Actual Critical Load Research Articles

Effect of Cold Formed Steel Column Arranged by Section C-81, C-75, and TH-35 on Compression Load

To reduce the possibility of local buckling of Cold formed steel structural components under compression, especially in column structures, it is necessary to use an arranged by some sections of Cold formed steel. The arranged some sections of Cold formed steel carries a higher load than a single section. This study aims to determine the effect of cold formed steel column that composed of a combination some sections of C-81, C-75, and TH-35 against compression load. There are two variations of the test specimen for column element, which are specimen type 1 in the form of C75 and TH-35, and specimen type 2 in the form of C-81 and C-75. The testing is conducted by applying pressure force on columns with lengths of 400, 600 and 800 mm. The results of specimen type 1 showed that the average actual critical load of column with lengths 400 mm, 600 mm and 800 mm are 51.47 kN, 75 kN, 78.5 kN. Meanwhile for specimen type 2 are 101.37 kN, 66.47 kN, and 85.60 kN. The column arranged by cold formed steel specimen type 2 has better compressive strength than specimen type 1.

Probabilistic buckling analysis of beam-column elements with geometric imperfections and various boundary conditions

Although the manufacturing process of structural members has been improved over the last decades, inherent geometric imperfections in slender beam-columns cannot be avoided altogether. Such imperfections largely affect the buckling behavior of the beam by modifying (reducing) the actual critical load in comparison with the theoretically perfect beam idealization. Whereas the deterministic characterization of such imperfections is a rather difficult task that should be performed from case to case, a more general approach would require dealing with imperfections within a probabilistic framework, i.e., treating them as random fields. The aim of this paper is to provide a simple but effective method to characterize probabilistically the response of imperfect beam-column elements based on the stochastic description of the imperfections. The proposed method assumes that the imperfections are expressed through series expansion wherein the critical shapes of the corresponding perfect beam represent the series polynomials. Once the continuum deflection field and the random imperfection field are discretized, some simple closed-form relationships between the imperfection parameters and the main response quantities (e.g. beam deflection and critical load) are established. In contrast to other methods, any type of boundary conditions can be incorporated without computational effort. A few boundary conditions are numerically investigated that clarify how the proposed method can be usefully employed to characterize the buckling response of imperfect beam-column elements probabilistically.

On the destabilizing effect of damping on discrete and continuous circulatory systems

The ‘Ziegler paradox’, concerning the destabilizing effect of damping on elastic systems loaded by nonconservative positional forces, is addressed. The paper aims to look at the phenomenon in a new perspective, according to which no surprising discontinuities in the critical load exist between undamped and damped systems. To show that the actual critical load is found as an (infinitesimal) perturbation of one of the infinitely many sub-critically loaded undamped systems. A series expansion of the damped eigenvalues around the distinct purely imaginary undamped eigenvalues is performed, with the load kept as a fixed, although unknown, parameter. The first sensitivity of the eigenvalues, which is found to be real, is zeroed, so that an implicit expression for the critical load multiplier is found, which only depends on the ‘shape’ of damping, being independent of its magnitude. An interpretation is given of the destabilization paradox, by referring to the concept of ‘modal damping’, according to which the sign of the projection of the damping force on the eigenvector of the dual basis, and not on the eigenvector itself, is the true responsible for stability. The whole procedure is explained in detail for discrete systems, and successively extended to continuous systems. Two sample structures are studied for illustrative purposes: the classical reverse double-pendulum under a follower force and a linear visco-elastic beam under a follower force and a dead load.

Postcritical Analysis of PrandtlPlane Joined-Wing Configurations

The postbuckling behavior of joined-wing configurations has not been fully addressed in the past. This topic is extensively discussed in this work. Starting from a baseline configuration, geometrical parameters as well as wings’ load repartition are varied to assess their influence on buckling occurrence. The snap-buckling phenomenon and postcritical pattern are investigated with the adoption of the arc-length technique. The complex load transferring through the joint induces a deformation shape that can no longer carry additional load after a critical point is reached. An abrupt snap to a configuration that is not continuously adjacent to the previous one is then observed. Comparison with the instability state obtained via eigenvalue analysis demonstrates that buckling prediction through linear-buckling analysis is inadequate, and often, the actual critical load is overestimated. This study shows that, for PrandtlPlane joined-wing configurations, increasing the height-to-wingspan ratio is beneficial as far as structural response is concerned. This provides an important design guideline because it is well known from previous studies that high height-to-wingspan ratio decreases the induced drag and fuel consumption. The lift repartition and its effects on buckling show that an improvement of the structural stability of the joined-wing airplane can then be achieved by increasing the amount of aerodynamic load on the upper wing while reducing the lift on the lower wing. The strong bending–torsion coupling typical of highly swept joined wings and its influence on the postcritical response are also investigated and discussed.

Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

The problem of the practical stability of structures is addressed in a modern way by considering the effects of both static and dynamic perturbations. The major historical contributions, due to Euler, Koiter and Thompson, are reviewed and illustrated by an archetypal model permitting to highlight the main mechanical and dynamical points. It is found that a global approach is necessary for a reliable safety estimation, especially in the neighborhood of (static) critical loads. Considering that the admissible load threshold has to account for robustness to finite perturbations, the Koiter critical load must be lowered, obtaining the so called Thompson critical load. It is shown how these two thresholds share some properties (e.g., both depend in a sensitive way on imperfections, which must be known for practical calculations), while having a deep different meaning: the former is related to static imperfections, and requires only a local analysis, while the latter is related to dynamical imperfections, and requires a global analysis. It is shown that PcritEuler≥PcritKoiter≥PcritThompson, i.e., that the advancement of knowledge leads to a lower estimation of the actual critical load.

New Spacer Grid to Enhance Mechanical/Structural Performance

A spacer grid (SG), which supports nuclear fuel rods (FRs) laterally and vertically with a friction grip, is one of the most important structural components in a PWR fuel. The forms of the grid straps and spring form are known to be closely related with the SG crush strength and the integrity of FR support, respectively. In this study, a new SG form and its manufacturing technology were suggested for enhancing the integrity of the FR support and the SG crush strength by using a systematic optimization technique without significantly increasing the pressure drop across the reactor core. Finite element (FE) analysis and crush strength tests on the new SG form were carried out to check the performance enhancement compared with commercial SGs. According to the results, it is estimated that the actual critical load enhancement of the SG is approximately up to 30%, and the actual contact area, when an FR is inserted into an SG cell, is more than double for the new SG form. And also, some design variables that affect the SG crush strength were classified and their effects on the crush strength were investigated by an FE analysis and a crush strength test.

Nonclassical Stability Problems: Instability of Slender Tubes under Pressure

Several nonclassical stability problems dealing with simple cantilever columns of practical engineering importance are comprehensively presented. The salient feature of these rather peculiar problems is that column instability with a tubular cross section filled with a liquid or subjected to gas pressure may occur while its cross section remains axially unstressed. Interesting subcases are also discussed where the static stability criterion of existence of two adjacent equilibria fails to predict the actual critical load. This leads to the erroneous conclusion that the undeformed configuration is the only equilibrium position, being stable irrespective of the level of external loading. Hence, the dynamic stability criterion which is of general validity must be employed for establishing the critical load. It has also been clarified that the hydrostatic pressure load, although nonconstant-directional, cannot be identified as nonconservative.

Criteria for Occurrence of Flutter Instability Before Buckling

The occurrence of utter instability through a Hopf bifurcation before static buckling in regions of divergence in nonconservative, nonself-adjoint, dissipative systems is thoroughly discussed using a qualitative analysis. This region where both Ziegler’s and static criterion may fail to predict the actual critical load is de® ned via two values (bounds)of the nonconservativenessloadingparameter ; the upperboundcorresponds to = 0.5 (being invariant with respect to all other parameters), whereas the lower bound corresponds to a double critical (divergence) point beyond which there are no adjacent equilibria. The location of the last point (lying always between = 0 and 0.5) depends on a stiffness parameter. It is also found that the region of nonexistence of adjacent equilibria becomes maximum(minimum)when the double critical point corresponds to = 0.5 ( = 0). The interaction of vanishing damping with various parameters leads to new phenomena related to point and periodic attractors as well as to a new type of dynamic bifurcation.

Criteria for occurrence of flutter instability before buckling in nonconservative dissipative systems

The occurrence of flutter instability through a Hopf bifurcation before static buckling in regions of divergence in nonconservative, nonself-adjoint, dissipative systems is thoroughly discussed using a qualitative analysis. This region where both Ziegler's and static criterion may fail to predict the actual critical load is defined via two values (bounds) of the nonconservativeness loading parameter η; the upper bound corresponds to η = 0.5 (being invariant with respect to all other parameters), whereas the lower bound corresponds to a double critical (divergence) point beyond which there are no adjacent equilibria. The location of the last point (lying always between η = 0 and 0.5) depends on a stiffness parameter. It is also found that the region of nonexistence of adjacent equilibria becomes maximum (minimum) when the double critical point corresponds to η = 0.5 (η = 0). The interaction of vanishing damping with various parameters leads to new phenomena related to point and periodic attractors as well as to a new type of dynamic bifurcation.

Design sensitivity analysis of critical load factor for nonlinear structural systems

A continuum formulation of sizing design sensitivity analysis of the critical load factor is developed for nonlinear structural systems that are subject to conservative static loadings. Both geometric and material nonlinear effects are considered. Sizing design variables, such as thickness and cross-sectional areas of structural components, are treated. The total Lagrangian formulation for an incremental equilibrium equation and one- and two-point linearized eigenvalue problems are utilized to estimate the critical load factor. A distributed parameter approach and the adjoint variable method are used to obtain design sensitivity expressions in continuum mechanics formulation. A numerical method is presented to evaluate design sensitivity expressions using established finite element codes. The proposed method is tested using several examples. The numerical results show that the proposed method for design sensitivity of the critical load factor is accurate. It is found that, as the applied load approaches the critical load, the design sensitivity of the critical load factor does not converge to the design sensitivity of the actual critical load, even though the estimated critical load converges to the actual critical load and the design sensitivity of the critical load factor is correct.

Asymptotic Stability Analysis of Viscoelastic Systems

Based on the equivalent energy stability criteria formulated elsewhere by the writer and by others, three approximate techniques are presented herein for the asymptotic stability analysis of the equilibrium of viscoelastic or damped elastic systems subjected to nonconservative and gyroscopic forces. Two of them give sufficient stability loads while the third, which is analogous to the familiar Rayleigh-Ritz method, provides approximate critical loads. It is shown that the Ritz method yields reasonably good results if dissipative (damping) forces are finite. However, further effort must be made to improve the lower bound of the approximate critical loads, which here are shown to be, in general, smaller, or at the most equal to the minimum actual critical load.

Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes

The author calculates the buckling load of a built-in free rod subjected to tangential end-thrust. As this stability problem is nonconservative, it cannot be solved by the usual static methods. Investigation of the free vibrations underP yields the buckling loadP k=20,05 α/l 2 (α: flexual rigidity,l: length). (Damping forces are not taken into account; they might diminish the actual critical load).