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.     

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.     

Thermomechanical Postbuckling Analysis of Cross-Ply Laminated Cylindrical Shell Panels

The postbuckling analysis of symmetric and antisymmetric cross-ply laminated cylindrical shell panels subjected to thermomechanical loading is examined in this paper. The formulation is based on an extension of Reissner’s shallow shell simplifications and accounts for parabolic distribution of transverse shear strains. Adopting a multiterm Galerkin’s method, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations. The nonlinear equilibrium paths through limit points are traced using the Newton–Raphson method in conjunction with Riks approach. Numerical results are presented for symmetric [?start0/90/0end?] and antisymmetric [?start0/90end?] cross-ply laminated cylindrical shell panels, that illustrate the influence of mechanical edge loads, lateral distributed load, initial imperfection, and temperature field on the limit loads and snap-through behavior.     

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%.     

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.     

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).     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

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.     

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.     

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.     

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.     

Buckling of a Weakened Infinite Beam on an Elastic Foundation

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

Hierarchical High-Fidelity Analysis Methodology for Buckling Critical Structures

A hierarchical high-fidelity analysis methodology for predicting the critical buckling load of compression-loaded thin-walled isotropic shells is described. This hierarchical procedure includes three levels of fidelity for the analysis. Level 1 assumes that the buckling load can be predicted by the classical shell solution with simply supported boundary condition, and with a linear membrane prebuckling solution. Level 2 includes the effects of a nonlinear prebuckling solution and the effects of traditional clamped or simply supported boundary conditions. Level 3 includes the nonlinear interaction between nearly simultaneous buckling modes and the effects of boundary imperfections and general boundary conditions. Various deterministic and probabilistic approaches are used to account for the degrading effects of unavoidable shell-wall geometric imperfections. The results from the three solution levels are compared with experimental results, and the effects of the assumptions and approximations used for the three solution levels are discussed. This hierarchical analysis approach can be used in the design process to converge rapidly to an accurate prediction of the expected buckling load of a thin-shell design problem.     

Nonlinear Buckling and Postbuckling of Cable-Stiffened Prestressed Domes

Nonlinear buckling phenomenon of a novel type of structure, namely, a prestressed dome, is investigated using a nonlinear finite-element model. A prestressed dome is formed by buckling its individual flat members into arch frame works, and then the structure may be stiffened by cable loops in the circumferential direction. A corotational formulation of a 3D beam element, and a cable element, which is modeled as a catenary between connected points in the dome, are used to develop an algorithm for nonlinear stability analysis of the system by considering large displacements and rotations. The incremental load technique using a Newton-Raphson iteration scheme in conjunction with Crisfield's modified arc-length method is utilized to trace the nonlinear path of equilibrium. Buckling of prestressed domes with different numbers and locations of cable stiffeners are studied, and the results show that skeletal domes with stiffeners buckle at much higher loads than the corresponding unstiffened domes.     

Tracing Secondary Equilibrium Paths of Elastic Framed Structures

For a framed structure that is subjected to bifurcation buckling, it may be useful to trace its secondary equilibrium path to gauge its sensitivity to geometric imperfections or to study the nature of load shedding from the buckled structure. For this purpose, a substantial number of branch-switching algorithms for tracing the secondary equilibrium paths of elastic structures have been proposed in the literature. However, virtually all of the published algorithms have heavy mathematical overtones that are not readily appreciated by practicing structural engineers. This paper presents a simple and efficient branch-switching algorithm that is explained in more easily understood terms. The proposed algorithm is demonstrated through numerical examples to be effective in tracing the secondary equilibrium paths of various framed structures with different types of postbuckling behaviors.     

Nonlinear Static Response and Free Vibration Analysis of Doubly Curved Cross-Ply Panels

In this paper, the nonlinear static and free vibration analyses of doubly curved cross-ply laminated panels subjected to thermomechanical loading are examined. The shell theory adopted in the present case is an extension of Reissner’s shallow shell simplifications that accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions. A multiterm Galerkin’s method is adopted to solve the governing nonlinear partial differential equations. The static equilibrium paths are traced using the Newton-Raphson method in conjunction with the Riks approach to overcome the limit points. The free vibration frequencies about a static equilibrium state of a deformed panel are reported by solving the linear eigenvalue problem. Analytical results are presented for symmetric [0/90/0] and antisymmetric [0/90] cross-ply laminated doubly curved panels that illustrate the influence of geometric properties, in-plane edge boundary conditions, lateral distributed load and temperature field on the nonlinear behavior and natural frequencies.     

Effects of Random Material Properties on Buckling of Composite Plates

Composites exhibit a considerable amount of variation in their material properties because their fabrication∕manufacturing process involves a large number of parameters that cannot be controlled effectively. In the present study, the material properties have been modeled as random variables for better prediction of the system behavior. The classical laminate theory and first-order and higher-order shear deformation theories have been employed in deriving the governing equations for buckling of laminated rectangular plates. A mean-centered first-order perturbation technique has been used to find the second-order statistics of the buckling load. The approach has been validated by comparison with results of Monte Carlo simulation. Typical results have been presented for a plate with all edges simply supported. The effectiveness of the theories in predicting the buckling load dispersions has been examined. The sensitivity of buckling loads to change in the standard deviation of random material properties and to system parameters—side-to-thickness ratio and aspect ratio—has been examined for cross-ply symmetric and antisymmetric laminates.     

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).     

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.