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.     

Spatial Stability of Shear Deformable Nonsymmetric Thin-Walled Curved Beams: A Centroid-Shear Center Formulation

An improved shear deformable curved beam theory to overcome the drawback of currently available beam theories is newly proposed for the spatially coupled stability analysis of thin-walled curved beams with nonsymmetric cross sections. For this, the displacement field is introduced considering the second order terms of semitangential rotations. Next the elastic strain energy is newly derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. Then the potential energy due to initial stress resultants is consistently derived with accurate calculation of the Wagner effect. Finally, equilibrium equations and force–deformation relations are obtained using a stationary condition of total potential energy. The closed-form solutions for in-plane and out-of-plane buckling of curved beams subjected to uniform compression and pure bending are newly derived. Additionally, finite-element procedures are developed by using curved beam elements with arbitrary thin-walled sections. In order to illustrate the accuracy and the practical usefulness of this study, closed-form and numerical solutions for spatial buckling are compared with results by available references and ABAQUS’ shell elements.     

Spatial Postbuckling Analysis of Nonsymmetric Thin-Walled Frames.?I: Theoretical Considerations Based on Semitangential Property

To present the spatial postbuckling analysis procedures of shear deformable thin-walled space frames with nonsymmetric cross sections, theoretical considerations based on the semitangential rotation and the semitangential moment are presented. First, similarity and difference between Rodriguez' rotations and semitangential rotations are addressed. Next, the improved displacement field is introduced using the second-order terms of semitangential rotations and rotational properties of off-axis loads and conservative moments are discussed based on the proposed displacement field. Finally, it is deduced that the resulting potential energy due to stress resultants corresponds to semitangential bending and torsional moments. In a companion paper, the elastic strain energy including bending-torsion coupled terms and shear deformation effects is newly derived and a clearly consistent finite-element procedure is presented based on the updated Lagrangian corotational formulation. Tangent stiffness matrices of the thin-walled space frame element are derived using Hermitian polynomials considering shear deformation effects, and a new scheme to evaluate incremental member forces and load correction stiffness matrices due to off-axis loads is presented and its physical meaning is addressed. Furthermore, finite-element solutions displaying spatial postbuckling behaviors are evaluated and compared with available solutions.     

Spatial Postbuckling Analysis of Nonsymmetric Thin-Walled Frames.?II: Geometrically Nonlinear FE Procedures

In a companion paper, transformation rules of Rodriguez' finite rotations and semitangential rotations are derived, respectively, and their rotational properties are discussed. The shear deformable displacement field of nonsymmetric thin-walled space frames is introduced based on semitangential rotations and the potential energy corresponding to semitangential internal moments are consistently derived using the proposed displacement field. In this paper, for spatial postbuckling analysis of thin-walled space frames, the elastic strain energy including bending-torsion coupled terms and shear deformation effects is derived using transformations of displacements and stress resultants defined at the centroid and the centroid-shear center, respectively. Tangent stiffness matrices of the frame element are derived by using Hermitian polynomials including shear effects, and an improved corotational formulation is presented by separating rigid body motions and pure deformations from incremental displacements, calculating the corresponding generalized forces and updating direction cosines of frame elements. In addition, a scheme to evaluate load correction stiffness matrices due to the off-axis loading and conservative moments is addressed. FE solutions for the postbuckling are presented and compared with results by ABAQUS shell models.     

Shear Lag of Thin-Walled Curved Box Girder Bridges

This note investigates the influence of shear lag for thin-walled curved box girders, including longitudinal warping. The longitudinal warping displacement functions of the flange slabs are approximated by a cubic parabolic curve instead of a quadratic curve of Reissner's method. On the basis of the thin-walled curved bar theory and the potential variational principle, the equations of equilibrium considering the shear lag, bending, and torsion (St. Venant and warping) for a thin-walled curved box girder are established. The closed-form solutions of the equations are derived, and Vlasov's equation is further developed. The obtained formulas are applied to calculate the shear lag effects for curved box girder bridges. Numerical examples are presented to verify the accuracy and applicability of the present method.     

Analytical and Experimental Study of Lateral and Distortional Buckling of FRP Wide-Flange Beams

A combined analytical and experimental evaluation of flexural-torsional and lateral-distortional buckling of fiber-reinforced plastic (FRP) composite wide-flange (WF) beams is presented. Based on energy principles, the total potential energy equations for instability of FRP WF sections are derived using the nonlinear elastic theory. For the analysis of lateral-distortional buckling, a fifth-order polynomial shape function is adopted to model the deformed shape of web panels. The models are validated by testing two geometrically identical FRP WF beams but with distinct material architectures produced by the pultrusion process. The beams are tested under midspan concentrated loads to evaluate their flexural-torsional and lateral-distortional buckling responses. To detect rotations of the midspan cross sections and onset of critical buckling loads, horizontal transverse bars are attached to the beam's flanges, and the bar ends are connected to linear variable differential transducers (LVDTs). For the same purpose, we use strain gauges bonded to the upper and lower surfaces near to the free edges of the top flange. A good agreement between the proposed analytical approach and experimental and finite-element analyses results is obtained, and simplified engineering equations for flexural-torsional buckling are formulated. The proposed analytical solutions can be used to predict flexural-torsional and lateral-distortional buckling loads for other FRP shapes and to formulate simplified design equations.     

Analytical Model of Curved I-Girder Web Panels Subjected to Bending

Curvature greatly complicates the behavior of curved plate girders used in bridges. The out-of-plane “bulging” displacement of the curved web results in an increase in stress, which must be considered in the design of plate girders with significant curvature. The currently used Guide Specifications for Horizontally Curved Bridges provides web slenderness reduction equations that account for curvature effects, but these equations were based on a regression of limited data from experimental and unit-strip analyses conducted in the early 1970s. This paper presents a theoretically pure analytical model that can be used to predict the transverse displacement and induced plate bending stresses of curved I-shaped plate girder web panels subjected to bending. Several boundary conditions are demonstrated and compared, and the finite-element method is used to verify the closed-form solutions. The effects of curvature on the elastic buckling behavior of curved web panels is also presented. Furthermore, a comprehensive literature review is presented, including numerous Japanese publications not readily available to American researchers.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

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.     

Lateral–Torsional Buckling of Singly Symmetric Tapered Beams: Theory and Applications

A general variational formulation to analyze the elastic lateral–torsional buckling (LTB) behavior of singly symmetric thin-walled tapered beams is presented, numerically implemented, validated and illustrated. It (1) begins with a precise geometrical definition of a tapered beam; (2) extends the kinematical assumptions traditionally adopted to study the LTB of prismatic beams; (3) includes a careful derivation of the beam total potential energy; and (4) employs Trefftz’s criterion to ensure the beam adjacent equilibrium. In order to validate and illustrate the application and capabilities of the proposed formulation, several numerical results are presented, discussed and, when possible, also compared with values reported by other authors. These results (1) are obtained by means of the Rayleigh–Ritz method, using trigonometric functions to approximate the beam critical buckling mode, and (2) concern the critical moments of doubly and singly symmetric web-tapered I-section simply supported beams and cantilevers acted by point loads. In particular, one shows that modeling a tapered beam as an assembly of prismatic beam segments is conceptually inconsistent and may lead to rather inaccurate (safe or unsafe) results. Finally, it is worth mentioning that the paper includes a state-of-the-art review concerning one-dimensional analytical formulations for the LTB behavior of tapered beams.     

Prediction of Maximum Section Flattening of Thin-walled Circular Steel Tube in Continuous Rotary Straightening Process

Cross-sectional ovalization of thin-walled circular steel tube because of large plastic bending,also known as the Brazier effect,usually occurs during the initial stage of tube′s continuous rotary straightening process.The amount of ovalization,defined as maximal cross section flattening,is an important technical parameter in tube′s straightening process to control tube′s bending deformation and prevent buckling.However,for the lack of special analytical model,the maximal section flattening was determined in accordance with the specified charts developed by experienced operators on the basis of experimental data;thus,it was inevitable that the localized buckling might occur during some actual straightening operations.New normal strain component formulas were derived based on the thin shell theory.Then,strain energy of thin-walled tube(per unit length)was obtained using the elastic-plastic theory.A rational model for predicting the maximal section flattening of the thin-walled circular steel tube under its straightening process was presented by the principle of minimum potential energy.The new model was validated by experiments and numerical simulations.The results show that the new model agrees well with the experiments and the numerical simulations with error of less than 10%.This new model was expected to find its potential application in thin-walled steel tube straightening machine design.     

Buckling of Vertical Cylindrical Shells Under Combined End Pressure and Body Force

This paper is concerned with the elastic buckling of vertical cylindrical shells under combined end pressure and body force. Such buckling problems are encountered when cylindrical shells are used in a high-g environment such as the launching of rockets and missiles under high-propulsive power. The vertical shells may have any combination of free, simply supported, and clamped ends. Based on the Goldenveizer-Novozhilov thin shell theory, the total potential energy functional is presented and the buckling problem is solved using the Ritz method. Highlight in the formulation is the importance of the correct potential energy functional which includes the shell shortening due to the circumferential displacement. The omission of this contributing term leads to erroneous buckling solutions when the cylindrical shell is not of moderate length (length-to-radius ratio smaller than 0.7 or larger than 3). New solutions for body-force buckling parameters are presented for stubby cylindrical shells to long tube-like shells that approach the behavior of columns. The effects of the shell thickness and length on buckling parameter are also investigated.     

Complementary Energy Based Formulation for Torsional Buckling of Columns

A unique formulation for the elastic torsional buckling analysis of columns is developed in this paper based on the principle of stationary complementary energy. It is well known that in displacement based numerical formulations, discretization errors lead to stiffer behavior; hence convergence from above. On the other hand, discretization errors in complementary energy based numerical formulations lead to softer behavior in linear elasticity problems, which is a desired feature from the engineering view point. However, complementary energy based formulations can only overpredict the buckling loads for the flexural buckling problems of columns unless the physical conditions are compromised. In this study a formulation based on the principle of stationary complementary energy is considered for the elastic torsional buckling analysis of columns. The complementary energy expression is obtained from the well known total potential energy functional by using Frederichs’ transformation. In contrast to flexural buckling analysis of columns, it is shown that when the principle of stationary complementary energy is used, the torsional buckling loads can be underpredicted. A mathematical proof is provided to elucidate this property. The convergence behavior of the approximate solutions is illustrated through numerical examples for several columns with different boundary conditions.     

Postbuckling of Elastic Beams Considering Higher Order Strain Terms

Nonlinear relations between the beam displacement and generalized strain measures, which have basic effects on postbuckling behavior of elastic beams, are presented. The complex coupling phenomena associated with the higher order strain terms is reviewed for the special case of planar and rectilinear pinned-pinned beams. Special consideration was made for the physical assumptions used in the various column-beam models. A natural hierarchy results yielding that all the higher order terms can, for a specific beam formulation, be steadily obtained by dissimilar polynomial approximations of the generalized strains. The asymptotic expansions method and the minimum energy criterion are used to perform analytical calculation of the postbifurcation equilibrium path at the neighborhood of a bifurcation point when only a unique buckling mode is assumed to occur. As a result, postbuckling branches are easily obtained even when accounting for both beam centerline extensional deformation and shear strain. They show that the critical load is scarcely affected by the higher order strain terms unlike the postbuckling paths which are found to be very sensitive to them.     

Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

This paper is concerned with the bending problem of micro- and nanobeams based on the Eringen nonlocal elasticity theory and Timoshenko beam theory. In the former theory, the small-scale effect is taken into consideration while the effect of transverse shear deformation is accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. General solutions for the deflection, rotation, and stress resultants are presented for transversely loaded beams. In addition, specialized bending solutions are given for beams with various end conditions. These solutions account for a better representation of the bending behavior of short, stubby, micro- and nanobeams where the small-scale effect and transverse shear deformation are significant. Considering particular loading and boundary conditions, the effects of small-scale and shear deformation on the bending results may be observed because of the analytical forms of the solutions.     

Buckling of Thin-Walled Eccentric Compressive Members Considering Shear Lag

Based on the displacement variational principle, this paper presents a general consistent method, called the spline finite member element method, for buckling analysis of thin-walled eccentric compressive members with arbitrary cross sections considering shear lag. A transformed B3-spline function presented in this paper is used to simulate the warping displacements along the cross section of the thin-walled member. Compared with the results from classical theory, the numerical results proposed in this paper demonstrate the versatility and accuracy of the proposed method.     

Elastic-Plastic Model of Pinned Beams Subjected to Impulsive Loading

An elastic-plastic dynamic analysis of simply supported beams with end membrane restraints subjected to impulsive loading is developed. The beam is elastically curved. The model takes into account elastic and plastic deformations and their effect on the stretch force and the distribution of inertia forces. Instantaneous plastification of the midspan zone due to bending and axial actions is considered. The effect of the length of the plastic zone is discussed. Equations of motion are derived by virtual work. Using the yield condition, the number of these equations is reduced by relating the variations of the displacement variables. This analysis is compared to the standard form of Lagrange's equation of motion showing that the latter is not energy conservative for this case where the deflection shape is nonlinearly dependent on the generalized displacements. Test results using a spring-powered apparatus are presented. Strain rate sensitivity is accounted for as plastic damping. The model's results are compared to test and rigid-segment model results. The comparison shows that the tests are sensitive to the curvature of the axis.     

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.     

Strength of Stiffened Flanges in Horizontally Curved Box Girders

Longitudinal stiffeners are often attached to increase the buckling strength of thin-walled box girder flanges. The minimum required rigidity for longitudinal stiffeners for curved box girder flanges is given by the AASHTO “Guide specifications for horizontally curved steel girder highway bridges.” However, this requirement is simply adopted from the current AASHTO specifications for straight stiffened flanges. The validity of this requirement has been questioned in a series of recent studies. The effect of important design parameters on the minimum required stiffener rigidity is investigated numerically in this study by examining the prebuckling stress distribution and elastic and inelastic buckling stresses of horizontally curved stiffened flanges. In order to characterize and quantify the analytically collected data, a series of parametric studies were performed. A new equation for the minimum required rigidity for the longitudinal stiffeners is derived from regression analyses. Through the evaluation of a few selected case studies and a design example, the validity and reliability of the proposed new equation is demonstrated.     

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.