Stability of cantilever on elastic foundation under a subtangential follower force via shear deformation beam theories

doi:10.1016/j.tws.2020.106853

Thin-Walled Structures

Volume 154, September 2020, 106853

https://doi.org/10.1016/j.tws.2020.106853 Get rights and content

Highlights

•
Stability of cantilever on elastic foundation via shear deformation beam theories.
•
Buckling load of cantilever subjected to a subtangential follower force.
•
Divergence instability of clamped-free rectangular or circular column.
•
Effect of the warping shapes of the cross-section on critical buckling load.

Abstract

The static stability of clamped-free columns resting on elastic foundation is investigated under a nonconservative subtangential follower force. The higher-order shear deformation beam theories are applied to treat structural instability of a clamped-free rectangular and circular beam resting on Winkler foundation. Based on Engesser's assumption, a single governing equation is derived for divergence instability of Beck cantilever under a subtangential follower force. For different warping shapes of rectangular and circular cross-sections, the critical divergence buckling loads are determined. Static buckling still occurs for a cantilever column subjected to a subtangential follower force with small nonconservativeness parameter and dynamic flutter instability occurs for large nonconservativeness parameter. In a free space, a unified relationship with the Euler buckling loads is given explicitly for the case of a dead load, and the buckling load has an apparent reduction due to shear deformation with different warping of the cross-section. The obtained results further modify the Euler buckling formula to suit for the columns with a broad range of slenderness. Elastic foundation nearly linearly increases the buckling load and a nonconservativeness parameter also raises the divergence buckling load. For shorter columns, the classical buckling loads are significantly overestimated. The effects of slenderness, warping shapes, nonconservativeness parameter, and spring stiffness coefficient are analyzed.

Introduction

Since Beck proved in 1952 that Euler's cantilever beam subjected to a tangential load at the free end was susceptible to instability, the stability problem of a cantilever beam subjected to a nonconservative generalized follower force or distributed loading along the beam has attracted much attention of researchers, and great progress has been made in numerous theoretical and experimental investigations [1,2]. For the stability of Beck's column, the effects of various types of damping including internal and external damping, tip masses, nonconservativeness parameter, rotatory inertia, thermomechanical coupling, elastic foundation, and support characteristics, etc. on the state of stability of Beck's columns have been analyzed. Langthjem and Sugiyama [3] gave a survey of the studies on simple and flexible structure subjected to nonconservative follower loads conducted before the last century. Elishakoff [4] also presented a critical review of pertinent papers on the dynamic stability of structures subjected to follower forces.

It is recognized that the stability loss of elastic structures arises from not only via an instantaneous and irreversible change of configuration (static divergence instability) but also via an exponential increasing motion with time (dynamic flutter instability). In theoretical aspects, Katsikadelis and Tsiatas [5] studied the dynamic stability of damped Beck's column with variable cross-section using the von Karman nonlinear kinematic relation. Marzani et al. [6] applied the generalized differential quadrature method to solve nonconservative stability problems of a nonuniform column subjected to an arbitrary distribution of compressive subtangential follower forces. Based on Engesser's and Haringx's assumptions, Attard et al. [7] utilized the finite element method to study dynamic stability of shear-flexible Beck's columns subjected to subtangential follower forces via the fifth-order Hermitian beam elements. Goyal and Kapania [8] also used the finite element method to treat dynamic stability of laminated beams subjected to nonconservative loading. Lazopoulos and Lazopoulos [9] adopted the gradient elasticity theory of beams to analyze the stability of an elastic beam compressed by nonconservative forces. Tomski and Uzny [10] analyzed free vibrations and stability of a new slender system subjected to a conservative or nonconservative load and obtained the load–natural frequency characteristic curves. Kim and Lee [11] employed the finite element method to handle divergence and flutter behavior of Beck's type of damped columns on a Winkler foundation that are subjected to sub-tangentially distributed follower forces. When multiple weak sections are present, Caddemi et al. [12] addressed the influence of an elastic end support on the dynamic stability of Beck's column. For Beck's nanocolumn, Li et al. [13,14] dealt with the surface effect on dynamic stability of micro/nanocantilevers in a free space and resting on an elastic foundation under a subtangential follower force and obtained the force–frequency interaction diagram. Xiao et al. [[15], [16], [17]] investigated the surface effects on the flutter instability of nanorod and nanoplate for different end restraints under generalized follower force. Humer and Pechstein [18] applied Reissner's geometrically exact relations for the planar deformation of beams to derive the solutions in terms of elliptic integrals for the buckling and postbuckling of a shear-deformable cantilever subjected to a follower force. Zhu et al. [19] adopted the nonlocal beam theory to cope with the stability of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation. Fazelzadeh et al. [20] employed the fully intrinsic beam model to consider the nonconservative stability analysis of columns with various loads and boundary conditions.

Although many researches have been reported for static and dynamic stability of cantilever beams/columns, those studies are mainly based on the Euler–Bernoulli and Timoshenko theories of beams. In other words, the shear stress is relaxed due to the assumption of vanishing or constant shear strain over the cross-section. There is little information on the study on the structural instability of cantilever beams under nonconservative loads via higher-order shear deformation beam theories, though buckling and free vibration of beams based on higher-order shear deformation beam theories were frequently analyzed with and without conservative dead loads [[21], [22], [23], [24], [25], [26]]. It is worth noting that for free vibration and buckling of rectangular beams with higher-order various shear deformation theories, a large number of papers have been published for shear deformation with various warping shapes, in particular for functionally graded beams [[27], [28], [29], [30]], sandwich or compoiste beams [[31], [32], [33], [34]], etc. In addition, the generalized beam theory was developed for a large variety of cross-sections including openun-branched, closed, circular, open, and arbitrary cross-sections [[35], [36], [37], [38], [39]].

In this paper, the stability of a cantilever beam under a nonconservative generalized follower force is studied using higher-order shear deformation beam theories for both rectangular and circular cross-section. Several new warping shape functions are presented for both cross-sections, respectively. A governing differential equation is derived and the characteristic equation for divergence buckling of the cantilever beam resting on elastic foundation is obtained. A necessary condition of the static stability of the cantilever beam is deduced. In a free space, an explicit expression for the buckling load for a conservative dead load is given. The effect of elastic foundation, nonconservativeness parameter, slenderness, and warping shape functions on the divergence buckling of cantilever beam is analyzed.

Section snippets

Governing equation

We consider instability of a uniform cantilever Beck column of length L under the action of a subtangential follower force P, as sketched in Fig. 1, where the cross-section of the column is either rectangular or circular. Width b and height h for rectangular cross-section and radius R for circular cross-section are denoted, respectively. We choose the axial direction of the column as the x axis. Based on Engesser's assumption, use of equations of motion allows us to write [[1], [2]]: $\frac{\partial Q}{\partial x} - N \frac{\partial^{2} w}{\partial x^{2}}$

Buckling load

In this section, we seek the divergence buckling loads of a cantilever Beck column under a nonconservative follower force. For convenience, the following dimensionless quantities are defined: $ξ = \frac{x}{L}, W (ξ) = \frac{w (x)}{L}, Ψ (ξ) = ψ (x),$ $p_{b} = \frac{P L^{2}}{E I}, p_{s} = \frac{P}{G A_{f}}, k = \frac{K L^{4}}{E I} .$

First, Eqs. (24), (25) can be rewritten as $p_{b} (Ψ' + W^{''}) - p_{s} p_{b} W^{''} - k p_{s} W = 0,$ $p_{s} I W^{'''} - p_{s} I_{f} (Ψ^{''} + W^{'''}) + p_{b} I (Ψ + W') = 0 .$

Following Li and Lee [14], we choose a new unknown function $F (ξ)$ to satisfy the governing equation: $(1 - \frac{p_{s} I_{f}}{I}) F^{I V} + (p_{b} - k \frac{p_{s} I_{f}}{p_{b} I}) F^{''} + k F = 0 .$

Thus, if we express

Numerical results and discussion

In the foregoing section, we have derived the divergence buckling load for instability of Beck columns under a nonconservative subtangential follower force based on higher-order shear deformation beam theories. Here numerical examples are given to show the dependence of divergence buckling load on the warping shape function of the cross-section.

First, in the absence of elastic foundation, $k = 0$ , the divergence buckling load satisfies Eq. (51). Clearly, for a conservative dead load, i.e. $η = 0$ , we

Conclusions

This paper investigated the structural stability of a Beck column resting on elastic foundation by using the higher-order shear deformation beam theories. For rectangular and circular cross-section, several new warping shape functions were suggested, which automatically meet the shear-stress-free surface conditions. A single governing equation was derived and the characteristic equation was deduced for clamped-free columns under nonconservative subtangential follower force. The influence of

Author statement

This is an original manuscript which has neither previously, nor simultaneously, in whole or in part, been submitted anywhere else for publication. All authors have approved the final version and agreed to submit it for publication.

Declaration of competing interest

The authors declare that they have no conflict of interest.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 11672336).

References (51)

M.A. Langthjem et al.
Dynamic stability of columns subjected to follower loads: a survey
J. Sound Vib.
(2000)
J.T. Katsikadelis et al.
Non-linear dynamic stability of damped Beck's column with variable cross-section
Int. J. Non Lin. Mech.
(2007)
A. Marzani et al.
Nonconservative stability problems via generalized differential quadrature method
J. Sound Vib.
(2008)
M.M. Attard et al.
Dynamic stability of shear-flexible Beck's columns based on Engesser's and Haringx's buckling theories
Comput. Struct.
(2008)
V.K. Goyal et al.
Dynamic stability of laminated beams subjected to nonconservative loading
Thin-Walled Struct.
(2008)
S. Caddemi et al.
Influence of an elastic end support on the dynamic stability of Beck's column with multiple weak sections
Int. J. Non Lin. Mech.
(2015)
X.-F. Li et al.
Resonant frequency and flutter instability of a nanocantilever with the surface effects
Compos. Struct.
(2016)
Q.-X. Xiao et al.
Flutter and divergence instability of rectangular plates under nonconservative forces considering surface elasticity
Int. J. Mech. Sci.
(2018)
B. Zhu et al.
Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation
Appl. Math. Model.
(2019)
Y. Huang et al.
Buckling of functionally graded circular columns including shear deformation
Mater. Des.
(2010)

Cited by (7)

Mathematical modelling, numerical analysis and damage of dams subjected to hydrodynamic pressure
2022, Ocean Engineering
Citation Excerpt :
Tornabene and FantuzziBacciocchi (2019) obtained a higher-order analytical formulation (zigzag theory) based on well-known Murakami's function for the free vibration analysis of composite beams and arches. Jiang et al. (2020) investigated the static stability of clamped-free columns resting on elastic foundation subject to a nonconservative subtangential follower force via higher-order shear deformation beam theories. Zhang et al. (2021) presented free vibration response of composite foils with various ply angles on the basis of beam theory.
Mathematical modelling and numerical analysis of a concrete dam with combined effects of water-sediment-foundation-earthquake are the main and novel contributions of this study. For this purpose, a concrete dam is modeled by exponential shear deformation beam theory (ESDBT). The hydro-dynamic pressure of water due to earthquake load is obtained by Helmholtz equation assuming various boundary conditions of reservoir and wave reflection coefficient. In addition, the forces of foundation and sediment are assumed by vertical and axial springs. On the basis of Hamilton's principle, the final motion equations are derived and solved by Harmonic differential quadrature (HDQ), Integral Quadrature (IQ) and with Newmark methods for calculating the dynamic deflection of the concrete dam. The Drucker–Prager criterion is applied for damage analysis of the concrete dam. The influences of various important parameters such as wave reflection coefficient, sediment type and its depth as well as foundation are investigated on the dynamic deflection, hydrodynamic pressure as a function of water depth, time history of hydrodynamic pressure and Drucker–Prager damage index of the concrete dam. The results indicated that with enhancing the sediment depth, the dynamic deflection and hydrodynamic pressure are decreased. In addition, the hydrodynamic pressure increases the dynamic deflection of the concrete dam about 35% with respect to hydrostatics pressure. With increasing the wave reflection constant, the absolute of Drucker–Prager damage index and the safety of concrete dam are reduced at the crest, middle and heel of the concrete dam.
Closed-form solutions for forced vibrations of a cracked double-beam system interconnected by a viscoelastic layer resting on Winkler–Pasternak elastic foundation
2021, Thin-Walled Structures
Citation Excerpt :
They found that the natural frequencies of nanoplates are significantly decreased as the nonlocal parameter rises, but such effect can be weakened by increasing foundation stiffness. In a very recent research by Jiang et al. [58], taking advantage of the high-order shear deformation theory, the static stability of clamped–free columns resting on elastic foundation subjected to a nonconservative subtangential follower force was studied. Green’s functions method is a powerful technique to solve forced vibrations of linear systems, which have been evidenced by plenty of researches [8,9,59–64].
A double-beam system, which consists of two parallel beams connected by a viscoelastic layer, has found broad application in practical engineering such as continuous dynamic absorbers and floating-slab tracks. During the structural service period, the crack is one of the most common defects, which poses a great threat to its normal operation and safety. As a first endeavor, this paper strives to obtain the closed-form solutions for steady-state forced vibrations of a cracked double-beam system resting on the Winkler–Pasternak elastic foundation subjected to harmonic loads. The mechanical properties of cracked cross-sections are characterized by the local stiffness model. Due to the existence of cracks, the cracked double-beam system is artificially divided into several intact segments, where fundamental dynamic responses of each segment are achieved by the Green’s functions method. Subsequently, the transfer matrix method is employed to obtain the steady-state responses of the whole cracked system via the compatibility conditions of cracked cross-sections and boundary conditions. Numerical calculations are performed to check the validity of the present solutions and to discuss the influences of some important parameters, such as crack geometries and connecting layer stiffness, on dynamic behaviors of cracked double-beam systems. Significant effects of the crack depth and location are revealed on the natural frequency and dynamic responses of the system. It is highlighted that based on the bending moment diagram of the mode shape of the intact double-beam system, the effect of the crack location on dynamic behaviors of its cracked system can be effectively predicted.
Free vibration of radially graded hollow cylinders subject to axial force via a higher-order shear deformation beam theory
2021, Composite Structures
Citation Excerpt :
Huang and Li gave the vibration and stability analyses of FGM circular cylindrical beams in which the influence of shear deformation and warping of the cross-section are fully considered [29,30]. Jiang et al. proposed two new warping shapes for circular cylindrical columns and treated divergence stability of a column resting on elastic foundation [31]. Recently, Ma et al. proposed the third-order shear deformation beam theory to analyze dynamic behaviors of uniform and radially graded hollow tube-beams [32,33].
The dynamic behavior of radially functionally graded tubes is studied when subjected to axial tensile and compressive forces. Differing from the Euler–Bernoulli/Timoshenko beam theories, a higher-order shear beam deformation model that does neither require a shear correction factor nor need a planar cross-section assumption after deformation. The cross-section’s warping is constructed to meet the shear-free condition on tube’s inner and outer surfaces. A governing partial differential equation is derived. Exact characteristic equations for determining the natural frequencies are given. For typical end supports including hinged-hinged, clamped–clamped, clamped-free, and clamped-hinged ends, the natural frequencies are calculated. By letting the frequency vanish, the critical buckling loads under compression can be exactly determined by solving the characteristic equation. A comparison of the present buckling loads and the natural frequencies with the classical ones and with finite element results is made and verifies the efficiency of the proposed model. The frequency-load interaction is plotted for typical boundary conditions. Axial tensile force causes the natural frequencies to increase and axial compressive force decreases the natural frequencies. The effects of the radial gradient, tube’s thickness and length, and the cross-sectional warping shape on the buckling loads and the natural frequencies are elucidated.
Effect of warping shape on buckling of circular and rectangular columns under axial compression
2021, Applied Mathematical Modelling
Citation Excerpt :
Recently, Ma et al. further put forth the third-order or higher-order shear deformation theory for hollow cylindrical beams and columns with annular cross-section [46,47]. Jiang et al. analyzed the influence of warping shapes on the stability of cantilever on elastic foundation via various shear deformation beam theories for rectangular and circular columns [48]. It is worth pointing out that if hollow cylindrical tubes are understood as circular cylindrical shells, a more adequate description on buckling of circular cylindrical shells and conical shells can be made based on more complicated and accurate shell model (see e.g. [49–52]).
The structural stability of a column with rectangular and circular cross-section under axial compression is studied based on various higher-order shear deformation beam theories. This paper has two-fold objectives. One is to introduce a transition parameter to describe the direction of axial load from Engesser’s hypothesis to Haringx’s one, then a unified method is presented to determine the critical load. The other is to introduce new cross-section warping shapes of rectangular and circular columns, then the buckling loads are exactly calculated and compared for various warping shapes. A governing equation for buckling of a column under axial compression is first derived. The buckling loads of a prismatic column with typical ends such as clamped-clamped, pinned-pinned, and clamped-free columns are determined and an explicit expression is obtained in terms of the Euler buckling load. The effects of the warping shape of the cross-section on the buckling loads are analyzed. The Haringx buckling load is greater than the Engesser buckling load that gives a conservative estimate of the critical load. A comparison of these buckling loads with Euler loads is made. The obtained results indicate that Euler buckling loads are both significantly overestimated for short columns or those with weak shear rigidity. The buckling loads are nearly not affected for very slender columns or those with high shear rigidity. The Euler loads are recovered from the present ones for columns in the case of shear locking. The buckling loads are also dependent on the warping shape of the cross-section.
Size-dependent vibrational behavior of embedded spinning tubes under gravitational load in hygro-thermo-magnetic fields
2022, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
Divergence and flutter instability of bending–torsional columns under different boundary conditions
2021, Journal of the Brazilian Society of Mechanical Sciences and Engineering

View all citing articles on Scopus

View full text

Stability of cantilever on elastic foundation under a subtangential follower force via shear deformation beam theories

Highlights

Abstract

Introduction

Section snippets

Governing equation

Buckling load

Numerical results and discussion

Conclusions

Author statement

Declaration of competing interest

Acknowledgement

J. Sound Vib.

Int. J. Non Lin. Mech.

J. Sound Vib.

Comput. Struct.

Thin-Walled Struct.

Int. J. Non Lin. Mech.

Compos. Struct.

Int. J. Mech. Sci.

Appl. Math. Model.

Mater. Des.

Compos. Struct.

Compos. Part B

Compos. Part B

Compos. Struct.

Int. J. Mech. Sci.

Compos. Struct.

Thin-Walled Struct.

Appl. Math. Model.

Int. J. Mech. Sci.

Compos. Struct.

Compos. Part B

Thin-Walled Struct.

Thin-Walled Struct.

Thin-Walled Struct.

Thin-Walled Struct.