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.     

Buckling of Variable Section Columns under Axial Loading

In this paper, the static stability of the variable cross section columns, subjected to distributed axial force, is considered. The presented solution is based on the singular perturbation method of Wentzel-Kramers-Brillouin and the column is modeled using Euler-Bernoulli beam theory. Closed-form solutions are obtained for calculation of buckling loads and the corresponding mode shapes. The obtained results are compared with the results in the literature to verify the present approach. Using numerous examples, it is shown that the represented solution has a very good convergence and accuracy for determination of the instability condition.     

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.     

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.     

Postbuckling of Beam Subjected to Intermediate Follower Force

This paper investigates the postbuckling behavior of a simple beam under an intermediate follower force acting in the tangential direction to the centroidal axis of the beam. One end of the beam is pinned, while the other end is attached to a roller support. Two approaches have been used in this study. The first approach is based on the elastica theory. The governing equations are derived and solved analytically for the exact closed form solutions that include the equilibrium configurations of the beam, equilibrium paths, and bending moment distribution of the beam. The exact solutions take the form of elliptic integrals of the first and second kinds. In the second approach, the shooting method is employed to solve a set of nonlinear differential equations with the boundary and intermediate conditions. The equations are integrated by using the Runge–Kutta algorithm. The error norms of the end and intermediate conditions are minimized to within a prescribed tolerance error. A comparison study between the analytical elliptic integral solutions and the numerical shooting method solutions show excellent agreement of results. Special features of the solutions are also highlighted.     

Nonlinear Analysis of Masonry Arches Strengthened with Composite Materials

The nonlinear behavior of masonry arches strengthened with externally bonded composite materials is investigated. A finite-element (FE) formulation that is specially tailored for the nonlinear analysis of the strengthened arch is developed. The FE formulation takes into account material nonlinearity of the masonry construction and high-order kinematic relations for the layered element. Implementation of the above concept in the FE framework reduces the general problem to a one-dimensional nonlinear formulation in polar coordinates with a closed-form representation of the elemental Jacobian matrix (tangent stiffness). A numerical study that examines the capabilities of the model and highlights various aspects of the nonlinear behavior of the strengthened masonry arch is presented. Emphasis is placed on the unique effects near irregular points and the nonlinear evolution of these effects through the loading process. A comparison with experimental results and a discussion of the correlating aspects and the ones that designate needs of further study are also presented.     

Nonlinear Analytical Solution for Cable Truss

In this technical note the nonlinear closed-form static solution of the suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span is presented. Irvine’s linearized forms of the deflection and the cable equations are modified because the effects of the nonlinear truss behavior needed to be incorporated in them. The concrete form of the system of two nonlinear cubic cable equations is derived and presented. From a solution of a nonlinear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The transformation analytical model serves to determine the response, i.e., horizontal components of cable forces and deflection of the geometrically nonlinear truss, due to the applied loading, considering effects of elastic deformations, temperature changes, and elastic supports. The deflection of asymmetric prestressed cable trusses has been compared with Irvine’s linear solution as well as the nonlinear finite element model results.     

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.     

Localized Buckling of a Bilinear Elastic Ring under External Pressure

A fully nonlinear finite element analysis for prediction of localization in moderately thick imperfect rings under applied hydrostatic pressure is presented. The present nonlinear finite element solution methodology includes all the nonlinear terms in the kinematic equations and utilizes the total Lagrangian formulation in the constitutive equations and incremental equilibrium equations. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. The driving factor behind this analysis is to determine the onset of localization arising out of the bilinear material behavior of the ring with modal imperfection. Numerical results suggest that material bilinearity is primarily responsible for the appearance of a limit or localization (peak pressure) point on the postbuckling equilibrium path of an imperfect ring.     

Tensegrity Structures with Buckling Members Explain Nonlinear Stiffening and Reversible Softening of Actin Networks

In this paper, a three-member tensegrity structure is used as a conceptual model for the dendritic actin network in living cells. The pre and postbuckling behavior of the tensegrity is analyzed basing on the energy method. Analytical simulations are carried out on the tensegrity by using the experimentally obtained scales and mechanic properties of actin-filaments for the structural members of the tensegrity. The model exhibits a stress stiffening regime followed by a stress softening regime in the load-stiffness relationship, which qualitatively tallies with the experimentally observed response of actin networks. Due to the simplicity of the model, there is only a single compressed member and the structure buckles abruptly, which results a softening regime much steeper than that observed in the actin network. To take the member length variety into account, we propose a conceptual large-scale tensegrity system with various member lengths, and its behavior is approximately estimated by the mean response of a large number of three-member tensegrity cells with their member length varying in the range of filament lengths. The obtained mean response exhibits a much better fitness to the response of actin networks than those exhibited by the single tensegrity model. The findings reported in this paper indicate that the dendritic actin network may work as a complex tensegrity system, when it is subjected to a stress.     

Theoretical and Experimental Analysis of GFRP Bridge Deck under Temperature Gradient

The temperature difference between the top and bottom of a glass fiber reinforced polymer (GFRP) composite deck, ～ 65°C ( ～ 122°F), is nearly three times that of conventional concrete decks ～ 23°C ( ～ 41°F). Such a large temperature difference is attributed to the relatively lower thermal conductivity of GFRP material. In this study, laboratory tests were conducted on two GFRP bridge deck modules (10.2 and 20.3?cm deep decks) by heating and cooling the top surface of the GFRP deck, while maintaining ambient (room) temperature at the deck bottom. Deflections and strains were recorded on the deck under thermal loads. Theoretical results (using macro approach, Navier-Levy, and FEM) were compared with the laboratory test data. The test data indicated that the GFRP deck exhibited hogging under a positive temperature difference (i.e., Ttop>Tbottom, heating test; Ttop and Tbottom are temperatures at top and bottom of the deck, respectively) and sagging under a negative temperature difference (i.e., Ttop相似文献   

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

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

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.     

Axisymmetric Vibrations of Reinforced Orthotropic Shallow Spherical Caps

Axisymmetric vibrations of reinforced shallow. spherical caps manufactured from orthotropic materials are considered. The closed form solution is obtained for the natural frequency of the cap with a clamped and immovable circular edge by assuming that the motion component parallel to the cap boundary plane (in plane) is negligible. Parametric studies are performed to assess the effect of various geometric and structural parameters on the natural frequency of the cap and, most importantly, to identify the most influencing parameters of the problem. From the generated data, it is concluded that the national frequency increases with increasing extensional stiffness and eccentricity of reinforcements and to a lesser extent with increasing bending stiffness of reinforcements. Other important parameters include the base circle radius and the initial rise of the cap.     

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.     

Numerical Analysis for the Nonlinear Free Vibration of Shallow Shells

A variational approach for the nonlinear free vibration of shallow shells having a quadrilateral boundary is presented in this paper. Natural coordinates ξ and η are used to map the prescribed geometry in the x–y plane. Displacement fields corresponding to u, v, w, β1, and β2 are expressed in terms of the product of two algebraic functions, the form of which is so chosen that the displacement boundary condition can be imposed by manipulating the coefficients. In arriving at the stiffness matrix, no simplification is applied to the nonlinear strains and the variation of the complete energy equation is considered. For the plate problems numerical results are obtained and compared with approximate analytical results by other researchers. Numerical results for the shallow shells are also presented and their characteristics are found to be significantly different from the results for the plates.     

Postbuckling of Moderately Thick Imperfect Rings under External Pressure

A fully nonlinear finite-element analysis for postbuckling response of a moderately thick imperfect ring under applied hydrostatic pressure is presented. The fully nonlinear theory employed here, in contrast to the von Karman approximation generally prevalent in the existing literature, for a moderately thick ring does not, on employment of the conventional Love–Kirchhoff hypothesis (originally developed for the small deflection regime), automatically guarantee vanishing of the transverse normal and shear strains in the large deflection regime. A curved six-node element, based on an assumed quadratic displacement field (in the circumferential coordinate), employs a two-dimensional hypothesis, known as linear displacement distribution through thickness theory, to capture the effect of the transverse shear/normal (especially, shear) deformation behavior. Numerical results show that even for a sufficiently thin ring, the conventional nonlinear theory, based on von Karman approximation, produces an error on the order of 10%.     

Nonlinear Analyses for Coupled Problem of Temperature and Seepage Fields in Cold Regions Retaining Walls

In this paper, a mathematical mechanical model and the governing differential equations of the coupled problem of temperature and seepage fields, with phase change, are derived from the theories of heat transfer and seepage. The finite-element formulation of this problem is then obtained using Galerkin’s method. Lastly, an illustrative example is provided. The example shows that the effect of seepage field on the temperature field of cold region retaining walls is large. The effect of this factor on cold region retaining walls should be taken into account in cold regions engineering design. Comparisons of the results of this approach with the measured data in the field have been made. The agreement is very good.     

Axisymmetric Nonlinear Stability of a Shallow Conical Shell with a Spherical Cap of Arbitrary Variable Shell Thickness

The axisymmetric nonlinear stability of a shallow conical shell with a spherical cap was studied using the point collocation method with the cubic B-spline function as a trial function. Formulas were set up to consider arbitrary variable shell thickness and different boundary and loading conditions. A FORTRAN program was written to determine the critical loads and trace the stable parts of the equilibrium paths of the shell by method of gradually applied load. The upper and lower critical loads obtained in this technical note for some specific cases with constant shell thickness are of very good accuracy and agree very well with known solutions and finite-element method results. The method was then expanded successfully to arbitrary variable shell thickness and different boundary and loading conditions.