Dynamic Critical Load Research Articles

Dynamic response and instability of frame structures

This paper presents a geometrically non-linear dynamic instability analysis for both two- and three-dimensional frames, which may be subjected to finite rotations. The finite element displacement method based on the beam–column approach is employed to derive the non-linear equations governing the behaviour of plane and spatial frames. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. The governing dynamic equilibrium equations are obtained from the static equations by adding the inertia and damping terms. The implicit Newmark time integration with the Newton–Raphson (NR) iteration method is employed. Dynamic critical loads are defined by the Budiansky–Roth criterion. Several numerical examples are illustrated to demonstrate the effectiveness of the present method.

Locality vs. criticality

Current memory hierarchies exploit locality of references to reduce load latency and thereby improve processor performance. Locality based schemes aim at reducing the number of cache misses and tend to ignore the nature of misses. This leads to a potential mis-match between load latency requirements and latencies realized using a traditional memory system. To bridge this gap, we partition loads as critical and non-critical. A load that needs to complete early to prevent processor stalls is classified as critical, while a load that can tolerate a long latency is considered non-critical. In this paper, we investigate if it is worth violating locality to exploit information on criticality to improve processor performance. We present a dynamic critical load classification scheme and show that 40% performance improvements are possible on average, if all critical loads are guaranteed to hit in the Ll cache. We then compare the two properties, locality and criticality, in the context of several cache organization and prefetching schemes. We find that the working set of critical loads is large, and hence practical cache organization schemes based on criticality are unable to reduce the critical load miss ratios enough to produce performance gains. Although criticality-based prefetching can help for some resource constrained programs, its benefit over locality-based prefetching is small and may not be worth the added complexity.

VARIABILITY IN THREE FINNISH INTEGRATED ACIDIFICATION MODELS

Integrated assessment models have been used to support ofnegotiations for further emission reductions of acidifyingcompounds in Europe. More attention is being paid to theuncertainties in integrated models. Data from three Finnishintegrated acidification models were compiled to estimate thevariation and relative importance of different modules. Themodels included site-specific and regional dynamic simulationsand steady-state critical load calculations for forest soilsand lakes. The main emphasis was on the variability ofemissions and the uncertainties in ecosystem effects. Althoughmaximum technically feasible emission reduction measures cantheoretically result in very low deposition, the variabilitybetween realistic scenarios is rather restricted. Thevariability of deposition loading is largely determined byreductions in nearby emission sources. The dynamicsimulations, which are often based on detailed input data,seem to retain larger variability than steady-state criticalload approaches. This study suggests that the uncertainties ineffects seem to be larger generally than other modules ofintegrated acidification models. The results indicate the needfor further work on uncertainty analysis for integrated modelsand the availability of useful model systems for furtherconfirmation of ecosystem effects.

Numerical analysis of dynamic buckling of rectangular plates subjected to intermediate-velocity impact

This paper numerically investigates the dynamic buckling of thin imperfect rectangular plates subjected to intermediate-velocity impact loads. From numerical results obtained, a dynamic buckling and a dynamic yielding critical condition are defined, and the corresponding critical dynamic loads are estimated. Numerical model employed in the present study is validated by experimental data reported earlier. Results from parametric study indicate that initial imperfection and load duration have significant influence on the dynamic buckling of the plates. The smaller the initial imperfection and the load duration, the higher the dynamic buckling critical loads of the plates. Moreover, different hardening ratios of plate material also affect the elastic–plastic dynamic buckling properties of the plates. If the plate buckles plastically, the dynamic buckling load increases as the hardening ratio of plate material increases. Unlike thin plates under high-velocity impact that buckling always occur after load application, plates under intermediate-velocity impact analyzed in the present study all buckle during the loading phase.

Initiation and growth of delamination in glass/epoxy composites subjected to static and dynamic loading by acoustic emission monitoring

Delamination is one of the most common failure modes of composite materials. It may result from imperfections in the production process or may be caused by service life conditions, such as impact by foreign objects. The presence of delamination in the composite material may reduce the overall stiffness. The geometrical parameters, material proprieties, and loading conditions are important factors which affect the initiation, development and growth of delamination (Chen HP, David L. Comp. Sci. and Technol. 1993; 46:325–33). Experimental results obtained from monotonic tests and instrumented drop tests have shown good agreement between monotonic and dynamic critical load and crack-growth data. Previous studies made on different composite materials (Meraghni F, Benmedakhane S, Benzeggagh ML. In: Poursartip A, Street K, editors. Proc. ICCM 10, vol. I Vancouver, Canada: Woodhead Publishing, 1995:359–66; Barre S, Benzeggagh ML. Comp. Sci. and Technol. 1994; 54:369–76; Benzeggagh ML, Benmedakhene S. Comp. Sci. and Technol. 1995; 55:1–11) have allowed us to set up a schematic model of acoustic emission. In this model the different levels of amplitude signals emitted by materials under different types of loading are assigned to different damage mechanisms. This experimental methodology has been applied to this work and has allowed continuous monitoring of damage growth through delamination tests. This type of analysis has also allowed us to highlight the complementary aspect of acoustic-emission amplitude distributions with microscopic observations.

Dynamic Buckling of Imperfect Cylindrical Shells Under Axial Compression and Bending Moment

The dynamic stability of circular metallic and laminated cylindrical shells is investigated. The cylinders are geometrically Imperfect and subjected to axial compression or pure bending moment. These loads are suddenly applied with constant magnitude and finite or infinite duration. The finite element method is employed to generate dynamic responses and the equations of motion approach to determine dynamic critical loads. The effects of load duration and imperfection amplitude on critical loads are discussed. It is found that the dynamic critical loads decrease with increasing load duration and converge to those for the load case of infinite duration. The convergence rate is related to the fundamental frequency of the cylinder. In addition, both the static and dynamic critical loads decrease with increasing imperfection amplitude.

Dynamic instability criteria for structures subjected to sudden step loads

Dynamic instability criteria for structures subjected to sudden step loads are presented. The emphasis is on developing effective methods to predict the critical dynamic loads for multi-degree-of-freedom finite element discrete structural systems. The present critical conditions are derived from the stability definitions given originally by Budiansky and Ruth about the stability of responses. Compared with other dynamic instability criteria, the present criteria can predict correct critical dynamic loads for both single- and multi-degree-of-freedom systems without excessively increasing the numerical effort. In addition, the method developed in the present studies can easily be implemented in general finite element dynamic analysis programs. Numerical examples are shown to illustrate the efficiency of the present method.

Nonlinear analysis of imperfect metallic and laminated cylinders under bending loads

The nonlinear buckling of imperfect metallic and laminated cylindrical shells under axial compression, bending, and a combination of compression and bending is investigated. The effects of length-to-rad ius ratio, radius-tothickness ratio, imperfection shape and amplitude, and boundary conditions on critical loads are considered. The finite element method is employed and the ANSYS computer code is selected to generate results. The effect of transverse shear deformation is included in the analysis. To acquire confidence in the solution methodology, results are produced and compared with those available by other solution techniques (analytical verification). Then, numerical results reflecting a wide range of length-to-radius ratios, radius-to-thickness ratios, and various imperfection shapes and amplitudes are presented and discussed. Among all conclusions, the most important one is that imperfection shapes substantially affect the critical loads. For each cylindrical configuration analyzed, certain imperfection shapes can be found, which make the cylinder extremely imperfection sensitive. HE primary consideration for designing and sizing jet engine casings is buckling under dynamic conditions. The loads that cause buckling are generated by the large forces resulting from rotor unbalance as a result of bladeout (loss of one or more blades during operation). These loads induce an end bending moment on the casing that is time dependent both in magnitude and direction. This bending moment is of finite but small duration. The time in which this moment reaches a peak value and vanishes is very short. The engine casing geometry can be approximated by a cylindrical shell. This configuration has numerous applications in real world structures. Airplane fuselages, submarine hulls storage bins, and missiles are but a few examples. Many times during service these systems are subjected to suddenly applied loads, such as blast loads on airplanes and submarine hulls and gust loads on storage bins and missiles. All of these situations can be simulated by cylindrical shells under dynamic bending moment or a combination of compressive and bending loads (Fig. 1). For jet engine casings, the current practice is to estimate the peak moment, consider it as a static bending moment, and compare it with the static critical condition (buckling) of cylindrical shell under bending moment. This procedure has led to designs that are very conservative according to data from experimental verification studies. The weight penalty in most cases is in the 30-50% range. Moreover, since a large number of new casings are made of polymeric matrix composite materials and since the static analytical methods for these configurations are of questionable value, the weight penalty can exceed the 30-50% range. Therefore, it is necessary to predict the static and dynamic critical loads of cylindrical shells under aforementioned loading conditions and to provide information that can be used in designs of cylindrical structures. In this paper, the static stability of metallic and laminated cylinders is studied and the following points are emphasized. 1) Cylindrical shells are subjected to axial compression, bending moment, or a combination of both loads. 2) Cylinders are geometrically imperfect. The effects of imperfect shapes and amplitudes on critical loads are discussed. Also,

Instability of cylindrical panels under combined static and dynamic loads

This paper presents a numerical study of the dynamic snap-through instability of cylindrical panels subjected to the combined loading of uniform pressure and a central concentrated load. All static and dynamic (sudden step load) combinations of the concentrated load and the uniform pressure are considered. The characteristics of the dynamic instability of the panel are described for the two cases of static pressure with dynamic concentrated load and dynamic pressure with static concentrated load. The critical load curves for both cases are presented. To explain the effects of the initial static loads on the critical dynamic loads, the static and dynamic critical loads corresponding to both static loads or both dynamic loads are also calculated. The Budiansky-Roth criterion is employed in all these studies to determine the critical dynamic loads. The numerical results show that the static critical load curve gives an upper-bound whereas the dynamic critical load curve offers a lower-bound. Both the compound critical load curves lie between the static and dynamic critical load curves.

Effects of damping on the parametric instability behaviour of plates under localized edge loading (compression or tension)

The parametric instability behaviour of a plate subjected to localized in-plane compressive or tensile periodic edge loading is studied, considering the effects of damping into the system. Different edge loading cases have been considered. Damping has been introduced in the form of proportional damping. Dynamic instability behaviour under compressive or tensile periodic edge loading shows that the instability regions are influenced by the load band width and its location on the edge. The effects of damping on the instability regions show that there is a critical value of dynamic load factor beyond which the plate becomes dynamically unstable. The critical dynamic load factor increases as damping increases. Damping generally reduces the widths of the instability regions.

Application of Hamilton's law to the stability analysis of a double pendulum under nonconservative loads

Selected results of a nonlinear, numerical analysis of the stability of a double pendulum under static and dynamic nonconservative loads are presented. The basis for the analysis—in contradistinction to numerical methods of the integration of equations of motion commonly used in this type of problem—was Hamilton's law. A numerical analysis of the convergence of the adopted approach was made both from the viewpoint of the applied polynomial approximation and integration step. Results concerning levels of static and dynamic nonconservative critical loads and the quasi-chaotic behaviour of systems with and without imperfections are presented in the form of phase portraits.

Initial and progressive failure analysis of laminated composite structures under dynamic loading

The problem of theoretical prediction of the initial failure and ply-by-ply failure processes in laminated composite structures under dynamic loading is under consideration. A history of deformation can be predicted at any point of a structure using the proposed analytical techniques. The phenometological. second-order tensor-polynomial and maximum stress failure criteria are used to calculate the lower bound of an applied dynamic load. This lower bound corresponds to a start of failure in a structural part. A ply-by-ply failure model is then developed. Using the model, some higher bound for a critical dynamic load impulse value, corresponding to the total exhaustion of a load-bearing capacity by all of the layers in a laminated structure can be predicted. The analysis is applied to thin-walled imperfect laminated graphite/epoxy cylindrical shells, loaded with a short-time impulse of axial compression or external pressure. A general approach to the 3D dynamic deformation analysis of a brick-type mosaic plate and its interaction with a rigid impactor is proposed The approach allows one to model both the initial and damage induced inhomogeneities in a composite structure under dynamic impulsive or impact loading cases.

Delamination growth in composite laminates with adhesive strips subjected to static and impact loading

Graphite/epoxy composites with adhesive strips have been subjected to static loading and dynamic low velocity impact loading to determine the adhesive crack arrest capability. A quasi-static analytical beam model was developed for the calculation of the critical strain energy release rate, G c a measure of the resistance to crack growth, for plain specimens and specimens with adhesive at a 0/90 interface. Experimentally determined static critical load and crack growth data for clamped [90 5/0 5/90 5] graphite/epoxy composite beams loaded at the mid-span was used with the beam model to determine G c. Results showed substantial increases in G c for specimens with adhesive. Experimental results obtained from instrumented drop tests indicated good agreement between static and dynamic critical load and crack growth data, substantiating the validity of the quasi-static assumptions for the beam model for impact problems when contact times were large.

Dynamic stability of an axially loaded beam on an elastic foundation with damping

The subject of this research is the dynamic stability of a simply supported beam which is subjected to a periodic axial load and supported laterally by an elastic foundation with damping. The Winkler, Hetényi and Pasternak foundation models are modified to include the viscous damping of the supporting medium. Stability studies are performed for the 1 2 subharmonic response by determining the conditions on the dynamic load which produce a periodic solution to the resulting differential equation of motion. Approximations to the critical dynamic loads and the regions of instability are employed to discuss the boundedness of the lateral displacements of the beam. Viscous damping is emphasized in the studies, which illustrate the effect of the foundation properties on the stability behavior. The results indicate that an increase in the foundation stiffness or damping increases the critical dynamic load and shifts the region of instability to a higher applied frequency. The mode of vibration associated with the smallest critical dynamic load is shown to vary with both the foundation stiffness and the flexural rigidity. A comparison of the Winkler model results with those obtained from the Pasternak or Hetényi models shows that the inclusion of the foundation's shear or flexural stiffness properties increases the effect of damping.

The dynamic elastic buckling of a circular arch with finite displacements and initial imperfections

The equilibrium equations for elastic circular arches are established using the principle of virtual work. The nonlinear partial differential equations of motion are solved using a finite difference method (Park's method for time difference). The dynamic stability of a hinged and a clamped elastic circular arch with a uniform step load is analysed with finite deformations and initial geometric imperfections. Results show that the buckling mode varies with the value of the arch half angle, θ 0. The boundary condition and initial imperfection amplitude also effect the buckling mode. A nearly perfect arch usually buckling with a “direct” buckling form, while an imperfect arch with an “indirect” buckling form. The effect of θ 0 on the ratio p d/ p s ( p d is the dynamic critical load and p s the static critical load) is shown for different initial imperfections and different boundary conditions.

Some new instability aspects for nonconservative systems under follower loads

The mechanism of instability of nonlinear nonconservative discrete systems under follower loads with or without pre-critical deformation is thoroughly re-examined with the aid of a complete nonlinear dynamic analysis. Considering the stability of motion in the large, in the sense of Lagrange, the critical (divergence or dynamic) load is defined as the minimum load for which an unbounded (divergent) motion is initiated. Regions which have been considered (on the basis of linearized analyses) as of flutter instability are found (using a nonlinear dynamic analysis) dynamically stable. Some new instability phenomena contradict existing findings which have been widely accepted. Moreover, it is established that the divergence buckling loads, obtained by static methods of analysis, coincide with the nonlinear dynamic loads only in the case of no pre-critical deformation. Cases of random-like (or chaotic-like) motions for certain values of the nonconservativeness loading parameter are also revealed for autonomous non-dissipative structural systems.

Stability of forced oscillations of composite shells of revolution under a load with audio frequencies

The stability of forced steady-state oscillations of composite shells of revolution subjected to an external uniform load with an audio frequency is investigated within the framework of the geometrically nonlinear theory of shells. The investigation is conducted using the method of the numeric construction of periodic solutions of systems of nonlinear differential equations describing the dynamic behavior of shells. A class of shells composed of cylindrical and toroidal bands connected to ellipsoidal and spherical bottoms and covers is examined. A number of problems of the stability of forced oscillations of these shells is solved, curves of critical dynamic loads versus frequency of steady-state ocillations are plotted, and the modes of stability loss of the oscillations are defined.

The effects of non-linearities and compressibility on the static and dynamic critical load of non-conservative discrete systems

The non-linear static and dynamic stability of a non-linearly elastic, two degree of freedom system under a partial follower load is thoroughly discussed. Conditions for divergence and flutter instability in the large, in the sense of Lagrange, are established. These conditions may be completely different from those corresponding to linearized analyses. It is found that the static method of analysis is applicable for the entire range of the non- conservativeness loading parameter provided that the material non-linearity is accounted for. The stability or instability of critical states depends on the amount of material non-linearity. The unexpected result that the non-linear static critical load may differ substantially from the corresponding non-linear dynamic critical load is deduced with the aid of a numerical integration of the original non-linear equations of motion. The effects of compressibility as well as of other parameters on such a difference are properly examined. New phenomena which contradict well-known results based on linearized analyses reveal the true instability mechanism of non-conservative systems

Dynamic Buckling of Guyed Stacks, Masts and Columns with Constant Inertia and Algebraic Polynomials Generated by use of Operator T n Attached to Bessel Functions

A new method of analysis of the dynamic critical load of columns, guyed stacks and masts with constant inertia, under the combined action of horizontal loads, axial load and uniformly distributed loads along the vertical axis is presented. The integro-differential formulation of the problem leads to fourth- and fifth- order partial differential equations. In addition to the solution of the equation of motion in expansion series, the new method of solution proposed to solve the fifth-order partial differential equation has led to the differential equation of the plate on elastic foundations in the old Winkler hypothesis. Moreover, the operator T n, attached to the Bessel functions J v, has been used to generate a new set of algebraic polynomials.

Static and Dynamic Buckling of Pressure-Loaded, Ring-Stiffened Cylindrical Shells

The analysis of imperfect, stiffened, thin circular cylindrical shells, subjected to suddenly applied pressure, is presented. Moreover, a solution scheme for the complete (including post-limit point behavior) static analysis is described. The concept of dynamic stability is discussed for suddenly loaded structures. In addition, a fairly complete review of the reported works on the subject is presented. Results are presented for a ring-stiffened geometry. These include critical dynamic loads, critical static loads (limit point loads), as well as minimum post-limit point loads.