Supported End Conditions Research Articles

Design Analysis for Buckling of Tanks and Silos

When subject to combinations of internal pressure and axial compression, experience shows that the often thin cylindrical shell walls of tanks and silos are prone to a local form of buckling failure that is essentially axisymmetric and is strongly influenced by the nature of the end or intermediate support conditions. This paper will outline the basis of an analytical solution to the inherently nonlinear elastic-plastic buckling into axisymmetric modes, in which end constraints and arbitrary radial pressure loading are modeled as essentially loading imperfections on an otherwise linear, classical, critical buckling formulation. Closed form analytical expressions for the imperfect behavior, including geometric imperfections, are then used to predict either first surface or full section plasticity, allowing buckling failure to be summarized in a form that is closely analogous to the Ayrton-Perry formula for the buckling of columns. It is suggested that a suitably simplified form of this general approach to axisymmetric buckling, when combined with similarly simple methods for the buckling into nonaxisymmetric modes, could provide an important alternative to the current procedures for the buckling design of thin-walled, cylindrical, tanks and silos.

Axisymmetric and beamlike vibrations of multiwall carbon nanotubes

Several vibration problems of multiwall carbon nanotubes (MWNTs) are studied in detail based on a multiple-elastic shell model. According to recent data available in the literature, an updated value of bending stiffness for single-wall carbon nanotubes (SWNTs) is suggested, which is in a much better agreement with atomistic model for phonon-dispersion relation of SWNTs. For axisymmetric vibrations (with circumferential wave number $n=0$), it is found that longitudinal $(L)$ modes of individual tubes of a MWNT have almost identical frequencies and are usually coupled with each other through Poisson-ratio effect-induced radial $(R)$ vibrations and interlayer van der Waals interaction. Especially in the transition zone of $R$- and $L$ modes, the significant Poisson-ratio effect leads to mixed $R--L$ modes with comparable longitudinal and radial displacements. On the other hand, for beamlike vibrations (with $n=1$), the present multiple-shell model is found to be in good agreement with the multiple-beam model for almost coaxial bending $(B)$ modes of large- and small-radius MWNTs and noncoaxial $B$ modes of small-radius MWNTs (e.g., of the outermost radius less than $2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$), with relative errors less than 10%. However, for high-order noncoaxial modes of large-radius MWNTs, the relative errors between the two models increase up to 50% in extreme cases due to larger non-beamlike deformation of the cross section while both models give similar overall vibration modes through the entire length of MWNTs. In particular, for lower circumferential wave numbers $(n=0--10)$, the lowest frequency always corresponds to the minimum half-axial wave number $m=1$ for simply supported end conditions. When the wave vector decreases, the lowest frequency decreases and the associated mode shifts from an $R$ mode with larger $n$ to a coaxial $B$ mode with $n=1$.

Characterisation of the rotational stiffness and strength of web-flange junctions of pultruded GRP WF-sections via web bending tests

Three-point flexure tests on the webs of 12 pultruded Glass Reinforced Plastic (GRP) Wide Flange (WF) specimens are described. The webs of the specimens were tested with their ends either simply or semi-rigidly supported. Dual knife-edge fixtures and rigid steel supports were designed and fabricated to simulate these support conditions. A series of low load tests were carried out on the webs of the WF specimens for both types of end conditions and deflections and strains were recorded at mid-span. Using simple beam analysis models in conjunction with the test data, both the transverse elastic modulus of the web and the rotational stiffness per unit length of the web-flange junctions of each specimen were determined. The webs of five of the specimens were failed under simply supported end conditions and the remaining seven under semi-rigid end conditions. From these failure tests the initial failure strengths of the web-flange junctions and the ultimate bending strengths of the webs were determined. The failure loads and modes of failure for both sets of end conditions have also been quantified.

Distortional buckling formulae for cold-formed steel C- and Z-section members: Part II—Validation and application

General GBT-based fully analytical formulae have been derived in a companion paper, which provides distortional bifurcation stress estimates in cold-formed steel C- and Z-section members acted by arbitrary applied stress distributions and displaying four end support conditions. This paper (i) addresses the implementation of the above general formulae, (ii) illustrates their application in detail, (iii) validates them, by means of comparisons with exact results, and (iv) compares their estimates with values yielded by other formulae available in the literature. After considering a wide range of (i) cross-section geometries and lengths and (ii) applied stress distributions, it is concluded that the GBT-based formulae are both accurate and universal.

Distortional buckling formulae for cold-formed steel C and Z-section members: Part I—derivation

This paper presents the derivation of generalised beam theory (GBT)-based fully analytical formulae to provide distortional critical lengths and bifurcation stress resultant estimates in cold-formed steel C and Z-section members (i) subjected to uniform compression (columns), pure bending (beams) or a combination of both (beam–columns), (ii) with arbitrary sloping single-lip stiffeners and (iii) displaying four end support conditions. These formulae incorporate genuine folded-plate theory, a feature which is responsible for their generality and high accuracy. After a brief outline of the GBT fundamentals and linear stability analysis procedure, the main concepts and steps involved in the derivation of the distortional buckling formulae are described and discussed. Moreover, the paper also includes a few remarks concerning novel aspects related to the distortional buckling behaviour of Z-section beams and C-section beam–columns, which were unveiled by the GBT-based approach. Finally, note that, in a companion paper [Thin-Walled Struct., 2004 doi: 10.1016/j.tws.2004.05.002], the formulae derived here are validated and their application, accuracy and capabilities are illustrated. In particular, the GBT-based estimates are compared with exact results and, when possible, also with values yielded by the formulae developed by Lau and Hancock, Hancock, Schafer and Teng et al.

Minimum weight of cold-formed steel sections under compression

This paper presents a combined theoretical and experimental study on the minimum weight and the associated optimal geometric dimensions of an open-channel steel section with given length subjected to a prescribed axial compressive load. Sections both with and without lips are analyzed. The results obtained using a nonlinearly constrained optimization method are compared with those estimated from a simple-minded optimization procedure that assumes the simultaneous occurrence of all failure modes in a minimum weight structure. The types of failure mode considered include yielding, flexural buckling, torsional–flexural buckling, and local buckling. The failure criterion is based purely on compressive strength, with other possible design constraints (e.g. bending stiffness, minimum gauge and cost) ignored. The effects of end support conditions and restraint on torsional buckling are examined. The load capacity of a C-section calculated according to the 1998 British Standard Institution’s specifications on Structural Use of Steelwork in Building is used to check the validity of theoretical predictions. Finally, two new C-sections with lips were designed and manufactured based on the optimal results, and tested. Test results confirm the analytical predictions, with the optimal C-sections performing much better than the existing ones.

On the application of beam-column interaction formulae to steel members with arbitrary loading and support conditions

An investigation on dealing with the application of the so-called Level 1 and Level 2 beam-column interaction formulae to isolated steel members with arbitrary loading and end support conditions is reported. The issues addressed concern mostly the importance, both in terms of safety and accuracy, of adopting appropriate “equivalent moment factor” (Cm) values. Only the in-plane behavior of members subjected to axial compression and major or minor axis bending is investigated, but several load combinations and support conditions are dealt with. The strength estimates provided by the Level 1 and Level 2 interaction formulae are presented, discussed and compared with “exact” values, obtained from second-order plastic zone analyses performed by means of the FEM code, ABAQUS. On the basis of such comparative studies, several conclusions are drawn concerning the safety and accuracy of the strength estimates yielded by the two interaction formulae and, in particular, it is shown that significant improvements can be achieved by adopting “exact” Cm values. Moreover, some attention is paid to issues related to the proper choice or determination of these “exact” Cm values.

GBT buckling analysis of pultruded FRP lipped channel members

The main aspects concerning an orthotropic second order generalised beam theory (GBT), previously developed by the authors to enables the structural analysis of thin-walled members displaying special orthotropy, are briefly presented and discussed in the first part of the paper. Special attention is paid to the symbolic and numerical computational aspects related to the implementation of the GBT equations to perform member linear stability analyses. For this purpose, a finite element formulation is developed, validated and applied to solve the eigenvalue problem associated with the GBT system of differential equilibrium equations. Then, the second order orthotropic GBT is employed to investigate the buckling behaviour of pultruded fiber reinforced plastic lipped channel members, namely columns, beams and beam-columns. In particular, with the objective of illustrating the concepts and procedures involved in the performance of a GBT analysis, a detailed in-depth study of simply supported lipped channel columns is presented, in which the relevant (local and global) buckling modes are identified and the corresponding bifurcation stress values are determined. In addition, the results of an investigation concerning the influence of the applied stress distribution, cross-section geometry, material properties and end support conditions on the member buckling behaviour are presented, which required the completion of a number of parametric studies. In these investigations, careful consideration was given to the occurrence and characterisation of local-plate, distortional and mixed flexural–distortional buckling modes.

Elasto-plastic buckling and postbuckling of arches subjected to a central load

This paper presented a rational finite curved beam-element model for 3D nonlinear elasto-plastic analysis of arches. The finite element model incorporates the effects of large twist rotations of the cross-section and has consistent sampling scheme of sampling points over the cross-section and so it can perform the elastic and elasto-plastic flexural–torsional buckling and postbuckling analysis of arches in combined bending and compression actions. Elastic and elasto-plastic flexural–torsional buckling and postbuckling behaviour of arches that are subjected to a central concentrated load has been investigated using the rational finite element model. It has been found that the slenderness, included angle, and the torsional parameter of an arch play important roles in the elastic and elasto-plastic buckling. In addition, the number of the inflexion points is important for the elastic and elasto-plastic buckling of fixed arch. Yielding is a significant factor for the elasto-plastic buckling and postbuckling behaviour. When the included angle of a stocky arch is not large, its elasto-plastic buckling load is much lower than its elastic buckling load. When the included angle a stocky arch is large, the end support condition plays an important role in the buckling and postbuckling behaviour of the arch. The elasto-plastic buckling load of a stocky pin-ended arch with a large included angle is equal to its elastic counterpart. The elasto-plastic buckling load of a stocky fixed arches with a large included angle is lower than its elastic counterpart. For a slender arch, its elasto-plastic buckling load is also equal to its elastic buckling load. In general, the elasto-plastic postbuckling load carrying capacity of an arch decreases when the arch continues to deform while its elastic postbuckling load carrying capacity can increase. For slender shallow arches, the elastic postbuckling response is stiffer because of the relaxation of axial load and moment distribution.

AXISYMMETRIC VIBRATION OF CYLINDRICAL SHELLS WITH INTERMEDIATE RING SUPPORTS

This paper treats the axisymmetric vibration of thin circular cylindrical shells with intermediate ring supports based on the Goldenveizer–Novozhilov thin shell theory. An analytical method is proposed, and new exact solutions are presented to study the axisymmetric vibration characteristics of the ring supported cylindrical shells. In the proposed method, the state-space technique is employed to derive a homogenous differential equation system for a shell segment, and a domain decomposition approach is developed to cater for the continuity requirements between shell segments. Exact frequency parameters are presented for circular cylindrical shells that have multiple intermediate ring supports and various combinations of end support conditions. These exact vibration frequencies may serve as important benchmarks against which researchers can validate their numerical methods for such circular cylindrical shell problems.

Vehicle–passenger–structure interaction of uniform bridges traversed by moving vehicles

An investigation into the dynamics of vehicle–occupant–structure-induced vibration of bridges traversed by moving vehicles is presented. The vehicle including the driver and passengers is modelled as a half-car planar model with six degrees-of-freedom, and the bridge is assumed to obey the Euler–Bernoulli beam theory with arbitrary conventional boundary conditions. Due to the continuously moving location of the variable loads on the bridge, the governing differential equations become rather complicated. The numerical simulations presented here are for the case of vehicle travelling at a constant speed on a uniform bridge with simply supported end conditions. The relationship between the bridge vibration characteristics and the vehicle speed is rendered, which yields into a search for a particular speed that determines the maximum values of the dynamic deflection and the bending moment of the bridge. Results at different vehicle speeds demonstrate that the maximum dynamic deflection occurs at the vicinity of the bridge mid-span, while the maximum bending moment occurs at ±20% of the mid-span point. It is shown that one can find a critical speed at which the maximum values of the bridge dynamic deflection and the bending moment attain their global maxima.

Dynamic interaction of vehicles moving on uniform bridges

The dynamic interaction problem of moving vehicles on uniform suspension bridges is studied. The resulting variable moving loads acting on the bridge are of great practical importance to both bridge and automotive engineers. The vehicle, including the occupants, is modelled as a planar half-car with six degrees of freedom, and the bridge is assumed to obey the Euler-Bernoulli beam theory with arbitrary conventional boundary conditions. However, the numerical simulations presented here are for the case of a vehicle travelling at a constant speed on a bridge with simply supported end conditions. Owing to the continuously moving location of the loads on the bridge, the governing differential equations will have time-varying coefficients and hence become rather complicated. The relationship between the bridge vibration characteristics and the vehicle speed is established, resulting in a search for a particular speed that determines the maximum values of dynamic deflection and the bending moment of the bridge.

Exact solutions for vibration of cylindrical shells with intermediate ring supports

This paper presents new exact solutions for vibration of thin circular cylindrical shells with intermediate ring supports, based on the Goldenveizer–Novozhilov shell theory (Theory of thin shells; The theory of thin elastic shells). An analytical method is proposed to study the vibration behaviour of the ring supported cylindrical shells. In the proposed method, the state-space technique is employed to derive the homogenous differential equation system for a shell segment and a domain decomposition approach is developed to cater for the continuity requirements between shell segments. Exact frequency parameters are presented in tables and design charts for circular cylindrical shells having multiple intermediate ring supports and various combinations of end support conditions. These exact vibration frequencies may serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shell problems.

Distortion and Warping at Beam Supports

When a doubly symmetric I-beam buckles elastically in a lateral-distortional mode, its web distorts, so that the two flanges have increased relative lateral displacements and may have different twist rotations. The centroid, shear center, and principal axes of the distorted cross section are all different from those of the undistorted cross section. The end connections to adjacent members may induce flange restraining moments that oppose warping deformations and modify the elastic buckling resistance. This paper investigates the effects on the lateral-distortional buckling of beams of web distortion and of the warping restraints induced by end-support conditions. It is found that a doubly symmetric I-section becomes unsymmetric during lateral-distortional buckling and that web distortion reduces the effective torsional and warping rigidities. The proposed approximations for the elastic lateral-distortional buckling resistances of beams without and with end warping restraints provide good predictions. Equations for the warping restraint stiffness induced by some commonly used end conditions are established. Comparisons with ABAQUS shell element results show that they provide good approximations.

Design Equation for Overlap Tubular K-Joints under Axial Loading

Using a previously presented finite-element study of axially loaded overlap tubular K-joints as a platform, this paper presents further work that aimed at extending and enhancing the strength estimation equation already presented. The current work includes extending the θ (brace angle) validity range to 30° ≤ θ ≤ 60°; investigating the effects of the yield stress/ultimate tensile strength ratio and the brace/chord yield stress ratio; exploring the influence of loading reversal; and quantifying the reduction in strength caused by the absence of the hidden weld. The end result of these studies is a new, extended capacity equation. The accuracy of the proposed equation to predict the axial capacity of test joints was established by assessing it using an experimental database. The applicability of the proposed equation for design practice was further ascertained by investigating strength sensitivities to material curve variations, different chord end support conditions, and joints with unequal brace inclinations. The proposed design equation is found to be both accurate and conservative and is therefore recommended to be used in practice.

The Buckling Behavior of Pipes and Its Influence on the Axial Force Transfer in Directional Wells

An experimental setup was built at the University of Tulsa to study buckling and post-buckling behavior of pipes constrained in straight horizontal and curved wellbores. Experiments were conducted to investigate the axial force transfer with and without static internal pressure. Different stages of buckling phenomena and their relation to the axial force, the pipe diameter (1/4 and 3/8 in.) and the pipe end-support conditions have also been investigated. Experimental results have shown that the buckling load is a strong function of the pipe diameter and the pipe end-support conditions. Static internal pressure appears to have insignificant influence on the buckling behavior of pipes. A brief review of recently developed mathematical models to predict buckling behavior of pipes in inclined, curved, and horizontal sections of wellbore is also presented. Applications of the current theory are presented by using recently developed computer simulator. Results of the theoretical analysis have confirmed the versatility and effectiveness of computer simulator for better understanding and solving buckling related problems in the field. [S0195-0738(00)00903-1]

Mode jumping in the buckling of struts and plates: a comparative study

A detailed investigation of the phenomenon of mode jumping in compressed struts on stiffening foundations and elastic plates of varying lengths is performed, with emphasis on the effects of altering boundary conditions. The variety of possible modal interactions is presented in a concise form using the parameter space of Arnol'd tongues, borrowed from non-linear dynamical systems theory. For the strut system, a full range of end conditions from simply supported to clamped is examined. For the plate, simply supported and clamped flexural conditions along both long (unloaded) and short (loaded) edges are considered, together with in-plane conditions ranging from free to pull in, to fully restrained. For each system, simply supported end conditions are found to provide protection against early mode jumping in a so-called “safety envelope”, but this is eroded as the end conditions are systematically altered from simply supported to clamped. For the plate system, mode jumping is induced at an earlier stage in the loading process by restricting the long (unloaded) edges against in-plane movement, but is delayed by clamping the same edges against rotation.

Cylindrical Indentation of Orthotropic Laminates on Elastic Foundations

Problems of cylindrical indentation of orthotropic laminated beams on an elastic foundation are studied. Exact solution methods are developed to assess the displacement field for both the simply supported and clamped-ended cases. A numerical iteration algorithm is established to detect the contact length and the associated contact stresses for a prescribed contact load. The induced interlaminar shear stresses through the thickness and the amount of indentation are evaluated subsequently. The objective is to investigate the effects of span, indentor size, end support conditions, and foundation stiffness on the local field responses and load-indentation relationships. The results show that the effects of indentor size and foundation stiffness are significant. A sharp indentor or stiff foundation tends to localize the contact phenomenon, regardless of the span and the end support conditions. On the other hand, the effects of both the boundary condition and span may be prominent if a blunt indentor and a relatively soft foundation are employed.

圧縮力を受ける一対の曲がり部材で構成された構造体の振動特性に関する研究

This paper presents dynamic characteristics of curved pair members tied with struts subjected to axial compression. Numerical calculation was performed by using the natural frequency analysis method (QR method) changing the end support conditions, rise-tospan ratio and the axial load to check the practical use of the proposed paired structures. The effects of the end support conditions of those structures on the natural circular frequencies and the natural mode shapes were clarified using non-dimensional parametric analysis. It was found that curved pair members with both “fixed” or “hinged” endssubjected an axial load have higher natural circular frequencies in comparison with the usual straight structures with under the same conditions.

Vibration of Thick Prismatic Structures With Three-Dimensional Flexibilities

This paper presents an investigation on free vibration of thick prismatic structures (thick-walled open sections of L, T, C, and I shapes). The derivation of a linear frequency equation based on an exact three-dimensional small-strain linearly elastic principle is presented. This formulation uses one and two-dimensional polynomial series to approximate the spatial displacements of the thick-walled open sections in three dimension. The proposed technique is applicable to vibration of thick-walled open sections of different cross-sectional geometries and end support conditions. In this study, however, we focus primarily on the cantilevered case which has high value in practical applications. The perturbation of frequency responses due to the variations of cross-sectional geometries and wall thicknesses is investigated. First-known frequency parameters and three-dimensional deformed mode shapes of these thick-walled open sections are presented in vivid graphical forms. The new results may serve as a benchmark reference to future research into the refined beam and plate theories and also for checking the accuracy of new numerical techniques.