Y.H. Kuo

Stability of a Timoshenko beam resting on a Winkler elastic foundation

The influences of a Winkler elastic foundation modulus, slenderness ratio and elastically restrained boundary conditions on the critical load of a Timoshenko beam subjected to an end follower force are investigated. The characteristic equation for elastic stability is derived. It is found that the critical flutter load for the cantilever Timoshenko beam will first decrease as the elastic foundation modulus is increased and when the elastic foundation modulus is greater than the corresponding critical value, which corresponds to the lowest critical load, it will increase, instead. In particular, if the elastic foundation modulus is large enough, the critical flutter load for the cantilever Timoshenko beam can be greater than that of the Bernoulli-Euler beam. For a clamped-translational or clamped-rotational elastic spring supported beam resting on an elastic foundation, there exists a critical value of the spring constant for each beam. At this critical point, the critical load jumps and the type of instability mechanism changes. The jump mechanisms for beams resting on elastic foundations of different modulus values are different.

Elastic stability of non-uniform columns

Abstract A simple and efficient method is proposed to investigate the elastic stability of three different tapered columns subjected to uniformly distributed follower forces. The influences of the boundary conditions and taper ratio on critical buckling loads are investigated. The critical buckling loads of columns of rectangular cross section with constant depth and linearly varied width ( T 1 ), constant width and linearly varied depth ( T 2 ) and double taper ( T 3 ) are investigated. Among the three different non-uniform columns considered, taper ratio has the greatest influence on the critical buckling load of column T 3 and the lowest influence on that of column T 1 . The types of instability mechanisms for hinged-hinged and cantilever non-uniform columns are divergence and flutter respectively. However, for clamped-hinged and clamped-clamped non-uniform columns, the type of instability mechanism for column T 1 is divergence, while that for columns T 2 and T 3 is divergence only when the taper ratio of the columns is greater than certain critical values and flutter for the rest value of taper ratio. When the type of instability mechanism changes from divergence to flutter, there is a finite jump for the critical buckling load. The influence of taper ratio on the elastic stability of cantilever column T 3 is very sensitive for small values of the taper ratio and there also exist some discontinieties in the critical buckling loads of flutter instability. For a hinged-hinged non-uniform column ( T 2 or T 3 ) with a rotational spring at the left end of the column, when the taper ratio is less than the critical value the instability mechanism changes from divergence to flutter as the rotational spring constant is increased. For a clamped-elastically supported non-uniform column, when the taper ratio is greater than the critical value the instability mechanism changes from flutter to divergence as the translational spring constant is increased.

Analysis of non-uniform beam vibration

Abstract In this paper a systematic development of the solution theory for the non-uniform Bernoulli-Euler beam vibration, including both forced and free vibrations, with general elastically restrained boundary conditions, is presented. The frequency equation and dynamic forced response, which is shown in closed integral form, are concisely expressed in terms of the fundamental solutions of the system. If the exact, closed form fundamental solutions are not available, then approximate fundamental solutions can be obtained through a simple and efficient numerical method. The present analysis can also be applied to the vibrational analysis of a beam with viscous and hysteretic damping.

Bending vibrations of a rotating non-uniform beam with tip mass and an elastically restrained root

Abstract The influence of taper ratio, elastic root restraints, tip mass, setting angle and rotating speed on the pure bending vibration of rotating non-uniform beams is investigated. The characteristic equation is derived in term of the four fundamental solutions of the system. The three different non-uniform beams considered are the tapered beams with linearly varying width and constant depth, constant width and linearly varying depth, and linearly varying width and depth. The results indicate that the effect of taper ratio on the first natural frequencies of the three different tapered beams is less significant than those on the second and third natural frequencies, besides the beam with linearly varying width, constant depth and without tip mass. The tip mass can depress the effect of taper ratio on the first natural frequencies. For the second and third natural frequencies, as the taper ratio is increased, there exists a cross-over between the natural frequencies of the rotating beams with and without tip mass. While changing the rotational and translational elastic spring constants, there may also exist a cross-over between the third natural frequencies of the rotating beam with and without tip mass.

Exact static deflection of a non-uniform Bernoulli-Euler beam with general elastic end restraints

Abstract With the assumption that the bending rigidity of a non-uniform beam is second-order differentiable with respect to the axial coordinate variable, the exact solution of the static deflection of a non-uniform Bernoulli-Euler beam with general elastically restrained boundary conditions is developed in closed integral form. The Greens functions for the beam with various kinds of loading and boundary conditions are presented and expressed in terms of the four normalized fundamental solutions of the system. Examples are given to illustrate the analysis.

Bending Frequency of a Rotating Beam With an Elastically Restrained Root

Upper and lower bounds of the fundamental bending frequency of a rotating uniform beam with an elastically restrained root are obtained by the Rayleigh’s and minimum principles, respectively. It is shown that the fundamental bending frequency of the rotating uniform beam with rotational flexibility only is always higher than that of the nonrotating beam. If the setting angle is not equal to zero, the fundamental bending frequency of the rotating uniform beam with translational flexibility can be less than that of the nonrotating beam, and the phenomenon of divergence instability (tension buckling due to the centrifugal force) may occur. Finally, the influence of hub radius, setting angle, rotational speed, and elastic root restraints on the fundamental bending frequency of the beam is also investigated numerically by the transfer matrix method.

Bending vibrations of a rotating non-uniform beam with an elastically restrained root

Abstract The influence of taper ratio, elastic root restraint, setting angle and rotational speed on the bending natural frequencies of a rotating non-uniform beam is investigated using a semi-exact numerical method. One observes that the influence of taper ratio on the second and third natural frequencies of a rotating beam with constant width and linearly varied depth and a double-tapered beam is greater than that of a beam with constant depth and linearly varied width. For a beam with rotational flexibility only, the first three natural frequencies of the rotating beam are greater than those of the non-rotating beam. The second and third natural frequencies of a rotating beam with translational flexibility are greater than those of the non-rotating beam. However, the fundamental natural frequency of the rotating beam can be less than that of the non-rotating beam. In particular, when the translational rigidity of the root is relatively low and the setting angle and rotational speed of the beam are relatively high, the value of the natural frequency becomes pure imaginary and the phenomenon of divergence instability occurs. The natural frequencies are decreased when the setting angle is increased, and the influence of the setting angle on the natural frequencies becomes very significant when the phenomenon of divergence instability is about to occur.

Static analysis of nonuniform timoshenko beams

Abstract With the assumption that the bending rigidity of a beam is second-order differentiable with respect to the coordinate variable, the exact static deflection of a nonuniform Timoshenko beam with typical kinds of boundary conditions are given in closed form and expressed in terms of the four fundamental solutions of the governing differential equation. Finally, the limiting cases are studied and the results are shown to be consistent with those in the existing literature.

Divergence-type stability of a non-uniform column

Abstract A simple and efficient method is presented to calculate the buckling loads of a divergence-type nonconservative system, that of the breadth taper column under distributed follower forces. The characteristic equation of the system is derived and concisely expressed in terms of the four normalized fundamental solutions of the governing differential equation. These four fundamental solutions can be obtained approximately through a newly developed algorithm which has been demonstrated to be efficient, convenient and accurate. The influences of the boundary conditions, distributed follower forces and breadth taper ratio on critical buckling load are investigated.

Exact solutions for the non-conservative elastic stability of non-uniform columns

Abstract The exact characteristic equations for the non-conservative elastic stability of general elastic restrained non-uniform columns governed by a fourth order ordinary differential equation with arbitrarily varying polynomial coefficients are derived. The results are compared with those in the existing literature. In addition, the influence of the boundary conditions, taper ratio and variation of linearly distributed tangential forces on the non-conservative elastic stability of two kinds of tapered columns subjected to linearly distributed tangential forces is investigated. It is shown that the parameter e of the linearly distributed tangential forces g (X) = g 0 (1−eX/l) has great influence on the instability mechanism for the clamped-clamped columns.