Imperfect Beams Research Articles

Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams

Abstract The non-linear non-planar dynamic responses of a near-square cantilevered (a special case of inextensional beams) geometrically imperfect (i.e., slightly curved) and perfect beam under harmonic primary resonant base excitation with a one-to-one internal resonance is investigated. The sensitivity of limit-cycles predicted by the perfect beam model to small geometric imperfections is analyzed and the importance of taking into account the small geometric imperfections is investigated. This was carried out by assuming two different geometric imperfection shapes, fixing the corresponding frequency detuning parameters and continuation of sample limit-cycles versus the imperfection parameter. The branches of periodic responses for perfect and imperfect (i.e. small geometric imperfection) beams are determined and compared. It is shown that branches of periodic solutions associated with similar limit-cycles of the imperfect and perfect beams have a frequency shift with respect to each other and may undergo different bifurcations which results in different dynamic responses. Furthermore, the imperfect beam model predicts more dynamic attractors than the perfect one. Also, it is shown that depending on the magnitude of geometric imperfection, some of the attractors predicted by the perfect beam model may collapse. Ignoring the small geometric imperfections and applying the perfect beam model is shown to contribute to erroneous results.

Non-linear stability analysis of imperfect thin-walled composite beams

Abstract The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton–Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).

A static and dynamic analysis of the postbuckling of geometrically imperfect composite beams

Abstract The static and dynamic response of geometrically imperfect composite beams is presented. The proposed model accounts for the midplane stretching of the beam that follows the classical beam theory and its curvature is assumed to be shallow. The assumption of a small strain, moderate deformation is used. The governing equations are nonlinear integral partial-differential equations. A unified approach is used to handle the postbuckling response of composite beams with and without imperfection. As a result, an analytical solution for the beam’s static response in terms of the applied axial load, imperfection, and lay up is obtained. Moreover, the free vibrations in the postbuckling domain are investigated. Having the beam’s imperfection as a control parameter, the beam’s static and dynamic response is investigated and presented. Results show that the imperfection has a significant effect on the static and dynamic response of composite beams.

Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—Bifurcation analysis under base excitations

The non-linear non-planar steady-state responses of a near-square cantilevered beam (a special case of inextensional beams) with general imperfection under harmonic base excitation is investigated. By applying the combination of the multiple scales method and the Galerkin procedure to two non-linear integro-differential equations derived in part I, two modulation non-linear coupled first-order differential equations are obtained for the case of a primary resonance with a one-to-one internal resonance. The modulation equations contain linear imperfection-induced terms in addition to cubic geometric and inertial terms. Variations of the steady-state response amplitude curves with different parameters are presented. Bifurcation analyses of fixed points show that the influence of geometric imperfection on the steady-state responses can be significant to a great extent although the imperfection is small. The phenomenon of frequency island generation is also observed.

Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: Equations of motion and experimental validation

The non-linear equations and boundary conditions of non-planar (two bending and one torsional) vibrations of inextensional isotropic geometrically imperfect beams (i.e. slightly curved and twisted beams) are derived using the extended Hamilton's principle. The assumptions of Euler–Bernoulli beam theory are used. The order of magnitude of the natural geometric imperfection is assumed to be the same as the first order of vibrations amplitude. Although the natural imperfection is small, in contrast to the case of straight beams (i.e. geometrically perfect beams), this study shows that the vibration equations are linearly coupled and have linear and quadratic terms in addition to cubic terms. Also, in the case of near-square or near-circular beams, coupling terms between lateral and torsional vibrations exist. Furthermore, a problem of parametric excitation in the case of perfect beams changes to a problem of mixed parametric and external excitation in the case of imperfect beams. The validity of the model is investigated using the existing experimental data.

Design rules for lateral torsional buckling of channel sections subject to web loading

AbstractChannel sections are widely used in practice as beams. However, design rules for eccentrically loaded (not through shear centre) beams with channel cross‐sections are not available in Eurocode 3. In this paper five proposed design rules are summarised, explained and their validity is checked by Finite Element analyses. The design rules yield ultimate loads that are compared to ultimate loads from geometrical and material nonlinear analyses of imperfect (GMNIA) beams with channel cross‐sections. A parameter study is performed by varying the dimensions of the cross‐section, the span length to section height ratio of the beams, the type of loading and the point of load application but is limited to load application through the web of the channels. Based on one of the proposed design rules, this study has led to a new design rule which conforms to Eurocode 3.

Non-linear interactions in imperfect beams at veering

Non-linear interactions in a hinged–hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force– and frequency–response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.

Shell Theory Accuracy with Regard to Initial Postbuckling Behavior of Cylindrical Shell

are the normal strains in the prebuckling state. The buckling anal- ysis for the nondebonded imperfect beams is then similar to that presented earlier for the free-vibration problem. In the case of a generally debonded beam, an exact buckling solution cannot be obtained and approximate treatments should be employed. How- ever, if the beam is with through-length delamination at the kth interface, for example, an exact solution can be obtained, just as it can for the nondebonded imperfect beam, but with K (k)

Moderately Large Vibrations of Imperfect Elastic-Plastic Composite Beams with Thick Layers

Moderately Large Vibrations of Imperfect Elastic-Plastic Composite Beams with Thick Layers

Nonlinear High-Order Response of Imperfect Sandwich Beams with Delaminated Faces

Delaminations within face sheets are often observed when a sandwich structure is exposed to impact loads. The buckling and postbuckling behavior of sandwich beams with delaminated faces is investigated. The governing nonlinear equations, boundary conditions, and continuity conditions are formulated through variational principles. The beam construction consists of two stiff composite laminated face sheets and a soft core of a foam or a low-strength honeycomb type. A higher-order theory is used for the core that accounts for the nonlinear distortion of the plane of the section of the core and the compressibility in the thickness direction. The delamination considered is an interface crack, in which the substrate includes the transversely flexible core. As an illustration, the results of numerical computation for a simply supported beam are presented to show the effects of a delamination inside the face sheets on the behavior of sandwich structures under compressive loading.

Nonlinear high-order response of imperfect sandwich beams with delaminated faces

Nonlinear high-order response of imperfect sandwich beams with delaminated faces

STABILITY DESIGN OF LONG PRECAST CONCRETE BEAMS.

& This paper presents the equations needed for design engineers to check the stability of precast concrete beams when simply supported, during transportation, when being lifted and while sitting on flexible bearings. It shows how the critical loads can be determined and how estimates can be made of the eAect of imperfections both in the beam itself and in the degree of levelness of its supports. It shows how the stresses induced by second-order eAects in imperfect beams can be determined. Various examples are given of the method in use.

On the Stability of Schapery Beams

The postbuckling behavior of imperfect beams made of a nonlinear viscoelastic material is investigated. The material is modeled according to the Schapery strain based representation of nonlinear viscoelasticity. Solution within a nonlinear geometrical theory are developed to calculate the growth of the deflection in time

Asymptotic behaviour of imperfect viscoelastic beams

In this paper the problem of the lateral stability of imperfect viscoelastic beams is analysed. The problem is examined by means of the quasi-static approach and leads to a system of integro-differential equations which can be resolved by a series expansions and the Laplace transforms. The viscous critical load is defined according to the asymptotic behaviour of the beam and can be evaluated by introducing the assumption of weak fading memory, solid material and thermodynamic compatibility. The critical load does not depend either on the type or the entity of the imperfections. Furthermore, some characteristic aspects of the problem, like the faster progress of the torque moment with respect to the progress of the lateral bending moment, are underlined. For a three-element model it is possible to reach a closed-form solution. In the case of the Poisson ratio constant in time and three-element model, an interesting analogy with the asymptotic behaviour of the imperfect column is observed.

Buckling of randomly imperfect beams

The qualitative behavior of buckled states of two different models of elastic beams is studied. It is assumed that random imperfections affect the governing nonlinear equations. It is shown that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation whose coeffficients are Gaussian random variables. The corresponding asymptotic expansion for the displacement is to lowest order a Gaussian stochastic process.

Imperfection-sensitivity of elastically supported beams and its relation to the double-cusp instability model

This paper deals with the two-mode initial postbuckling analysis of geometrically imperfect beams on elastic foundation under compression. By adjusting the beam length to the characteristic length ratio, it is found that the resulting two-mode stability problem may be classified as the double-cusp model in catastrophe theory. The appropriate postbuckling coefficients are computed with the use of Koiter’s theory of elastic stability. The equilibrium paths and the critical sets are plotted and analysed. Eight independent control parameters are suggested. The paper is the first in the open literature to apply the double-cusp model to a two-mode stability problem which displays imperfection-sensitivity.

An approach to the design of imperfect slender beams

A method is described for deriving design curves which provide a smooth transition through the various modes of failure of steel beams and are therefore valid over the whole range of beam slendernesses. Comparison with available test results and the theory for perfect beams demonstrates the validity of the approach which leads to more manageable equations than previous methods.

Deformations of Geometrically Imperfect Beams

The effects of a number of representative types of imperfection on the elastic deformations of simply supported I-beams with either equal end moments or central concentrated load are investigated, and it is shown that each of these imperfections can be approximated by an equivalent sinusoidal initial crookedness or twist. The results of this investigation are then used in an examination of a linear plot method of extrapolating for the elastic critical load devised by Massey, and it is concluded that the plots obtained by this method may be nonlinear. An improved method of extrapolation is suggested and the critical load estimates made by this method from a number of experimental measurements are compared with the Massey plot estimates and the theoretical predictions.