Influence of Shear Deformation on Buckling of Pultruded Fiber Reinforced Plastic Profiles

Theoretical studies of the influence of shear deformation on the flexural, torsional, and lateral buckling of pultruded fiber reinforced plastic (FRP)-I-profiles are presented. Theoretical developments are based on the governing energy equations and full section member properties. The solution for flexural buckling is consistent with the established solution based on the governing differential equation. The new solutions for torsional and lateral buckling incorporate a reduction factor similar to that for flexural buckling. The solution for lateral buckling also incorporates the influence of prebuckling displacements. Closed form solutions for a series of simply supported, pultruded FRP I-profiles, based on experimentally determined full section flexural and torsional properties, indicate the following conclusions. For members subjected to axial compression, shear deformation can reduce the elastic flexural and torsional buckling loads by up to approximately 15% and 10%, respectively. For members subjected to bending, prebuckling displacements can increase the buckling moments by over 20% while shear deformation decreases the buckling moments by less than 5%.     

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.     

Euler Critical Force Calculation for Laced Columns

The paper presents a method of solving the buckling problem of laced column as a statically indeterminate structure without analyzing determinants of high order. The flexural and torsional buckling problems of laced column are reduced to the two-point boundary value problem for a difference equation system. The value of Euler critical load is determined as a result of analyzing the fourth order determinant for column with any degree of static indeterminacy. The solution is based on the method of initial values. Stability of columns with any types of lattice (crosswise, serpentine, with batten struts); with any number of lattice panels and with variable lattice spacing can be examined by this manner. The analogy between the flexural and torsional buckling of the laced column is established. It enables one to use the same relations for consideration of both kinds of buckling. The obtained numerical results show that the Euler critical loads calculated by this method can be substantially differed from those based on the approximated Engesser’s approach. A PC program for checking stability of laced column by designer can be developed on the basis of the present method.     

Equilibrium-Based Finite-Element Formulation for the Geometrically Exact Analysis of Planar Framed Structures

This paper addresses the development of a hybrid-mixed finite-element formulation for the geometrically exact quasi-static analysis of elastic planar framed structures, modeled using the two-dimensional Reissner beam theory. The proposed formulation relies on a modified principle of complementary energy, which involves, as independent variables, the generalized vectors of stress resultants and displacements and, in addition, a set of Lagrange multipliers used to enforce the stress continuity between elements. The adopted finite-element discretization produces numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the static boundary conditions. It consists, therefore, in a true equilibrium formulation for arbitrarily large displacements and rotations. Furthermore, as it does not suffer from shear locking or any other artificial stiffening phenomena, it may be regarded as an alternative to the standard displacement-based formulation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions are compared with those obtained using the standard two-node displacement-based formulation. Numerical analyses of convergence of the proposed finite-element formulation are also included.     

Novel Formulation to Study Thermal Postbuckling of Circular Plates with Edges Elastically Restrained against Rotation

A novel formulation is used to study the thermal postbuckling behavior of circular plates, with the edges supported to not have lateral deflection and elastically restrained against rotation. The elastic restraint is mathematically represented by an elastic rotational spring. The circular plate is subjected to a uniform edge compressive radial load, developed because of a uniform temperature rise. The formulation is on the basis of on the radial tensile load developed in the plate because of the large deflections of the plate with edges immovable in the plane normal to the edge and the linear buckling load corresponding to the uniform edge radial compressive load. The developed radial tensile load is obtained by using Berger’s approximation. The numerical results obtained from the present investigation in terms of the ratios of the postbuckling to the buckling loads for several rotational spring stiffness values compare well with those obtained by using the versatile finite-element analysis.     

Tangent Stiffness Equations for Laterally Distributed Loaded Members

Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Thus, the tangent stiffness equations take into consideration axial forces, bowing effect, and laterally distributed loads. As a numerical example, elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated to compare the results obtained from the present method with those from the conventional matrix method in which the distributed loads are considered as a series of concentrated loads at additional intermediate nodes of a member. Furthermore, buckling tests were carried out to confirm the equations derived as well as to clarify the buckling behavior of space frame structures. In conclusion, it can be said that the new equations can provide a good efficient way of estimating the equilibrium paths and buckling loads. They can also lead to a significant savings in core storage and computing time required for the analysis of space frame structures.     

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.     

Behavior of Corrugated Web I-Girders under In-Plane Loads

A theoretical formulation of the linear elastic in-plane and torsional behavior of corrugated web I-girders under in-plane loads is presented. A typical corrugated web steel I-girder consists of two steel flanges welded to a corrugated steel web. Under a set of simplifying assumptions, the equilibrium of an infinitesimal length of a corrugated web I-girder is studied, and the cross-sectional stresses and stress resultants due to primary bending moment and shear are deduced. The analysis shows that a corrugated web I-girder will twist out-of-plane simultaneously as it deflects in-plane under the action of in-plane loads. In the paper, the in-plane bending behavior is analyzed using conventional beam theory, whereas the out-of-plane torsional behavior is analyzed as a flange transverse bending problem. The results for a simply supported span subjected to a uniformly distributed load are presented. Finally, finite element analysis results are presented and compared to the theoretical results for validation.     

Large-Deflection Stability of Slender Beam-Columns with Both Ends Partially Restrained against Rotation

The large-deflection analysis and postbuckling behavior of laterally braced or unbraced slender beam columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by the senior writer. The proposed method can be used in the large-deflection elastic analysis and postbuckling behavior of slender beam columns with rigid, semirigid, and simple flexural connections and both ends. Only bending strains are considered, i.e., the effects of axial and shear strains are neglected. An example is included that shows the effects of flexible connections at both ends on the large-deflection analysis and postbuckling behavior of slender beam columns.     

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.     

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.     

Closed-Form Solutions for Buckling of Long Anisotropic Plates with Various Boundary Conditions under Axial Compression

Closed-form solutions for buckling of long plates with flexural/twist anisotropy with the short edges simply supported and with the longitudinal edges simply supported, clamped, or elastically restrained in rotation under axial compression are presented. An energy method (Rayleigh–Ritz) is employed to obtain the critical buckling loads. The critical buckling loads are expressed in terms of minimum nondimensional buckling coefficients and stiffness parameters. The new closed-form solutions show an excellent agreement when compared to existing solutions and finite-element analysis. Due to their simplicity and accuracy, the new closed-form solutions can be confidently used as an alternative to computationally expensive structural analysis to assess buckling in the preliminary design phase of composite structures.     

Static Stability of Beam-Columns under Combined Conservative and Nonconservative End Forces: Effects of Semirigid Connections

Stability criteria that evaluate the effects of combined conservative and nonconservative end axial forces on the elastic divergence buckling load of prismatic beam-columns with semirigid connections is presented using the classic static equilibrium method. The proposed method and stability equations follow the same format and classification of ideal beam-columns under gravity loads presented previously by Aristizabal-Ochoa in 1996 and 1997. Criterion is also given to determine the minimum lateral bracing required by beam-columns with semirigid connections to achieve “braced” buckling (i.e., with sidesway inhibited). Analytical results obtained from three cases of cantilever columns presented in this paper indicate that: (1) the proposed method captures the limit on the range of applicability of the Euler’s method in the stability analysis of beam-columns subjected to simultaneous combinations of conservative and nonconservative loads. The static method as proposed herein can give the correct solution to the stability of beam-columns within a wide range of combinations of conservative and nonconservative axial loads without the need to investigate their small oscillation behavior about the equilibrium position; and (2) dynamic instability or flutter starts to take place when the static critical loads corresponding to the first and second mode of buckling of the column become identical to each other. “Flutter” in these examples is caused by the presence of nonconservative axial forces (tension or compression) and the softening of both the flexural restraints and the lateral bracing. In addition, the “transition” from static instability (with sidesway and critical zero frequency) to dynamic instability (with no sidesway or purely imaginary sidesway frequencies) was determined using static equilibrium. It was found also that the static critical load under braced conditions (i.e., with sidesway inhibited) is the upper bound of the dynamic buckling load of a cantilever column under nonconservative compressive forces. Analytical studies indicate the buckling load of a beam-column is not only a function of the degrees of fixity (ρa and ρb), but also of the types and relative intensities of the applied end forces (Pci and Pfj), their application parameters (ci, ηj, and ξj), and the lateral bracing provided by other members (SΔ).     

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.     

Buckling Reversals of Axially Restrained Imperfect Beam-Column

The stability and postbuckling analysis of an axially restrained prismatic beam-column with single symmetrical cross section and an initial imperfection (camber) is presented. The proposed model is that by Timoshenko but including the effects of small camber of any form and any transverse loading. This model can be used to (1) determine the prebuckling elastic response and initial buckling load; (2) explain the postbuckling elastic behavior including the phenomena of snap-through, snap-back, and reversals of deflections; and (3) determine the effects of high modes of buckling on the stability behavior of beam-columns with small camber. In addition, closed-form equations corresponding to the transverse and axial deflections caused by any transverse loads on a partially restrained beam-column are developed as well as the bending stress along its span. It is shown that the prebuckling, stability, and postbuckling behavior of a beam-column depends on (1) the cross section and material properties (area, inertia, and elastic modulus); (2) the magnitude of the end restraints; and (3) the type and lack of symmetry about the beam-column midspan of the applied transverse loads and initial camber or imperfection. For transverse loads that are not symmetrical with respect to the beam-column midspan, the pre- and postbuckling criterion given by Timoshenko might yield significant errors in both the critical load and deflections. Three examples are presented that show the effectiveness and validity of the proposed equations and the limitations of Timoshenko's criteria.     

Boundary Condition Effects in Buckling of “Soft” Core Sandwich Panels

The effects of boundary conditions on the critical load level and the corresponding deflection mode shape of sandwich panels with a “soft” core due to in-plane loads are presented. The study is conducted using a closed-form high-order linearized buckling analysis that includes the influence of the transverse flexibility of the core as well as of the localized effects on the overall sandwich panel behavior, and allows the use of different boundary conditions for the upper and lower skin at the same section. The panel construction is general and consists of two skins (not necessarily identical), metallic or composite-laminated symmetric, and a soft core made of foam or a low-strength honeycomb. The closed-form high-order analysis yields the general buckling behavior of the structure, which means that the solutions obtained allow for interaction between the skins and the core. The solutions are general and are not based on separation of the buckling response on several types of uncoupled buckling modes, such as overall buckling, skins wrinkling, etc., as commonly used in the literature. The numerical scheme consists of finite differences to approximate the governing equations of the closed-form high-order formulation and to transform the set of linearized governing differential equations into an eigenvalue problem that is solved using the deflated iterative Arnoldi procedure. The influence of a general type of boundary conditions, including different conditions throughout the height of the same section and nonidentical conditions at the upper and lower skin, as well as of the core properties, on the buckling behavior of the sandwich panels is considered. The discrepancy between the Timoshenko-Reissner model and the present formulation is discussed. In particular, a partial fixity phenomenon due to the existence of the pinned boundary conditions, i.e., simply supported conditions, at the upper and lower skins at the edge is demonstrated. It is shown that the core properties affect the buckling loads and the corresponding modes of the panel in such a way that the structures with identical boundary conditions but with different cores may undergo different types of buckling such as overall and local as well as interactive loss of stability. The effect of an edge concentrated moment, induced by a couple and exerted on the skins only is also studied.     

Buckling Analysis of Nonuniform and Axially Graded Columns with Varying Flexural Rigidity

In this paper, we present a novel analytic approach to solve the buckling instability of Euler-Bernoulli columns with arbitrarily axial nonhomogeneity and/or varying cross section. For various columns including pinned-pinned columns, clamped columns, and cantilevered columns, the governing differential equation for buckling of columns with varying flexural rigidity is reduced to a Fredholm integral equation. Critical buckling load can be exactly determined by requiring that the resulting integral equation has a nontrivial solution. The effectiveness of the method is confirmed by comparing our results with existing closed-form solutions and numerical results. Flexural rigidity may take a majority of functions including polynomials, trigonometric and exponential functions, etc. Examples are given to illustrate the enhancement of the load-carrying capacity of tapered columns for admissible shape profiles with constant volume or weight, and the proposed method is of benefit to optimum design of columns against buckling in engineering applications. This method can be further extended to treat free vibration of nonuniform beams with axially variable material properties.     

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.     

Torsional Stiffness of Prestressing Tendons in Double-T Beams

When a prestressed double-T beam is subjected to torsion, a pair of prestressing tendons resists torsional rotation because of the restoring action of the displaced prestressing tendons. A comprehensive formulation to account for the torsional restoring action of double-T beams is presented, based on Vlasov’s hypothesis of considering warping displacement in an open-section. The deformation energies of prestressing tendons and reinforcing bars are calculated based on the deformed geometry to obtain the total potential energy. A two-noded beam element with seven degrees of freedom per node approximates an axial displacement, two translations, two flexural, and one torsional rotations, and a warping displacement to derive the finite-element equilibrium equations by minimizing the potential energy function. The role of prestressing forces of the tendons on the torsional resistance and the limitations of the traditional transformed section approach are addressed when it is applied to torsional problems. As a numerical example, an existing three-span continuous double-T beam is analyzed, and the bimoment and angle of twist are compared to those calculated using conventional three-dimensional finite-element analysis and the analytical solution of governing differential equations.     

侧向冲击荷载下钢筋混凝土墩柱的性能

采用数值仿真技术建立了足尺钢筋混凝土墩柱精细有限元模型, 分析了侧向冲击荷载下墩柱的动态响应和抗冲击性能, 提出了一种基于截面损伤因子的损伤评估方法, 讨论了不同碰撞参数对钢筋混凝土墩柱破坏模式和损伤机理的影响.结果表明: 冲击荷载下钢筋混凝土墩柱的耗能主要分为接触区域局部耗能和构件整体耗能; 当冲击体的初始动能恒定时, 冲击质量和冲击速度的不同组合会导致钢筋混凝土墩柱损伤破坏机理的显著差异; 基于截面损伤因子的损伤评估方法可以比较准确地描述墩柱的破坏状态.轴压力对墩柱抗撞能力的有利贡献比较有限, 且墩柱随着轴力的增大更易发生剪切破坏; 冲头刚度对碰撞力和墩柱动态响应的影响十分显著.