations of the bending deflection [i. Notice that the net effect of the rotatory inertia is to reduce the total axial load capacity of the column. The solution to Eq. Once Eq. Dynamic stability of a cantilever column subjected to gravity and fixed-line forcess. Using Eqs. Then, Eqs. Consequently, the dynamic stability of the Beck column corresponds also to the Reut column.

Effects of gravity load, translational and rotational inertias on the dynamic stability of reut and beck columns. Figures 2 , 3 and 4 show the influence of the gravity load, translational and rotational inertias on the buckling loads of both columns of figures 1 a and 1 b. Each peak in the curves of figures 2 a-g corresponds to the phenomenon of flutter when the natural frequencies of the first and second modes of vibration become identical to each other.

Note that the curve corresponding to the static case i. This curve is identical to that presented recently by [16] in terms of effective length factor K. Therefore, the effects of the gravity load on the dynamic stability of the columns of figures 1 a and a are coupled together with the translational and rotational inertias of the column. The effects of an end gravity force, translational and rotational inertias along the member on the stability of Reut and Beck columns were presented and discussed using the dynamic formulation.

The proposed method and eigenvalue equations general capturing the static buckling or divergence as well as the dynamic instability ''flutter'' of slender cantilever columns and subjected to any combination of gravity and non—conservative fixed-line or follower axial forces applied at the free end.

Analytical results obtained from the two cases presented Reut and Beck columns indicate that: 1 the characteristic equations that include the effects of an end gravity force, translational and rotational inertias of Reut and Beck columns are identical to each other; 2 the dynamic method, as proposed herein, gives the correct solution to any combinations of gravity and non—conservative forces.

Important features of the effects of end gravity force, translational and rotational inertias on the stability of Reut and Beck columns were fully discussed herein. Non-conservative Problems of the Theory of Elastic Stability. The Macmillan Company. New York, USA. Panovko, I. Selected Problems and Questions in Strength of Materials. MIR Publishers.

Moscow, Russian. Langthjem, Y. Sugiyama, S. Ryu, M. Applied Mechanics Reviews. Sugiyama, K. Katayama, S. Ryu, Y. Computers and Structures.

International J. Zuo, H. Elishakoff, N. Andersen, J. Rao, G. Viola, A. Engineering Fracture Mechanics. Hernandez, J. Zapata, L. Arboleda, J. Clough, J. Dynamics of Structures. McGraw - Hill Book Co. Prentice Hall. New Jersey, USA. Timoshenko, J. Theory of Elastic Stability. Engineering Societies Monographs. McGraw-Hill Book Company. Services on Demand Article. English pdf Article in xml format Article references How to cite this article Automatic translation Send this article by e-mail.

Structural model Consider a prismatic element that connects points A perfectly clamped end and B free end , see figure 1. Summary and conclusions The effects of an end gravity force, translational and rotational inertias along the member on the stability of Reut and Beck columns were presented and discussed using the dynamic formulation. References 1. Ciudad Universitaria Calle 67 No. How to cite this article.

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Reut and Beck columns: effects of end gravity force, translational and rotational inertias. Columnas de Beck y Reut: efectos de una fuerza de gravedad en el borde libre, inercias traslacionales y rotacionales. Luis G. Arboleda Monsalve 1 , David G. Room A Evanston, IL

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