Elastic Buckling Stress Research Articles

Local and distortional buckling of thin-walled short columns

This paper assesses the applicability of Eurocode 3 (EC3) to the prediction of the compression capacity of short fixed-ended columns with different cross-sections. This compression capacity is determined by combining the effective width of plane elements due to local buckling and the effective stiffener thickness due to distortional buckling. Numerical calculations have been carried out in order to compare alternative methods for determining the minimum elastic distortional buckling stress in compression. The method given in EC3 does not correlate as well as Lau and Hancock's method with the results given by Generalized Beam Theory (GBT). The end boundary conditions have a significant influence on the distortional buckling strength, and thus also on the compression capacity of short columns. Selected experimental results from compression tests on C-, Hat- and rack upright-sections are compared with the predictions given by EC3. The procedure in EC3 was modified by determining the distortional buckling stress using GBT, taking into account the actual column length and the end boundary conditions. This lead to better agreement between the experimental results and the theoretical predictions.

Laterally Braced Cold-Formed Steel Flexural Members with Edge Stiffened Flanges

The moment capacity of a laterally braced cold-formed steel flexural member with edge stiffened flanges (e.g., a channel or zee section) may be affected adversely by local or distortional buckling. New procedures for hand prediction of the buckling stress in the local and distortional mode are presented and verified. Numerical investigations are employed to highlight postbuckling behavior unique to the distortional mode. Compared with the local mode, the distortional mode is shown to have (1) heightened imperfection sensitivity, (2) lower postbuckling capacity, and (3) the ability to control the failure mechanism even in cases when the elastic buckling stress in the local mode is lower than in the distortional mode. Traditional design methods do not explicitly recognize distortional buckling, nor do they account for the observed phenomena in this mode. A new design method that integrates distortional buckling into the unified effective width approach, currently used in most cold-formed steel design specifications, is presented. For each element a local buckling stress and a reduced distortional buckling stress are compared to determine the effective width. Comparison with experimental tests shows that the new approach is more consistent and reliable than existing design methods.

Piezoelectric behavior of wood under combined compression and vibration stresses II: Effect of the deformation of cross-sectional wall of tracheids on changes in piezoelectric voltage in linear-elastic region

This study investigated and clarified the relation between the piezoelectric voltage and microscopic fracture of hinoki (Chamaecyparis obtura Endl.), in particular the deformation of the cross-sectional wall of the tracheid in linear-elastic regions under combined compression and vibration stresses. The piezoelectric voltage-deformation (P-D) curve consisted of a linear region starting from the origin followed by a convex curved region. The linear region of theP-D curve was only about 60% of that of the load-displacement (L-D) curve. By applying combined stresses to a specimen, the cross-sectional walls of the tracheid were deformed mainly at the radial walls. When a tracheid was regarded approximately as a hexagonal prism, the elastic buckling stress of the radial wall was estimated from scanning electron microscope images and our method based on a modification of the Gibson and Ashby method. As a result, it was estimated that the elastic buckling stress was only about 80% of the stress at the proportional limit of theP-D curve. It is found that there are two consecutive regions before the proportional limit of theP-D curve: One is the region up to the spot where the radial cell wall generates the elastic buckling, and the other is the region starting from the end of the aforementioned region up to the proportional limit of theP-D curve.

Design for distortional buckling of flexural members

For thin-walled flexural members composed of high-strength steel and/or slender elements in the section, a mode of buckling at half-wavelengths intermediate between local buckling and flexural—torsional or flexural buckling can occur. The mode is most common for edge-stiffened sections such as C- and Z-purlins, and involves rotation of the flange and lip about the flange—web junction. The mode is commonly called distortional buckling. This paper presents a design method for distortional buckling of flexural members recently drafted for use in the Australian/New Zealand Standard for Cold formed Steel Structures. Methods for computing the elastic buckling stress, and design curves for determining the distortional buckling strength are presented. Comparisons of different methods for computing the elastic distortional buckling stress are made with accurate solutions based on the finite strip method of structural analysis.

Distortional Buckling of Thin-Walled Beams/Panels. I: Theory

The introduction of the standing seam panel for roof systems provided the building industry with an efficient solution to watertightness problems. These panels have two major advantages over the conventional trapezoidal through-fastened panel: an elevated seam and concealed “hold-down” clips. The relatively small depth-to-width ratio of the panel guarantees that overall, the panel is laterally stable under gravity load. The capacity of laterally stable standing-seam panels may, however, be reduced due to lateral instability of the unsupported compression flange. This mode of instability has been referred to in the literature as distortional buckling—a form of instability for unsupported lengths between those for local and lateral buckling. In the present paper, an analytical expression, based on a beam on elastic foundation model, is presented for estimating the elastic distortional buckling stress in sections with laterally unsupported compression flanges. The beam is assumed to comprise the outstanding leg, and the elastic foundation is provided by the rotational restraint of the tension flange and the web. A comparison of the analytical results with the results from elastic finite-strip buckling analyses shows good agreement between the two methods of analysis.

Local Buckling by Complex Finite Strip Method Using Bubble Functions

Bubble functions are finite element shapes that are zero on the boundary of the element, but are nonzero at the other points. They have been hitherto used to augment finite element formulations to obtain rapid convergence. The paper augments bubble functions to the ordinary semi‐analytical complex finite strip treatment in order to calculate the elastic local buckling stress of plate assemblies. The results show that the use of bubble functions improves significantly the convergence of finite strip method in terms of strip subdivision, and leads to much smaller storage required for the structure stiffness and stability matrices. Numerical studies are included, including long plates in compression and shear, a channel section in pure compression and shear, a stiffened plate, and an I‐section with a longitudinal stiffener. These studies illustrate the power of the bubble function based method for studying the elastic stability of plates and plate assemblies.

亜鉛めっき橋梁の溶融亜鉛めっきぜい化割れ発生に関する熱弾塑性解析

The aim of the present study is to clarify the mechanical behavlor of a bridge girder during hot galvanizing. The mechanical behavior is analyzed using finite element method (FEM) analysis in which the constitutive equation is given by Bodner's viscoplastic model. The mechanism of zinc induced cracking is also discussed in detall. A shape and size parameter, Rσmax. is proposed and discussed. Rσmax is the ratio of the maximum compressive thermal stress to the elastic buckling stress of the panel and is related to zinc induced cracking. The parameter Rσmax correlates well with the final tensile strain calculated by FEM analysis. Critical strain for crack initiation, εc, was obtained from bending tests and from FEM analysis for several steels. Each value of Rσmax at crack initiation was estimated from comparison with the final tensile strain and εc. The estimated critical value of Rσmax is nearly the same as the critical value of Rσmaxobtained by calculation of the transition from non-cracking to cracking behavior of bridge girders.

Load Shortening in Plastic Buckling of Cylinders

It is shown that for moderate deflection theory, the behavior of a long pressurized elastic‐perfectly plastic cylinder with radially restrained ends under axial compression depends on only two parameters: the ratio of elastic buckling stress to yield stress, σ¯CL=σCL/σy, and the ratio of nominal circumferential stress to yield stress, p¯=pR/(σyt). This makes it possible to present results covering almost all conceivable values of these two parameters. Equations fitting these results are developed. Their application in the simplified analysis of unanchored tanks under seismic lateral loads is illustrated by an example. This indicates that the lateral load capacity of such tanks can be substantially larger than the lateral load at which plastic buckling (also referred to as elephant‐foot bulging) of the tank wall first occurs.

Tests of Cold‐Formed Channels with Local and Distortional Buckling

Thin‐walled channel sections may undergo a mode of buckling, called distortional, in which a lip‐stiffened flange of the section rotates about the flange‐web junction. If the sections are composed of high‐strength steel, then there may be a significant postbuckling reserve of strength beyond the elastic distortional buckling stress in a similar manner to that which normally occurs for local buckling. The results of compression tests of thin‐walled channel section columns formed by brake‐processing are described. Two different section geometries, a simple lipped channel and a lipped channel with an intermediate stiffener in the web, were tested between fixed‐end boundary conditions. The geometries and yield strength were chosen to ensure that a substantial postbuckling strength reserve occurred in the distortional mode and that local buckling might occur simultaneously at a shorter wavelength. Distortional or mixed local‐distortional buckling stresses obtained from testing are compared with theoretical buc...

Discussion: Ultimate load capacity of square shear plates with circular perforations

The problem of web plates with holes, particularly in the elastic-plastic and elastic ranges of buckling stress, still requires a convincing solution, and the authors have made a useful contribution. It is unfortunate that they impose the basic constraint of a uniform stress along the boundaries. Although this is assumed in the classical analysis to determine the initial buckling stress, it is far from realistic for any shear web in which there are initial imperfections or after the inital buckling stress is exceeded (Marsh 1985). A correct solution can be found only by imposing a shear deformation on an imperfect web and allowing the computer program to determine the distribution of the stresses. For high b/t ratios the authors' analysis would give very conservative results. On the other hand, the idea that elastic buckling stress can be equated to the yield stress to give the transition from yielding to buckling can be seriously in error for slendernesses at and below this value, as the full yield shear stress over the entire plate section is not sustained, particularly if there are significant residual stresses or imperfections. It would have been useful if the author had given the values of the yield stress and elastic modulus used in the analysis. Definitions of and slenderness ratio would also have been of value (Fig. 1).

Elastic buckling analysis of ideal thin-walled structures under combined loading using a finite strip method

The details of an application of the finite strip method to the elastic buckling analysis of thin-walled structures with various boundary conditions and subjected to single or combined loadings of longitudinal compression, transverse compression, bending and shear are presented. The presence of shear loading is accounted for by modifying the displacement functions which are commonly used in cases when shear is absent. A program based on the finite strip method was used to obtain the elastic buckling stress, buckling plot and buckling mode of thin-walled structures and some of these results are presented.

Folding of oceanic lithosphere

Folding of the lithosphere just south of the Bay of Bengal appears as (1) undulations in acoustic basement topography and (2) as linear geoid undulations in the Seasat altimeter data. From the Seasat data we find that the east‐west trending folds have wavelengths ranging from 130 to 250 km and clustering about 190 km. The horizontal gravity disturbances due to the folds range in amplitude from 15 to 50 mGal. Elastic models of oceanic lithosphere have, in the past, been used to demonstrate the implausibility of lithosphere buckling, or folding, in response to compression. These elastic models typically predict that compressive stresses of about 5 GPa are required to buckle oceanic lithosphere with an age comparable to that of the northeastern Indian Ocean (40–70 Ma). These stresses exceed the strength of lithospheric rock. We use an elastic‐plastic model to show that oceanic lithosphere of this age should have a net compressive strength equal to about 12% of the elastic buckling stress. We further demonstrate that loads approaching the net compressive strength cause the lithosphere to fold with a wavelength about 200 km, i.e., the wavelength observed from Seasat. Our results reinforce earlier speculation that this folding may be related to the Himalayan orogeny.

Buckling and Ultimate Strength of Offshore Framed Structures

The development of oil and gas is increasing in deeper and severer waters. Accordingly, it is getting very important to estimate the buckling and ultimate strength of offshore framed structures such as jackup rigs and platforms.The following methods are introduced in this paper from the design point of view to estimate the buckling strength of braces and ultimate strength of offshore framed structures as both accurately and simply as possible. The method to estimate the buckling strength uses the simple column buckling equation with initial deflection in combination with the elastic buckling stress obtained from eigen value analysis of framed structures. Simplified non linear analysis method which consists of iterative procedure of structural model modification/linear analysis/ buckling and plastification criteria check is used to estimate the ultimate strength.Four pieces of 3-dimentional 1/7 scale leg model of actual jackup rig were tested and compared with the above mentioned methods and obtained the following results. (1) The calculated buckling strength is within the error of 8% comparing to experimental results and the method to estimate the buckling strength seems accurate in spite of the simplicity of the buckling equation. (2) Simplified non linear method is found useful for design work by comparing the calculated and experimental results. (3) With using two kinds of brace material, one is high tensile steel and the other is mild, the effect of material on the buckling, the ultimate strength and the mechanism of failure is confirmed both by experiments and calculations.

Orthotropic Plates under Varying Compression

An equation for predicting the elastic buckling stress of orthotropic plates is presented. The plate analyzed is simply supported along both transverse edges and elastically restrained along both longitudinal sides. Also, the plate is subjected to compressive stress that varies linearly in the direction of loadings and shear stress.

External Pressure Tests of Ring-Stiffened Fabricated Steel Cylinders

Summary This paper summarizes the results of 20 hydrostatic external pressure tests on ring-stiffened fabricated steel cylinders. The objectives of these tests were to develop a comprehensive data base for fabricated cylinders and to determine the adequacy of the current API design rules for fixed offshore platforms. platforms. The test specimen parameters were selected to represent (1) typical offshore platform member sizes (D/t of 32 to 128), (2) commonly used platform steels (A36 and A572), and (3) routine platform fabrication procedures. The specimens were designed to fail at different stress levels. Fourteen specimens were designed to fail by local buckling of the shell between rings and six were designed to fail by general instability. Out-of-roundness measurements were taken at the middle of each bay and adjacent to each stiffening ring. The out-of-roundness values exceeded the API tolerance limit of 1% for many of the test specimens. The test results are compared with theoretical methods used to predict the buckling pressure based on measured out-of-roundness values. pressure based on measured out-of-roundness values. In general, poor correlation is found. On the basis of the test results, empirical reduction formulas to account for out-of-roundness effects are proposed. Introduction The buckling criteria for cylinders under external pressure vary widely with different specifications. pressure vary widely with different specifications. The ASME Boiler and Pressure Vessel Code rules are based on an allowable pressure equal to one-third of the theoretical buckling pressure. For inelastic buckling, the tangent modulus is used in place of Young's modulus. The initial geometry of the cylinder must meet two requirements. Overall out-of-roundness is limited by D max - D min less than or equal to 0.01D nom where D max, D min, and D nom are the maximum, minimum, and nominal diameters, respectively. The maximum deviation from a true circular arc over a distance corresponding to one-half of the theoretical buckle wave length is limited to a value between 0.3 t and 0.1 t, where t is shell thickness. The specified out-of-roundness criterion is intended to limit the reduction in buckling stress to 80% of theoretical. The resulting factor of safety is 3 x 0.8 = 2.4.The British Standard Inst. (BSI) BS5500 rules for unfired pressure vessels are based on an allowable pressure equal to one-third of the theoretical pressure equal to one-third of the theoretical buckling pressure for elastic buckling. For inelastic buckling, a lower bound curve on test data was determined and a factor of safety of about 1.5 was used to determine the allowable pressure. Cylinders must be circular to within 0.5% on radius measured from the true center.The Det norske Veritas rules for offshore structures are based on an elastic buckling stress equal to the theoretical buckling stress reduced by a factor that varies with the shell geometry. This factor varies from a maximum of 1.0 to a minimum of 0.6. An empirical reduction factor is applied for inelastic buckling. The out-of-roundness requirements are the same as the ASME code for maximum deviation from a true circle measured over one-half of a wave length and the same as the BSI rules for overall out-of-roundness.The hydrostatic collapse criterion given in the tenth edition (and several earlier editions) of API RP 2A was based on the theoretical equation developed by Timoshenko to account for out-of-roundness. API Spec. 2B rules limit the out-of-roundness to D max - D min less than or equal to 0.01 D nom [or 0.25 in. (6.35 mm) for cylinders up to 48 in. (1.22 m) in diameter or 0.5 in. (12.7 mm) for cylinders greater than 48 in. (1.22 m) in diameter, whichever is less]. JPT P. 2528

Ultimate Compressive Strength of Wide Rectangular Plates

It is reported that several cargo ships have suffered from buckling failures of deck or bottom plates stiffened transversely. The ultimate compressive strength of rectangular plates is studied by analyzing a simplified model which clarifies the effects of yielding on the strength.Approximate formulas for prediction of the buckling strength are proposed in terms of the aspect ratio and the ratios of yield stress to elastic buckling stress and initial deflection to thickness. The accuracy of the formulas is verified by the detailed analysis using the combined technique of the finite-element and energy methods.

Analysis and Stability of Floating Roofs

An oil storage tank floating roof, of the type considered, consists essentially of a circular membrane stiffened around its edge by a compartmented buoyant ring or pontoon. A method is presented for the static stress analysis of this type of roof. Solution of the membrane large deflections equations is accomplished numerically by assuming a range of starting values for the boundary parameters. The unique values which correspond to equilibrium and compatibility at the pontoon/membrane interface are established by cross-plotting those parameters that are common to both at their connection point. Elastic buckling stresses are derived for the pontoon on the basis of small displacements, and the complete analysis procedure is illustrated by application to a particular roof design. The effects of nonsymmetrical normal loading are examined and related to the probability of various types of collapse. In this context, it is demonstrated that partial loss of pontoon buoyancy is likely to seriously reduce a floating roof operating safety factor.

On the Ultimate Strength of Corrugated Plates Subjected to Compressive Load

In this paper, the authors propose two new formulas for the plastic buckling of plates as well as the plastic buckling of corrugated plates, one of which is called T-formula (isotropic tangent modulus formula) and another is nai icd Orthogonal Anisotropic Tangent Modulua Formula. The former which gives the lower limit of plas lc buckling stress of plates, corresponds to “Shanley load” in case of column instability, and Gt =Et/2 (1-ν). The latter is obtained from a plane stress problem in which the yielded plate subjected to comp essive load is regarded as a orthotropic corrugated plate, and Gt=E Et/ E+ (1+2ν) Et. These two formulas tabulated in Table 1 with other famous formulas, showed a satisfactory agreement with the test results (Fig. 10).The authors also established the general calculating method to obtain the ultimate strength of corrugated plates in plastic range (using τ-formula explained above) as well as in elastic range for design purposes, including the determination of all the rigidities of corrugated plate. Especially, the torsional rigidity Dxy, was checked by a test which coincided with the calculated value.The behavior of elastic buckling stress (σ) of orthotropic plates is also studied in this paper. Provided Dx, Dy=αDx, H=βDx and α =β2 (where α= anisotropic coefficient for bending and β= anisotropic coefficient for torsion), it is shown in Fig. 7 that σ decreases gradually as βdecreases from 1 to 0 and the corresponding buckling mode also changes as σ varies.

Plastic Buckling of Simply Supported Plates Subjected to Combined Shear and Bending or Eccentric Compression in Their Plane

Formulas are derived for computing the plastic buckling stress of long simply supported plates that are subject to a combination of shear and longitudinal bending in their plane. The energy method and oblique coordinates are used. The integration in the plastic regions is carried out partially by using Simpson's rule. The more general case of combined shear and eccentric compression is also considered. After evaluating these computations for some definite cases, simple design formulas are presented for computing the plastic reduction factor, with which the elastic buckling stress for the so-called associated elastic case has to be multiplied in order to obtain the plastic buckling stress. An approximate formula is presented for calculating the elastic buckling stress for combined shear and eccentric compression.

Theory of Plastic Buckling of Plates and Application to Simply Supported Plates Subjected to Bending or Eccentric Compression in Their Plane

Abstract After some general considerations on the plastic buckling of plates, the plastic buckling stresses are calculated for long plates, subject to longitudinal bending or eccentric compression in their plane, and simply supported at their unloaded edges. The solutions are based on the author’s theory of plastic plate buckling and are obtained by reducing the governing partial differential equation to ordinary finite-difference equations. Second-order finite differences are used, with a spacing equal to one ninth of the plate width. A simple design formula is presented for the plastic reduction factor with which the elastic buckling stress has to be multiplied for obtaining the plastic buckling stress.