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%.     

Shear Band Analysis in the Biaxial Test

Simulation of a biaxial test by finite-element calculations may lead to unexpected localized deformations even for an elastic-perfect plastic Mohr–Coulomb model. The effects of localization lead to difficulties in computations. This paper describes a simple model for the strength development after occurrence of localization. The localization model describes the postfailure behavior assuming elastoplastic behavior inside the shear band and elastic behavior of the material outside the shear band. The resulting two nonlinear equilibrium equations are analyzed with a rigorous nonlinear scheme. The nonlinear equations show snap-through and snap-back behavior. Snap-back occurs due to energy release of unloading of the material outside the shear band. The simulations show that localization in shear bands is possible for orientations between the Roscoe and Mohr–Coulomb direction. Shear bands with orientations close to the Roscoe direction show a more gradual decrease of strength in comparison to bands with orientations close to the Coulomb direction. The flexibility of the shear band is shown to be important for postbifurcation behavior.     

Stability of Anisotropic Laminated Rings and Long Cylinders Subjected to External Hydrostatic Pressure

The problem of buckling of rings under external pressure has attracted interest since the late 1950s; however, the formulations developed, to date, to obtain the critical pressure are limited to special cases of orthotropic laminated construction. In this work, analytical and numerical treatments are carried out to provide results on the buckling of thin and moderately thick anisotropic rings and long cylinders. A generalized closed-form analytical formula for the buckling of thin anisotropic laminated rings is developed. Standard energy-based formulation and classical lamination theory are used to obtain the equilibrium equations assuming an intermediate class of deformation. The constitutive equations are statically condensed, in terms of the ring’s boundary conditions, to produce the effective axial, coupling, and flexural rigidities. In addition, a three-dimensional (3D) tube finite-element model is developed for nonlinear analysis of anisotropic laminated composite rings or long cylinders. The element accounts for prebuckling ring twist and first-order shear deformations. Fourier series expansions are used to express the in-plane and out-of-plane components of deformation and geometry at the three nodes of the cylindrical element. Isoparametric quadratic shape functions are used to interpolate the displacement field in?between. Comparisons of the analytical and numerical results show excellent agreement for thin rings. Parametric studies are also conducted to address the effects of lamination, shell thickness, and initial out-of-roundness imperfection on the external buckling pressure.     

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.     

Postbuckling of Pressure-Loaded Functionally Graded Cylindrical Panels in Thermal Environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to lateral pressure in thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher-order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. The two straight edges of the panel are assumed to be simply supported and two curved edges are either simply supported or clamped. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflection in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of simply supported, pressure-loaded, perfect and imperfect, functionally graded cylindrical panels with two constituent materials under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections, are studied.     

Tilting Analysis of Rectangular Elastic Layers Bonded between Rigid Plates

A theoretical approach to determine the tilting stiffness of a rectangular elastic layer bonded between two rigid plates is presented. On the basis of two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the tilting stiffness of the bonded rectangular layer is then established in an explicit single-series form. Whereas the finite element method can be applied to calculate the stiffness, the series solution provides a convenient way for parametric studies. Through the obtained pressure expressions, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear traction on the bonding surface can be found. The error of using the approximate shear boundary condition is negligible for the tilting stiffness, but becomes significant for the horizontal displacements and bonding shear stresses near the edges of the rectangular layers.     

Propagation of Localization Instability Under Active and Passive Loading

The mechanical response of a solid continuum changes drastically as the deformation evolves from a diffuse state to a highly localized state. For this reason the subject of strain localization has received much research attention lately. This paper investigates the impact of strain localization in the form of strong discontinuity, or displacement jump, on the limit strengths of retaining walls supporting an elastoplastic backfill. The analysis focuses on the propagation of strong instability in active and passive loading using a recently developed strong discontinuity finite element model where the elements are enhanced to accommodate the presence of displacement jumps. Specifically, the analysis applies to dilative frictional material that is susceptible to shear banding. For the retaining wall problem, strong instability is shown to initiate at the ground surface and propagate downward at an angle that depends on the state of stress at the onset of localization.     

Postbuckling of Axially Loaded Functionally Graded Cylindrical Panels with Piezoelectric Actuators in Thermal Environments

A compressive postbuckling analysis is presented for a functionally graded cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electrical, and thermal loads. The temperature field considered is assumed to be of uniform distribution over the panel surface and through the panel thickness and the electric field considers only the transverse component EZ. The material properties of the presently considered functionally graded materials (FGMs) are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, whereas the material properties of the piezoelectric layers are assumed to be independent of the temperature and the electric field. The governing equations are based on a higher-order shear deformation theory with a von Kármán-Donnell-type of kinematic nonlinearity. A boundary layer theory for shell buckling is extended to the case of hybrid FGM cylindrical panels of finite length. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect FGM cylindrical panels with fully covered piezoelectric actuators, under different sets of thermal and electrical loading conditions. The effects due to temperature rise, volume fraction distribution, applied voltages, panel geometric parameters, in-plane boundary conditions, as well as initial geometric imperfections are studied.     

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.     

Influence of Micromaterial Heterogeneity on Strain Localization in Granular Materials

Lade’s constitutive model was modified to incorporate the couple stress and the particle’s rotation within the framework of the Cosserat continuum. The finite element equations were implemented in the finite-element program (ABAQUS) to predict the strain localization (shear bands) in granular materials. Material spatial heterogeneity such as local void ratio, particle size, surface roughness and shape indices was mapped into the finite element mesh to account for the local heterogeneity of the material properties. The model was found to respond well to such spatial heterogeneities and the results compare well with experiments. The material spatial distributions were generated using scanning electron microscope and optical microscope images. The surface roughness and the shape indices were found to affect the shear band thickness; a parametric study was performed and such effects were found to be significant. The shear band thickness was found to increase as the surface roughness of the particles, particle size, and the particle angularity index increase while it tends to decrease as the particle sphericity, initial density and the confining pressure increase.     

Nine-Node Resultant-Stress Shell Element for Free Vibration and Large Deflection of Composite Laminates

The newly constructed formulation of an element-based resultant-stress nine-node composite shell element is presented for the solution of free vibration and large deflection problems of isotropic and composite laminates. In this paper, the effectiveness of this new formulation is investigated in the static and free vibration analysis. The strain–displacement relationship of the shell could be explained from the point of the new element-based Lagrangian finite element formulation. The newly added terms between bending strain and displacement reflect the contributions of displacements to the curvature. Natural coordinate-based strains, stresses, and constitutive equations are used throughout the element-based Lagrangian formulation of the present shell element which offers significant implementation advantages compared with the traditional Lagrangian formulation. Using the assumed natural strain method the present shell element generates neither membrane nor shear locking behavior, and such an element performs very well as much as shells become thin. The arc-length control method is used to trace complex load–displacement paths and the Lanczos method is employed in the calculation of the eigenvalues of shells. A number of numerical analyses are presented and discussed in order to explore the capabilities of the present shell element. The test results showed very good agreement.     

Higher-Order Finite Strip Method for Postbuckling Analysis of Imperfect Composite Plates

Postbuckling analysis is essential to predict the capacity of composite plates carrying considerable additional load before the ultimate load is reached, and manufacturing-induced geometric imperfections often reduce the load-carrying capacity of composite structures. A higher-order finite strip method based on the higher-order shear deformation plate theory is developed for postbuckling analysis of laminated composite plates with initial geometric imperfection subjected to progressive end shortening. The arbitrary nature of initial geometric imperfection induced during manufacturing is accounted for in the analysis. Nonlinear equilibrium equations are solved by a Newton-Raphson procedure. Examples of postbuckling analyses of unsymmetric cross-ply, angle-ply, and arbitrary laminates are presented, and the accuracy and performance of the method are examined. The numerical higher-order finite strip method presented can be used as an accurate and efficient tool for postbuckling analysis of imperfect composite plates.     

Nonlinear Analysis of Angle‐Ply Composite and Sandwich Laminates

A refined higher order shear deformation theory for linear and geometrically nonlinear behavior of fiber‐reinforced angle‐ply composite and sandwich laminates is established. Laminae material is assumed to be linearly elastic, homogeneous and isotropic/orthotropic. The theory accounts for nonlinear quadratic variation of transverse shear strains through the thickness of the laminate and higher order terms in Green's strain vector in the sense of von Karman. A simple C0 finite‐element formulation of this theory is then presented with a total Lagrangian approach, and a nine node Lagrangian quadrilateral element is chosen with nine degrees of freedom per node. Numerical results are presented for linear and geometrically nonlinear analyses of multilayer angle‐ply composite and sandwich laminates. The theory is shown to predict displacements and stresses more accurately than first‐order shear deformation theory. The results are compared with available closed‐form and numerical solutions of plate theories and three‐dimensional finite‐element solutions. New results are also generated for future evaluations.     

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

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

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

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

基于有限元极限平衡法的三维边坡稳定性

提出了一种基于有限元弹塑性应力场和极限平衡状态的三维边坡稳定分析方法——三维有限元极限平衡法。首先,考虑三维空间中滑动方向,提出滑动面上一点在滑动方向上的极限平衡条件,并证明滑动面上土体整体达到极限平衡状态与滑动面上土体各处在滑动方向上处于极限平衡状态等价。再通过刚体极限平衡假定计算主滑方向和滑动面上各点滑动方向。最后,定义局部安全系数为抗剪强度与滑动方向上剪应力投影的比值,基于三维边坡整体极限平衡条件将局部安全系数通过积分中值定理转变为整体安全系数。该方法计算简单,消除了剪应力比形式定义安全系数滑动面形状限制,具备合理性与有效性。算例验证结果表明,该方法滑动方向假设合理,安全系数与严格三维极限平衡法结果一致。      

Transverse Shear Effect on Flutter of Composite Panels

The effect of transverse shear deformation on the supersonic flutter of composite panels has been investigated using the finite element method. First‐order shear‐deformation laminated‐plate theory and quasi‐steady aerodynamic theory are employed for the analysis. The total displacement of the plate is expressed as the sum of the displacement due to bending and the displacement due to shear deformation. Thus, the aerodynamic pressure induced by the plate motion is also the sum of the pressure induced by bending deformation and the pressure induced by shear deformation. Numerical results show that the transverse shear deformation may have a significant effect on the flutter boundary if aerodynamic damping were small or neglected in the determination of flutter boundary.     

Role of ligaments and facets in lumbar spinal stability

STUDY DESIGN: The issue of segmental stability using finite element analysis was studied. Effect of ligament and facet (total and partial) removal and their geometry on segment response were studied from the viewpoint of stability. OBJECTIVES: To predict factors that may be linked to the cause of rotational instabilities, spondylolisthesis, retrospondylolisthesis, and stenosis. SUMMARY OF BACKGROUND DATA: The study provides a comprehensive study on the role of facets and ligaments and their geometry in preserving segmental stability. No previous biomechanical study has explored these issues in detail. METHODS: Three-dimensional nonlinear finite element analysis was performed on L3-L4 motion segments, with and without posterior elements (ligaments and facets), subjected to sagittal moments. Effects of ligament and facet (partial and total) removal and their orientations on segment response are examined from the viewpoint of stability. RESULTS: Ligaments play an important role in resisting flexion rotation and posterior shear whereas facets are mainly responsible for preventing large extension rotation and anterior displacement. Facet loads and stresses are high under large extension and anterior shear loading. Unlike total facetectomy, selective removal of facets does not compromise segmental stability. Facet loads are dependent on spatial orientation. CONCLUSIONS: Rotational instability in flexion or posterior displacement (retrospondylolisthesis) is unlikely without prior damage of ligaments, whereas instability in extension rotation or forward displacement (spondylolisthesis) is unlikely before facet degeneration or removal. The facet stress and displacement distribution predicts that facet osteoarthritis or hypertrophy leading to spinal stenosis is most likely under flexion-anterior shear loading. Selective facetectomy may restore spinal canal size without compromising the stability of the segment. A facet that is more sagittally oriented may be linked to the cause of spondylolisthesis, whereas a less transversely oriented facet joint may be linked to rotational instabilities in extension.     

Effect of Stress‐Dependent Base Layer on the Superposition of Flexible Pavement Solutions

Flexible pavement structural analysis for design usage must consider (as a minimum) multiple wheel/axle loading configurations, seasonal variations of material layer properties, and the nonlinear behavior of unbound materials. Although these requirements are all easily within the capabilities of three‐dimensional finite element analysis, the required computation times may be impracticably long for routine design. Compromises between analytical rigor (e.g., three‐dimensionality) and analysis features (e.g., multiple wheels, seasonal property variations, material nonlinearity) must be made. One compromise is to retain seasonal property variations and material nonlinearity within an axisymmetric single wheel finite element model and to approximate multiple wheel effects via superposition. Although this superposition of nonlinear solutions is undeniably invalid from a rigorous theoretical viewpoint, the errors may be well within acceptable magnitudes for practical design. The paper investigates this issue by comparing superimposed nonlinear solutions against computationally rigorous three‐dimensional nonlinear solutions and evaluating the discrepancies in key pavement response quantities. The results suggest that the errors from superimposing nonlinear solutions are acceptably small for key pavement response quantities. Moreover, these errors are substantially smaller than those resulting from neglect of nonlinear unbound material behavior, a modeling compromise that is common in pavement structural analysis today.     

Finite Element Response Sensitivity Analysis of Steel-Concrete Composite Beams with Deformable Shear Connection

The behavior of steel-concrete composite beams is strongly influenced by the type of shear connection between the steel beam and the concrete slab. For accurate analytical predictions, the structural model must account for the interlayer slip between these two components. In numerous engineering applications (e.g., in the fields of structural optimization, structural reliability analysis, and finite element model updating), accurate response sensitivity calculations are needed as much as the corresponding response simulation results. This paper focuses on a procedure for response sensitivity analysis of steel-concrete composite structures using displacement-based locking-free frame elements including deformable shear connection with fiber discretization of the cross section. Realistic cyclic uniaxial constitutive laws are adopted for the steel and concrete materials as well as for the shear connection. The finite element response sensitivity analysis is performed according to the direct differentiation method. The concrete and shear connection material models as well as the static condensation procedure at the element level are extended for response sensitivity computations. Two steel-concrete composite structures for which experimental test results are available in the literature are used as realistic testbeds for response and response sensitivity analysis. These benchmark structures consist of a nonsymmetric, two-span continuous beam subjected to monotonic loading and a frame subassemblage under cyclic loading. The new analytical derivations for response sensitivity calculations and their computer implementation are validated through forward finite difference analysis based on the two benchmark examples considered. Selected sensitivity analysis results are shown for validation purposes and for quantifying the effect and relative importance of the various material parameters in regards to the nonlinear monotonic and cyclic response of the testbed structures.