On dynamic buckling of elastic–plastic beams

doi:10.1016/S0020-7462(99)00054-2

International Journal of Non-Linear Mechanics

Volume 35, Issue 4, July 2000, Pages 721-734

https://doi.org/10.1016/S0020-7462(99)00054-2 Get rights and content

Abstract

A theoretical study of dynamic buckling of elastic–plastic beams under compressive axial forces is presented. It is assumed that the loading process is so slow that axial inertia effects may be neglected. Four types of axial loading are considered. It is supposed that the beam's material has linear strain hardening. Equations of motion are derived from Hamilton's principle. A numerical method of integrating these equations is presented. The main aim of the paper is to shed light on the problem how the deflection shapes develop in the post-critical stage. For this purpose six examples are presented.

Introduction

The dynamic buckling of elastic structures has drawn the researchers attention over the past 50 years. At this point we refer to two classic papers on this topic. In 1949, Lavrentiev and Ishlinsky [1] showed that in the case of the dynamic buckling of elastic beams higher model forms may be realized (Fig. 1). In 1951, Hoff published a paper [2] in which the buckling of an elastic beam under constant axial velocity was discussed.

In our days dynamic elastic buckling has been the subject of extensive research (for details see [3], [4], [5], [6], [7]). From more recent works let us cite the paper by Ari-Gur and Eliashakoff [8] in which the axial load was taken in the form of a half-sine pulse. In comparison, up to now plastic structures have been moderately studied. For analyzing the behavior of a structure in the post-critical stage some authors have recommended to neglect the elastic-unloading and thus to proceed out from the rigid–plastic model (see [5], [6]). Such an approach has been applied also by Lepik [9]. Elastic unloading is taken into account in paper [10] by Lee where the upper end of the beam moves downward with a prescribed velocity. The equations of motion are deduced from the quasibifurcation theory. The results are valid if the strains are small enough.

The stress–strain diagram with elastic unloading and secondary plastic loading has been used also in paper [11] by Sugiura et al. in which the motion of the beam struck axially with a mass is discussed.

Now a days, the following three main approaches to analyzing dynamic buckling problems may be stated:

In the present paper the last approach is put into use. In our opinion the advantage of this method lies in the fact that the modal structure of the solution is maintained. This circumstance enables us to render the mechanics involved more understandable. In many papers it has been assumed that the axial force T is constant along the beam. It follows from the equations of motion (see Eq. (9a)) that in this case the axial inertia effects may be neglected. Such an approach, called the case of slow excitation (see Ref. [13]), is used also in this paper: it significantly simplifies the solutions and makes it easier to interpret the obtained numerical results. We proceed from the Bernoulli–Euler beam model, since calculations with more exact models (e.g., Timoshenko-type models) have shown that the effects of rotatory inertia and transverse shear are small [8], [11].

In the present paper, the equations of motion are derived from Hamilton's principle. An algorithm of solution, which is applicable to an arbitrary number of DoF, is presented. Numerical examples are given. We hope that these results help us clarify the question how the deflection shapes develop in the post-critical stage (this problem is not only of theoretical value, the obtained results may also be useful in analyzing experimental data).

The present work and the previous papers of the author have certain similar points. In Ref. [14], elastic–plastic vibrations of a buckled beam under harmonic excitation are discussed: the beam is compressed by an axial load, then it is fixed in the compressed position and a transverse load is suddenly applied. In Ref. [15] a method for calculating the buckling threshold and spectrum of bifurcation is proposed.

Section snippets

Variational equation

Let us consider a beam with rectangular cross-section; $B, h$ and L are the width, thickness and height of the beam, respectively. The $x^{∗}$ -axis is directed along the beam, its origin is located in one end-section of the beam. At $x^{∗} =0$ an axial compressive load $p^{∗} (t)$ is applied (Fig. 1a).

The following notations are introduced: $t^{∗}$ — time, ϱ — density, $u^{∗}$ — axial displacement, $w^{∗}$ — deflection, $w_{0}^{∗}$ — initial deflection, E — Young's modulus, σ_s — yield stress, $T^{∗}$ — axial force, $M^{∗}$ — bending moment.

Constitutive equations

We confine ourselves to the case where the beam material has linear strain-hardening. Elastic unloading and secondary plastic loading are taken into account according to the ideal Bauschinger effect. The stress–strain diagram is presented in Fig. 2, where the parameter λ=dσ/de characterizes strain-hardening (for an elastic material λ=0). The diagram in Fig. 2 consists of the following line segments.

(i)	Segment 1 corresponds to elastic loading, here $σ=e/e_{s} .$
(ii)	Segment 2 corresponds to

Types of axial loading

For the sake of concreteness we consider the following types of loading:

Type 1:	Here a constant axial load of infinite duration acts (Fig. 3a).
Type 2:	Constant load of finite duration (Fig. 3b).
Type 3:	Exponentially decreasing load $p(t)=p_{0} e^{−αt}$ (Fig. 3c).
Type 4:	The end section of the beam $x=0$ moves with a constant axial speed $v_{0}$ .

It is assumed that all external loads are compressive and p₀ exceeds the critical load.

Critical loads

For applying the method of solution, proposed in Section 6, we do not need the critical loads. On the other hand, the knowledge of these loads is of great help to us in interpreting the achieved numerical results.

In the bifurcation theory of elastic–plastic structures a complicated problem is to determine the size and form of the unloading zone. There have been several discussions about this problem, nevertheless, we do not have yet a mathematically correct solution.

Therefore, we make use of

Method of solution

We derive the equations of motion from variational equation (10). By assuming that the axial stress σ is uniformly distributed over the end-sections x=0 and 1, the boundary conditions M(0,t)=M(1,t)=0 can be replaced by w″(0,t)=w″(1,t)=0. All boundary conditions are satisfied if we seek the solution in the form $u(x,t)=u_{0} (t)(1−x),$ $w(x,t)= ∑ k=1 s f_{k} (t) sin kπx.$

Let us develop the initial deflection w₀(x) into the Fourier series $w_{0} (x)= ∑ k=1 s a_{k} (t) sin kπx.$

The variations $δw, δw′, δw″$ can be found from Eq. (17b) by

The elastic case

The proposed solution is greatly simplified if the stresses do not exceed the yield stress. Now σ=e/e_s and Eq. (19) can be rewritten in the form $f ̈_{k} =π^{2} k^{2} [(p−p_{cr}^{(k)})f_{k} +pa_{k}],$ where $p_{cr}^{(k)} = π^{2} k^{2} 12e_{s} h L^{2} .$

Let us consider the case p=p₀=const (loading type 1). If we introduce the symbol $Ω_{k} =πk (p−p_{cr}^{(k)})$ then the solution of Eq. (25), satisfying the initial conditions $f_{k} (0)= f ̇_{k} (0)=0,$ is $f_{k} = pa_{k} p−p_{cr}^{(k)} (ch Ω_{k} t−1) if p>p_{cr}^{(k)}, pa_{k} p_{cr}^{(k)} −p (1− cos Ω_{k} t) if p<p_{cr}^{(k)} .$

This result has been thoroughly analyzed in many papers (see

Examples

In order to get an insight into the behavior of the beams in the post-critical stage some examples are presented. In all these cases for material parameters the values $λ=0.95, e_{s} =0.004$ were chosen. The modal imperfections were taken in the form a_k=a₀/k², where a₀=0.001. Integrals , were evaluated by ten-point Gauss integration. In the case of integrals (20) the Simpson formula with a stepsize Δx=0.01 was applied. The time increment was Δt=0.002. Modal amplitudes versus time are shown in Fig. 6;

Discussion

In order to simplify the analysis of dynamic plastic buckling some authors have recommended to use the rigid–plastic material conception (see e.g. [5], [6]). In this case, elastic deflections are neglected and we find ourselves all the time on the segment 2 in Fig. 2. , , hold if we evaluate the critical load form Eq. (14). For the purpose of estimating the applicability of this simple solution, some calculations were carried out. One typical result is shown in Fig. 6, Fig. 9. The beam and

Concluding remarks

In the present paper, method for analyzing dynamic buckling of elastic–plastic beams is proposed. Accordingly, the deflections are expanded into a Fourier series by the eigenmodes. In this way, the modal structure of the solution is maintained. The last circumstance makes it easier to interpret the results. The recommended method is applicable to systems with an arbitrary number of DoF. To make things simpler in this paper only the case of slow excitation is discussed.

In order to demonstrate

Acknowledgements

Support from the Estonian Science Foundation under Grant ETF-3380 is gratefully acknowledged.

References (17)

J. Ari-Gur et al.
Dynamic instability of a transversely isotropic column subjected to a compression pulse
Comput. & Struct.
(1997)
L.H.N. Lee
Dynamic buckling of an inelastic column
Int. J. Solids and Struct.
(1981)
D. Karagiozova et al.
Dynamic elastic–plastic buckling phenomena in a rod due to axial impact
Int. J. Impact Engng.
(1996)
Ü. Lepik
Elastic–plastic vibrations of a buckled beam
Int. J. Non-Linear Mech.
(1995)
Ü. Lepik
A contribution to bifurcation analysis of elastic–plastic beams
Int. J. Impact Engng.
(1998)
M.A. Lavrentiev, A.Y. Ishlinsky, Dynamic forms of loss of the stability of elastic systems, Dokl. USSR 64 (1949)...
N.J. Hoff
The dynamics of the buckling of elastic columns
J. Appl. Mech.
(1951)
V.V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San-Francisco, CA...

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