Etiology of idiopathic scoliosis. Computational study

A review of the literature on the mechanical aspects of the etiology for idiopathic scoliosis reveals that the buckling hypothesis has been presented as a purely mechanical phenomenon. In an attempt to confirm the buckling hypothesis, a numerical simulation of growth and the resulting buckling phenomena was done by means of finite element analysis. It previously was observed that growth was induced in the T4 to T10 vertebrae. Only the sacrum was assumed to be stationary. From the growth analysis, a deformation process that mitigated thoracic kyphosis was obtained as observed in healthy children during early adolescence. From the buckling analysis, the first to the fourth buckling modes that correspond to the first side bending, first forward bending, first rotation, and second side bending modes were obtained. The shape of the fourth buckling mode (second side bending mode) was in good agreement with the clinical shape. Considering the potential for controlling these modes by posture change, it is concluded that the second bending mode in the coronal plane is one of the most likely etiologic candidates in the mechanics of thoracic idiopathic scoliosis.     

Asymptotic-Numerical Method to Analyze the Postbuckling Behavior, Imperfection-Sensitivity, and Mode Interaction in Frames

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an “internally constrained” beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to validate and illustrate the application and potential of the proposed method, several numerical results are presented, concerning (1) four validation examples (Euler column and three simple frames—two or three members), for which there exist some (perfect frame) analytical and numerical asymptotic results reported in the literature; (2) a single-bay pitched-roof frame with partially restrained column bases; and (3) a three-bay frame with two leaning columns. These results comprise (1) the initial postbuckling behavior of perfect frames (individual and coupled buckling modes) and (2) geometrically nonlinear equilibrium paths describing the behavior of frames containing initial geometrical imperfections or primary bending moments. In the latter case, most of the semianalytical results are compared with fully numerical values, yielded by finite element analyses performed in the commercial code ABAQUS.     

Lateral-Torsional Buckling of Stepped Beams with Continuous Bracing

Continuous span multibeam steel bridges are common along the state and interstate highways. The top flange of the beams is typically braced against lateral movement by the deck slab, and in many bridges the cross section is stepped at discrete points along the span. Design equations for lateral–torsional buckling (LTB) resistance in the American Association of State Highway and Transportation Officials “Load and resistance factor design bridge design specifications” are for prismatic beams and ignore the lateral restraint provided by the bridge deck. A new design equation is proposed that can be applied to I-shaped stepped beams with continuous top flange lateral bracing. By including the effects of the change in cross section size and the continuous top flange bracing, the calculated LTB resistance is significantly increased. Critical bending moment values from the proposed equation are compared to values from finite element method buckling analyses. The new equation is sufficiently accurate for use in design and in the evaluation of existing bridges.     

Postbuckling Response Simulations of Laminated Anisotropic Panels

Numerical simulations are used in conjunction with experiments to study the buckling and postbuckling responses and failure initiation of flat, unstiffened composite panels. The numerical simulations are conducted using two‐dimensional shear‐flexible finite elements. The effect of the laminate stacking sequence on the buckling and postbuckling responses is studied. Correlation between numerical and experimental results is good through buckling, but the numerical models overestimate the postbuckling stiffness of the panels when nominal values of the material properties are used. To explain the discrepancies in the postbuckling stiffnesses, analytic sensitivity derivatives are calculated and used to study the sensitivity of the buckling and postbuckling responses to variations in different material and lamination parameters. Experimental results indicate that failure occurs along a nodal line. Numerical results show that the location of failure initiation corresponds to that of the maximum transverse shear‐strain energy density in the panel, which occurs at the edge of the panel at a nodal line. However, the transverse shear deformation has a negligible effect on the global response characteristics of the panel.     

Interaction between Local and Lateral Buckling on Pultruded I-Beams

This paper presents the results of an experimental work involving pultruded beams. The tests developed attempt to observe the interaction between local and global buckling in open-section beams. A modified three-point bending test with both ends clamped has been used in order to reduce the slenderness of the structural elements. By means of a finite-element model the critical bending moment has been calculated. Special care has been taken to obtain an accurate correspondence between the real test and the finite-element model. The comparison made between test results and critical bending moments showed that the above-mentioned interaction clearly reduces the lateral buckling load in the low slenderness range. Based on the experimental data, Dutheil’s formulation has been adjusted leading to a new design equation. The proposed equation showed a good correlation in the low slenderness range, but did not match well with experimental data from literature developed in the high slenderness range. For high slenderness values, using the critical bending moment seems to be the best design method. Therefore, more experimental work has to be done on pultruded beams in order to establish a suitable formulation to describe the transition between the low and high slenderness ranges’ behaviors.     

Spatial Stability of Nonsymmetric Thin-Walled Curved Beams. II: Numerical Approach

In a companion paper, for spatial stability of nonsymmetric thin-walled curved beams, a general formulation was derived based on a displacement field considering the second-order terms of semitangential rotations. Closed-form solutions were newly derived for in-plane and out-of-plane buckling of simply supported curved beams with monosymmetric sections subjected to pure bending or uniform compression. In this paper, to get numerical solutions for the buckling of thin-walled curved beams subjected to general loadings, finite-element procedures are developed using thin-walled curved beam elements and straight frame elements with nonsymmetric sections. Numerical examples for the spatial buckling of doubly symmetric, monosymmetric, and nonsymmetric thin-walled circular beams are presented and compared with previously published solutions to illustrate the accuracy and the practical usefulness of the analytical solutions and numerical procedures.     

Nonlinear Thermoelastic Buckling of Pin-Ended Shallow Arches under Temperature Gradient

This paper presents a nonlinear thermal buckling analysis of circular shallow pin-ended arches that are subjected to a linear temperature gradient field in the plane of curvature of the arch. The linear temperature gradient produces axial expansion and curvature changes in the arch. The bending action produced by the curvature change and the axial compressive action produced by the restrained axial expansion may lead the arch to buckle suddenly in the plane of its curvature. The end reactions resulting from the restrained axial expansion also produce bending actions that are opposite to that produced by the temperature differential and tend to produce deflections on the convex side of the arch. A geometrically nonlinear analysis for thermoelastic buckling has been carried out based on a virtual work technique, and analytical solutions for the critical temperature gradients for the in-plane limit instability and bifurcation buckling are obtained. It is found that antisymmetric bifurcation is the dominant buckling mode for most shallow arches that are subjected to a linear temperature gradient. The limit instability is possible only for slender and shallow arches. It is also found that a characteristic value of the arch geometric parameter exists and that arches whose geometric parameter is less than this characteristic value show no typical buckling behavior. The formula for this characteristic value of the arch geometric parameter is derived.     

Spatial Stability of Nonsymmetric Thin-Walled Curved Beams. I: Analytical Approach

An improved formulation for spatial stability of thin-walled curved beams with nonsymmetric cross sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov's assumptions and invoking the inextensibility condition, the total potential energy is derived from the principle of linearized virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis so that the coupled terms of bending and torsion are added to the elastic strain energy. Also, the potential energy due to initial stress resultants is consistently derived corresponding to the semitangential rotation and moment. Analytical solutions are newly derived for in-plane and lateral-torsional buckling of monosymmetric thin-walled curved beams subjected to pure bending or uniform compression with simply supported conditions. In a companion paper, finite-element procedures for spatial buckling analysis of thin-walled circular curved beams under arbitrary boundary conditions are developed by using thin-walled straight and curved beam elements with nonsymmetric sections. Numerical examples are presented to demonstrate the accuracy and the practical usefulness of the analytical and numerical solutions.     

Investigation of Microstructure, Natural Frequencies and Vibration Modes of Dragonfly Wing

In the present work, a thorough investigation on the microstructural and morphological aspects of dragonfly wings was carried out using scanning electron microscope. Then, based on this study and the previous reports, a precise three-dimensional numerical model was developed and natural frequencies and vibration modes of dragonfly forewing were determined by finite element method. The results shown that dragonfly wings are made of a series of adaptive materials, which form a very complex composite structure. This bio-composite fabrication has some unique features and potential benefits. Furthermore, the numerical results show that the first natural frequency of dragonfly wings is about 168 Hz and bending is the predominant deformation mode in this stage. The accuracy of the present analysis is verified by comparison of calculated results with experimental data.This paper may be helpful for micro aerial vehicle design concerning dynamic response.     

Numerical Analysis and Experimental Study of Buckling Behavior of Steel Cylindrical Panels

The effects of the length, sector angle and different boundary conditions on the buckling load and post buckling behavior of cylindrical panels have been investigated using experimental and numerical methods. The experimental tests have been performed using a servo hydraulic machine and for numerical analysis, Abaqus finite element package has been used. The numerical results are in good agreement with the experimental tests.     

Deflection Prediction of Steel and FRP-Reinforced Concrete-Filled FRP Tube Beams

This paper presents the results of experimental and theoretical investigations that study the flexural behavior of reinforced concrete-filled fiber-reinforced polymer (FRP) tubes (RCFFTs) beams. The experimental program consists of 10 circular beams [6 RCFFT and 4 control reinforced concrete (RC) beams] with a total length of 2,000?mm, tested under four-point bending load. The experimental results were used to review and verify the applicability of various North American code provisions and some available equations in the literature to predict deflection of RCFFT beams. The measured deflections and the experimental values of the effective moment of inertia were analyzed and compared with those predicted using available models. The results of the analysis indicated that the behavior of steel and FRP-RCFFT beams under the flexural load was significantly different than that of steel and FRP-RC members. This is attributed to the confining effect of the FRP tubes and their axial contribution. This confining behavior in turn enhanced the overall flexural behavior and improved the tension stiffening of RCFFT beams. For that, the predicted tension stiffening of steel and FRP-RCFFT beams using the conventional equations (steel or FRP-RC member) underestimates the flexural response; therefore, the predicted deflections are overestimated. Based on the analysis of the test results, the Branson’s equation for the effective moment of inertia of RC structures is modified, and new equations are developed to accurately predict the deflection of concrete-filled FRP tube (CFFT) beams reinforced with steel or FRP bars.     

Use of Mixed-Mode Fracture Interfaces for the Modeling of Large-Scale FRP-Strengthened Beams

In this study, numerical models of fiber-reinforced polymer (FRP)-strengthened beams were developed using nonlinear fracture mechanics for the modeling of the concrete-FRP (longitudinal and U-wrap) interfaces. Mode 1, Mode 2, and mixed-mode interfacial behaviors were considered. Results from the finite-element models were compared with experimental tests of large-scale strengthened beams using FRP U-wraps as anchors. The numerical program assessed the effect of the interfacial modeling in the global and local responses. A parametric study was conducted to determine the effect of additional longitudinal FRP sheets in strengthened beams with and without FRP U-wraps. Results from this study indicate that the use of a mixed-mode concrete-FRP interface is a robust numerical approach for the prediction of the global and local responses of large-scale FRP-strengthened beams. The parametric study shows that the use of FRP U-wraps could improve the strength and ductility of the FRP-strengthened beams by changing their failure mode and deflection response. Appropriate modeling of the concrete-FRP interfaces is needed to successfully predict these effects.     

Matrix Analysis of Shear Lag and Shear Deformation in Thin-Walled Box Beams

This paper presents an initial value solution of the static equilibrium differential equations of thin-walled box beams, considering both shear lag and shear deformation. This solution was used to establish the related finite element stiffness matrix and equivalent nodal forces vector. In the procedure a special shear-lag-induced bimoment is introduced, so that the analysis of shear lag and shear deformation of thin-walled box beams is admitted into the program system of the matrix-displacement method. The present procedure can be used to analyze accurately the shear lag and shear deformation effects for thin-walled box beams, especially for some complex structures (such as continuous box girders and box beams with varying cross section, etc.). The numerical results obtained by the present procedure are consistent with the results of model tests and predictions of the finite shell element method or finite difference approach.     

Recurrent Single Delaminated Beam Model for Vibration Analysis of Multidelaminated Beams

Free vibration analysis of a through-width multidelaminated beam is performed in the present study. Multiple delaminations are assumed to spread from the top through the thickness direction of the beam. The natural frequencies of the multidelaminated beams are obtained from a recurrent single delaminated beam (RSDB) model, which is the subsingle delaminated beam from the top surface of a global beam. Each frequency equation for the RSDB with unknown boundary conditions is obtained through continuity conditions. Then this result is updated to the next one. With these sequential operations, the final frequency equation of the multidelaminated beams is obtained for both end boundary conditions of the global beam. The numerical results for the beams are compared with those of finite element analysis to give the reliance on the proposed model and to investigate the effects of the shape, number, and size of multidelaminations on the natural frequency. It was shown that the variations in the natural frequency for the multidelaminated beams were significantly affected by the delamination length.     

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.     

Experimental Evaluation of Compressive Behavior of Orthotropic Steel Plates for the New San Francisco–Oakland Bay Bridge

Compression tests were conducted on two reduced-scale orthotropic plates to verify the design strength of steel box girders for the new San Francisco–Oakland Bay Bridge. The first specimen was composed of three longitudinal closed ribs and a top deck plate. It failed in global buckling, followed by local buckling in the deck plate and ribs. The second specimen, which was composed of four longitudinal T-shaped ribs and a bottom deck plate, experienced global buckling as well as local buckling in the ribs and the deck plate. The ultimate strength and failure mode of both specimens were evaluated by two bridge design specifications: the 1998 AASHTO load and resistance factor design specification and the 2002 Japanese JRA specification. Findings from code comparisons showed that: (1) Sufficient flexural rigidity of ribs were provided for both specimens; (2) the JRA specification slightly overestimated the ultimate strength of both specimens; and (3) neither specifications predicted the observed buckling sequence in Specimen 2. A general-purpose nonlinear finite element analysis program (ABAQUS) was used to perform correlation study. The analysis showed that the ultimate strength and postbuckling behavior of the specimens could be reliably predicted when both the effects of residual stresses and initial geometric imperfections were considered in the model.     

Shear Buckling Resistance of GFRP Plate Girders

Thin webs of glass-fiber-reinforced polymer (GFRP) girders are sensitive to shear buckling, which can be considered an in-plane biaxial compression-tension buckling problem, according to the rotated stress field theory. An extensive experimental study was performed, which shows that an increasing transverse tension load significantly increases the buckling and ultimate loads caused by a decrease in the initial imperfections and additional stabilizing effects. The stacking sequence also greatly influenced the buckling behavior. Higher bending stiffness in the compression direction increased the buckling and ultimate loads, while higher bending stiffness in the tension direction changed the buckling mode shape. The general solution obtained using the Fok model accurately modeled the experimental results, while the simplified solution (modified Southwell method) provided accurate results only at higher tension loads.     

Shear-Flexural Buckling of Cantilever Columns under Uniformly Distributed Load

Approximate buckling formulas for shear–flexural buckling of cantilever columns subjected to a uniformly distributed load are derived, based on Timoshenko’s energy method. In this method the deflection curve at buckling is approximated by a trial function. Instead of trying to describe all possible buckling modes with one trial function, two trial functions are used: one to describe shear dominated localized buckling, another to describe bending dominated global buckling. It is investigated whether the bending dominated global buckling modes can best be described using polynomial functions, trigonometric functions, or a function defined by the lateral (flexural and shear) deflection of the cantilever column under uniformly distributed lateral load. The results of the derived formulas are compared to the exact solution and other approximate buckling formulas found in the literature. Attention is drawn to the fact that the shear–flexural buckling load cannot exceed the shear buckling load.     

p-y Criterion for Rock Mass

Drilled shafts socketed in rock mass have been used frequently as a foundation system to support both vertical and lateral loads. Traditionally, the lateral interaction between the drilled shaft and the surrounding rock medium has been characterized by means of nonlinear p-y curves; however, there is a lack of well verified p-y criterion for rock mass. In this paper, a hyperbolic p-y criterion is developed based on both theoretical derivations and numerical (finite element) parametric analysis results. The methods for determining pertinent rock parameters needed for constructing the proposed p-y curves are presented in the paper. Two full-scale lateral load tests on large diameter, fully instrumented drilled shafts socketed in rock conducted by the writers, together with additional four load test results reported by Gabr et al. were used to validate the applicability of the proposed hyperbolic p-y curves for rock mass. The comparisons between the computed shaft responses (both deflections and bending moments) and the actual measured responses are considered acceptable.     

Postbuckling of Shear Deformable Laminated Cylindrical Shells

A postbuckling analysis is presented for a shear deformable laminated cylindrical shell of finite length subjected to compressive axial loads. The governing equations are based on Reddy’s higher-order shear deformation shell theory with a von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, cross-ply laminated cylindrical shells. The effects of transverse shear deformation, shell geometric parameters, total number of plies, fiber orientation, and initial geometric imperfections are studied.