In-Plane Elastic Stability of Arches under a Central Concentrated Load

This paper is concerned with the in-plane elastic stability of arches with a symmetric cross section and subjected to a central concentrated load. The classical methods of predicting elastic buckling loads consider bifurcation from a prebuckling equilibrium path to an orthogonal buckling path. The prebuckling equilibrium path of an arch involves both axial and transverse deformations and so the arch is subjected to both axial compression and bending in the prebuckling stage. In addition, the prebuckling behavior of an arch may become nonlinear. The bending and nonlinearity are not considered in prebuckling analysis of classical methods. A virtual work formulation is used to establish both the nonlinear equilibrium conditions and the buckling equilibrium equations for shallow arches. Analytical solutions for antisymmetric bifurcation buckling and symmetric snap-through buckling loads of shallow arches subjected to this loading regime are obtained. Approximations for the symmetric buckling load of shallow arches and nonshallow fixed arches and for the antisymmetric buckling load of nonshallow pin-ended arches, and criteria that delineate shallow and nonshallow arches are proposed. Comparisons with finite element results demonstrate that the solutions and approximations are accurate. It is found that the existence of antisymmetric bifurcation buckling loads is not a sufficient condition for antisymmetric bifurcation buckling to take place.     

Equivalent Beam-Column Method to Estimate In-Plane Critical Loads of Parabolic Fixed Steel Arches

The objective of this paper is to investigate the characteristics of critical loads for parabolic fixed steel tubular arches. An advanced nonlinearity finite-element program, taking into account the geometric and material dual nonlinearity, is employed. The influence of nonlinearity and initial crookedness on arch critical load is discussed. It is found that the effect of rise-to-span ratio on the critical load of arch is significant. Therefore, a new equivalent beam-column method is proposed for estimating the corresponding in-plane critical loads of arch, in which a buckling factor K1 is employed to consider influence of rise-to-span ratio and a reduction factor K2 to consider the effect of initial crookedness. Pragmatic formulas and tabulated data are provided based on the present different Chinese design codes. It is proved that the presented method is sufficiently accurate to predict the in-plane critical load of parabolic fixed steel arch subjected to compression or to both bending and compression.     

Ketorolac (Toradol) as an analgesic in swine following transluminal coronary angioplasty

The arch forms of 38 cases (53 nonextraction and 23 extraction arches) in which expansion, while maintaining arch form, was the objective of the practitioner, were analyzed before treatment, after treatment, and an average of 6 to 8 years after retention. The cubic spline was used to fit a curve representing arch form. By superimposing the spline curves, changes in arch form were analyzed with the variables rebound change (RC), rebound index (RI), rebound number (RN), and stability number (SN). Traditional linear intraarch dimensions were also analyzed. Analysis of variance was used to determine differences between the maxillary and mandibular arches and between the extraction and nonextraction cases. Pearson correlation coefficients between spline variables and arch width variables were also computed. There was significantly more expansion in the maxillary arch than the mandibular arch during treatment, irrespective of extraction or nonextraction strategies. In the nonextraction cases, a greater amount of net expansion was achieved for all dimensions for the maxillary arch as compared with the mandibular arch. Overall, a relatively high stability in arch form was found. The findings suggest that stability may not be related to the amount of change produced during treatment. Significant expansion can be gained throughout the premolar regions and may be expected to be stable. The order of greatest net arch width gained was for the second premolars followed by first premolars, molars, and then the canines. The intercanine widths for both arches decreased toward pretreatment values, but were more stable in the maxillary arch in nonextraction cases. The cubic spline permits measurement of change in arch form both during treatment and retention periods.     

In-Plane Nonlinear Buckling Analysis of Deep Circular Arches Incorporating Transverse Stresses

Large discrepancies exist among current classical theories for the in-plane buckling of arches that are subjected to a constant-directed radial load uniformly distributed around the arch axis. Discrepancies also exist between the classical solutions and nonlinear finite-element results. A new theory is developed in this paper for the nonlinear analysis of circular arches in which the nonlinear strain-displacement relationship is based on finite displacement theory. In the resulting variational equilibrium equation, the energy terms due to both nonlinear shear and transverse stresses are included. This paper also derives a set of linearized equations for the elastic in-plane buckling of arches, and presents a detailed analysis of the buckling of deep circular arches under constant-directed uniform radial loading including the effects of shear and transverse stresses, and of the prebuckling deformations. The solutions of the new theory agree very well with nonlinear finite-element results. Various assumptions often used by other researchers, in particular the assumption of inextensibility of the arch axis, are examined. The discrepancies among the current theories are clarified in the paper.     

On the Flexural Analysis of Sandwich and Composite Arches through an Isoparametric Higher-Order Model

A higher-order arch model with seven degrees of freedom per node is proposed to study the deep, shallow, thick, and thin composite and sandwich arches under static loads. The strain field is modeled through cubic axial, cubic transverse shear, and linear transverse normal strain components. As the cross-sectional warping is accurately modeled by this theory, it does not require any shear correction factor. The stress-strain relationship is derived from an orthotropic lamina in a three-dimensional state of stress. The proposed formulation is validated through models with various curvatures, aspect ratios, boundary conditions, materials, and loading conditions.     

Exact Solution of Out-of-Plane Problems of an Arch with Varying Curvature and Cross Section

This paper deals with the exact solution of the differential equations for the out-of-plane behavior of an arch with varying curvature and cross section. The differential equations include the shear deformation effect. The cross section of the arch is doubly symmetric. Due to the double symmetry, in-plane and out-of-plane behavior will be uncoupled. However, a coupling of the out-of-plane bending and the torsional response will exist and will be discussed in this study. The governing differential equations of planar arches loaded perpendicular to their plane are solved exactly by using the initial value method. The analytical expressions of the fundamental matrix can be obtained for some cases. It is also possible to use these analytical expressions in order to obtain the displacements and the stress resultants for an arch with any loading and boundary conditions. The examples given in the literature are solved and the results are compared. The analytical expressions of the results are given for some examples.     

Buckling Mode Interaction in Fixed-End Column with Central Brace

The postbuckling behavior of an elastic fixed-end column with an elastic brace at the center is investigated. Attention is focused on those of brace stiffness near its threshold value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. We show that, for the brace stiffness greater than the threshold value, there are precisely two secondary bifurcation points on each primary postbuckling path bifurcating from one of the least two classical buckling loads, and the corresponding secondary postbuckling paths connect all of these secondary bifurcation points in a loop. For the brace stiffness less than the threshold value, no secondary bifurcation occurs. The asymptotic expansions of the primary and secondary postbuckling paths are constructed. The stability analysis indicates that, when the brace stiffness goes beyond its threshold value, the primary postbuckling path with a node in the center becomes unstable from stable by means of the secondary bifurcation (i.e., secondary buckling occurs).     

Flutter of an Arch Bridge via a Fully Nonlinear Continuum Formulation

A fully nonlinear parametric model for wind-excited arch bridges is proposed to carry out the flutter analysis of Ponte della Musica under construction in Rome. Within the context of an exact kinematic formulation, all of the deformation modes are considered (extensional, shear, torsional, in-plane, and out-of-plane bending modes) both in the deck and supporting arches. The nonlinear equations of motion are obtained via a total Lagrangian formulation while linearly elastic constitutive equations are adopted for all structural members. The parametric nonlinear model is employed to investigate the bridge limit states appearing either as a divergence bifurcation (limit point obtained by path following the response under an increasing multiplier of the vertical accidental loads) or as a Hopf bifurcation of a suitable eigenvalue problem (where the bifurcation parameter is the wind speed). The eigenvalue problem ensues from the governing equations of motion linearized about the in-service prestressed bridge configuration under the dead loads and wind-induced forces. The latter are expressed in terms of the aeroelastic derivatives evaluated through wind-tunnel tests conducted on a sectional model of the bridge. The results of the aeroelastic analysis—flutter speed and critical flutter mode shape—show a high sensitivity of the flutter condition with respect to the level of prestress and the bridge structural damping.     

Flotation to Trigger Lateral Buckles in Pipelines on a Flat Seabed

Triggering lateral buckles in subsea pipelines can be an effective way to ensure that the thermal expansion is spread over several buckles at a controlled spacing, rather than concentrating in a single buckle that may become overstressed, or causing excessive displacements at the ends of a pipeline where spool-piece connections or jumpers may become overstressed. For a given location where a buckle is to be triggered one calculates that axial load that would develop if the buckle fails to form, and ensures the buckle would be triggered at or below this axial load. This paper gives simple analytical expressions for triggering of such lateral buckles by applying buoyancy to the pipeline, e.g., by inflatable parachute-like bags, which can later be removed. The expressions apply for a single buoyancy load, two equal buoyancy loads, or distributed buoyancy over a specified length. They explicitly solve the design problem of establishing the required amount of buoyancy for a given axial load at which the buckle must be triggered. It is shown quite generally that bifurcation buckling into a lateral buckling mode occurs before vertical instability, and at moderate uplift displacements. The analytical results are based on a flat seabed, the theory of moderately large deflections, and elastic pipe behavior. An example involving a single point buoyancy load is also solved by finite-element analysis based on large deflection theory to verify the accuracy of the approach.     

Asymmetric Collapse Modes of Pipes under Combined Bending and External Pressure

The subject of pipe buckling and collapse under combined external pressure and bending is revisited in order to investigate the causes of angled buckles observed in recent experiments and the associated “scatter” in the critical loads. Stainless steel 304 tubes with D∕t = 18.3 are bent to collapse under various values of external pressure. Tubes bent at pressures higher than 0.72P0 exhibited angled buckles which were oriented at 20° to 45° to the axis of bending. In this pressure regime the scatter in the results was larger than usual. At lower pressures, the tubes buckled in the expected mode with the flattening being along the axis of bending. A previously developed formulation is extended so that it can handle asymmetric imperfections and buckling modes. The analysis is first used to reproduce the experimental results and subsequently to study the problem parametrically. The orientation of the initial ovality is found to play a role in the final buckling mode and in the value of the critical curvature for bending under high values of pressure. In addition, residual stresses can interact with the initial ovality affecting the critical curvature both positively and negatively. For lower pressures these effects are small.     

Alternative Approach to Modify a Discrete Kirchhoff Triangular Element

In this paper, the existing modified discrete Kirchhoff triangular bending element is further modified, using the least-squares method, to enable the analysis of arbitrary thin plate with different material and geometric properties. In the vibration and buckling analyses, the combined mass and combined geometric stiffness matrices are employed to improve the calculations of natural frequency and buckling load. A comparison between the proposed element and some existing elements shows that the former is a high precision element for thin plate bending, vibration, and buckling analyses.     

Plastic Buckling Transition Modes in Moderately Thick Cylindrical Shells

Moderately thick perfect cylindrical shells under axial compression first exhibit an axisymmetric buckling mode, where a localization of buckling patterns, referred to as an elephant foot bulge, is caused by the first plastic bifurcation. However, the transition from the axisymmetric buckling mode to a nonaxisymmetric buckling mode, referred to as a diamond buckling mode, may occur due to the next bifurcation if we continue the loading under displacement control. Herein, this phenomenon is examined, based on a rigorous plastic bifurcation analysis. As a result, it is observed that the circumferential wave number of the diamond buckling mode increases with the decrease of the wall thickness. The boundary conditions also considerably influence the occurrence of diamond buckling. It is found that the strain concentration is intensified for the diamond buckling modes, compared with the axisymmetric modes.     

Shapes of Submerged Funicular Arches

In this paper, the analytical forms of the shapes of submerged funicular arches are expressed in terms of elliptic integrals. The initial radius of curvature at the apex is related to the water depth at the apex and the axial compressive force. The shape of the submerged funicular arches depends on the ratio between this initial radius of curvature and the water depth at the apex. Using the analytical expressions, the maximum span and height of the submerged funicular arches can be determined explicitly. Besides, the analytical expressions are useful in determining the design parameters for the submerged funicular arches accurately.     

Approximate Derivation of Critical Buckling Load

This paper proposes an approximate derivation for the critical buckling load of a column, based on the application of a uniformly loaded beam's midspan moment and deflection to the buckled column's rotational equilibrium. The curvature of a pin-ended member, when it buckles under axial load, is similar to the curvature assumed by the same member when it deflects under a uniformly distributed load applied transversely along its entire length. Euler's famous equation for critical buckling load is based, of course, on the former assumption, in which the deflected column assumes the shape of a sine curve. However, dividing a uniformly loaded beam's midspan moment by its deflection provides a conservative result for the critical buckling load, within 3% of Euler's value, that can be derived solely on the basis of these commonly used beam equations.     

Local Buckling of Elastically Restrained Fiber-Reinforced Plastic Plates and its Application to Box Sections

An analytical study of local buckling of rectangular composite plates rotationally restrained elastically along unloaded edges and subjected to nonuniform in-plane axial action at simply supported loaded edges is presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and by using combined harmonic and polynomial buckling deformation functions, which satisfy all the restrained boundary conditions, the explicit solution of plate local buckling coefficients is obtained. The explicit formulas for local buckling strength of orthotropic plates are simplified to the cases of isotropic plates, which are consistent with classical solutions. The elastically rotationally restrained plates are further treated as discrete plates or panels of fiber-reinforced plastic (FRP) box shapes, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling strength of FRP box shapes is predicted. The theoretical predictions are in good agreement with transcendental solutions and finite-element eigenvalue analyses for local buckling of FRP box columns. The present explicit formulation can be applied to determine local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.     

Behavior of Curved I-Girder Webs Subjected to Combined Bending and Shear

Two previous papers by the writers described the buckling and finite-displacement behavior of curved I-girder web panels subjected to pure bending, presented a theoretically pure analytical model, and presented equations that describe the reduction in strength due to curvature. This paper describes the buckling and finite-displacement behavior of curved web panels under combined bending and shear. Unlike straight girder web panels, the addition of shear in curved panels is shown to increase the transverse “bulging” displacement of the web prior to buckling. The accompanying decrease in moment carrying capacity is analyzed in a manner similar to that used for the combined bending and shear nominal strength interaction for straight girder design. Preliminary recommendations are made toward forming design criteria for curved webs.     

Analytical Load Rating of an Open-Spandrel Arch Bridge: Case Study

The live load structural capacity of open-spandrel arch bridge structures is difficult to quantify. In addition to live and dead loads, geometric nonlinear effects, temperature effects, and material behavior play key roles in the design and load rating of such a structure. This paper is a case study that illustrates the effect these variables have on load rating a two-span shallow concrete arch bridge. Presented are load ratings of the structure’s arch ribs using a three-dimensional finite-element model with American Association of State Highway and Transportation Officials publications. As a result of this study, a refined analysis is recommended for load rating arch bridges.     

Buckling of Cylindrical Tunnel Liners

Buckling of thin cylindrical shell liners used for the stabilization of soft ground tunnels is treated as the buckling of an elastic ring restrained radially and tangentially by an infinite surrounding elastic medium. Stiffness components for the elastic medium are derived and used to provide various levels of approximation for the elastic critical loads and associated modes of the liner when subject to overburden pressure loading. For most practical tunnel liners, elastic buckling is found to occur in modes having relatively short circumferential wavelengths. In these circumstances an approximation introduced into the critical pressure analysis allows both the lowest critical pressure and its associated mode shape to be represented explicitly in terms of a single “soft ground tunnel buckling parameter”; this single composite parameter encapsulates all the relevant ground and liner geometric and material properties. It is this closed-form analytical representation of elastic critical buckling that provides a particularly convenient basis for predicting elastic-plastic failure. When imperfections are introduced, liner collapse and the various forms of ground failure can be modeled by methods analogous to the Ayrton-Perry expression for columns. Two generalized imperfection parameters emerge: one for liner collapse and the other for each of three possible, soft ground, failure modes. It is suggested that the analytical simplicity of the approach should make it an attractive alternative basis for at least the initial, rational, design of soft ground tunnel liners.     

Analytical Study of Bifurcation and Nonlinear Behavior of Sandwich Columns

Buckling behavior of axially compressed sandwich columns exhibits a variety of interesting phenomena. The column can buckle in a long wave “overall mode” or a short-wave “wrinkling mode,” the latter involving severe bending of the facings and transverse deformation of the core. The full-range nonlinear behavior of these structures is investigated in the elastic range taking typical examples. Columns can be, though not always, imperfection sensitive when wrinkling is the principal mode of buckling. The columns buckling in the overall mode collapse by the formation of localized wrinkles in the postbuckling range. An interesting new finding of this study is that, for certain combinations of core compliance and facing thickness, there can be a bifurcation from the prebuckling path at a load smaller than that predicted by the linear stability analysis. In this scenario, if wrinkling is the dominant mode of buckling, the column turns out to be imperfection sensitive.     

MR angiography of the supra-aortic arteries using a dedicated head and neck coil: image quality and assessment of stenoses

Our purpose was to evaluate a dedicated head and neck coil for demonstration of supra-aortic arteries with optimised magnetic resonance angiography techniques. We performed 47 examinations with a 1.5-T system. We used coronal 3D fast imaging with steady precession (FISP), axial 3D tilted optimised nonsaturating excitation (TONE) and 2D fast low-angle shot (FLASH) for the carotid bifurcation, axial 3D TONE with or without magnetisation transfer (MT) for intracranial arteries, and axial 3D FISP or TONE for the aortic arch. Evaluation included visual assessment of image quality and grading of stenoses near the carotid bifurcation; digital subtraction angiography was used as the reference method. Axial 3D TONE gave superior image quality at the carotid bifurcation, MT-TONE intracranially, and 3D FISP for the aortic arch vessels. Nevertheless, sensitivity and specificity for detection of significant stenoses were similar with coronal 3D FISP (96.3%, 94.0%), axial 3D TONE (92.6%, 92.5%) and axial 2D FLASH (96.3%, 86.6%). Image quality at the aortic arch needs further improvement.