Nonlinear analysis of thin walled bars of open cross-section

Energy equations governing the goemetrically nonlinear behaviour of thin walled bars of open cross-section when subjected to axial, flexural and torsional displacements, are derived. The equations are based on the assumption of small displacements and linear elastic material behaviour. Shear deformations due to nonuniform bending, and distortion of the cross-section are not included in the analysis. Solutions of the governing nonlinear equations are obtained by the finite element method, making use of linearised mid-increment stiffness matrices. The results of the geometrically nonlinear analysis are combined with an approximate failure criterion to predict the failure loads of I beams having initial imperfections. The predicted failure loads are compared, and show satisfactory agreement, with existing experimental results.     

Elasticity solution for an axially compressed and encased cylindrical specimen

We present the solution of the linear elasticity equations governing the deformation of an elastic cylinder encased in a tube and subjected to uniform compression on the flat ends. The solutions for the stresses, strains, and displacements in the encased and compressed cylinder are all systematically determined from the basic solution of Lamé's classical elasticity problem of the long tube subjected to internal and external pressures. We first derive the general elastostatic analysis for an encased hollow cylinder, stress-free at the cavity, and later particularize the solution to a solid cylindrical specimen. The effective modulus E_eff of the encased sample is found to be a function of the bulk modulus k and Poisson's ratio ν of the material. E_eff differs from k except for nearly incompressible materials, where E_eff approaches the bulk modulus value. In the incompressible case, we also show how a load applied on the cylinder's flat ends is equivalent to, and can be replaced by, the same load acting on the curved surface. For compressible materials, a more general expression for E_eff is found that also accounts for the case deformation. These results explain the deformation of an axially compressed and encased cylindrical specimen tested in compressibility measuring devices such as those described by Matsuoka and Maxwell [Response of linear high polymers to hydrostatic pressure. Journal of Polymer Science 1958; 32:131–59]. The present analysis thus contributes to a better understanding of how this device works and to the interpretation of measurements taken with it.     

The plane-strain general yielding of notched members due to combined axial force and bending—II: Deep circular notches

The slip-line fields proposed by Green for the plane-strain general yielding of notched plates in pure bending are generalized to provide solutions for notched plates subjected to combined axial force and bending. Deep symmetrical circular notches and single circular notches are considered. For notched plates subjected to arbitrary combinations of axial force and bending, two constraint factors are obtained, T₁ due to the axial force and M₁ due to the bending moment. These constraint factors are shown to have characteristic relationships on the T₁−M₁ plane for the single notched plate and the symmetrically notched plate. The derivations of these general yielding loci are described in detail and the statical and kinematical limitations of the slip-line fields are discussed. Finally it is demonstrated that the corresponding strain-rate vectors are normal to these loci and that, therefore, these loci can be regarded as plastic potentials.     

Effects of prebuckling deformations on the elastic flexural-torsional buckling of laterally fixed arches

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial.     

The plane-strain general yielding of notched members due to combined axial force and bending—I: Wedge-shaped notches with large flank angles

The slip-line field proposed by Green for the general yielding of notched plates in pure bending is generalized to provide solutions for the general yielding of notched plates subjected to combined bending and axial force. For symmetrically notched plates subjected to arbitrary combinations of axial force and bending moment, two constraint factors are obtained, T₁ due to the axial force and M₁ due to the bending moment. These constraint factors are shown to have a characteristic locus expressed by two parabolae in the T₁ − M₁ plane. In one extreme case where both flank angles are equal to π, i.e. a flat plate, the locus is the same as that obtained by Prager.The general yielding loci of single notched plates and plates containing symmetrical notches of the same flank angle are derived as special cases of the general problem and some practical problems involving single notches are considered.     

Effect of deep wedge-shaped notches of small flank angle on plastic failure

Green's slip-line field for the plane-strain plastic bending of a plate with a wedge-shaped notch of small flank angle is generalized to take into account axial force. Two constraint factors are derived, T₁ due to the axial force and M₁ due to the bending moment, where the moment is taken about the mid-point of the ligament length. These constraint factors are mapped in constraint factor space providing general yielding loci. It is shown that the slip-line field solution becomes inadmissible for certain combinations of axial force and bending moment and Rice's upper bound solution is used under these circumstances.     

Structural stability of slender aerospace vehicles: Part II: Numerical simulations

The methodology presented in Part I of this paper is used to obtain the numerical results for different cases. First, the predictions on stability regimes for a vehicle subjected to end thrust are validated by comparing results from published literature. Finite element simulation results involving aeroelastic effects are compared with the published literature. For numerical simulations, the analysis is carried out for assumed state of translational velocities. Also, the vehicle is assumed to be moving at zero angle of attack and the angular velocity is assumed to be small and it is neglected. Numerical simulations are carried out for a representative vehicle to determine the instability regimes with vehicle speed and thrust as the parameters, neglecting the aerodynamic forces. Simulations are carried out for a typical nose-cone launch vehicle configuration to analyze the aeroelastic stability at two different Mach numbers. Phenomenon of static instability (divergence) and dynamic instability (flutter) are observed. Vibrational mode shapes of the vehicle for different thrust and speed are analyzed. In the end, axial responses (displacements and velocities) of a typical vehicle subjected to axial thrust are determined using direct integration of the equations of motion. The axial displacements along the length of the vehicle due to two different thrust histories are compared. The coupling of rigid body motion with the elastic displacements is illustrated.     

Plastic buckling of circular tubes under axial compression—part I: Experiments

Elastic buckling of cylindrical shells due to axial compression results in sudden and catastrophic failure. By contrast, for thicker shells that buckle in the plastic range, failure is preceded by a cascade of events, where the first instability and failure can be separated by strains of 1–5%. The first instability is uniform axisymmetric wrinkling that is typically treated as a plastic bifurcation. The wrinkle amplitude gradually grows and, in the process, reduces the axial rigidity of the shell. This eventually leads to a limit load instability, beyond which the cylinder fails by localized collapse. For some combinations of geometric and material characteristics, this limit load can be preceded by a second bifurcation that involves a non-axisymmetric mode of deformation. Again, this buckling mode localizes resulting in failure.The problem is revisited using a combination of experiments and analysis. In Part I, we present the results of an experimental study involving stainless steel specimens with diameter-to-thickness ratios between 23 and 52. Fifteen specimens were designed and machined to achieve uniform loading conditions in the test section. They were subsequently compressed to failure under displacement control. Along the way, the evolution of wrinkles was monitored using a special surface-scanning device. Bifurcation buckling based on the J₂ deformation theory of plasticity was used to establish the onset of wrinkling. Comparison of measured and calculated results revealed that the wrinkle wavelength was significantly overpredicted. The cause of the discrepancy is shown to be anisotropy present in the tubes used. Modeling of the postbuckling response and the prediction of the limit load instability follows in Part II.     

Buckling analysis of FGM plates under uniform,linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.

     

Rigid-plastic modelling of blast-loaded stiffened plates—Part II: Partial end fixity, rate effects and two-way stiffened plates

A new rigid-plastic analysis of stiffened plates subjected to uniformly distributed blast loads is developed. In this first part of a two-part paper, a uniform one-way stiffened plate with clamped ends is modelled as a singly symmetric beam, comprised of one stiffener and its tributary plating. Rigid-plastic analysis is then applied to this beam using an idealized piecewise linear bending moment-axial force capacity interaction relation or yield curve. Two solutions to the response are developed. The first solution is in closed form and is based on the solution of the resulting linearized differential equations. The second solution is obtained by approximating the response as a sequence of instantaneous mode responses, where the mode shapes are determined by an extremum principle which maximizes the rate of change of the kinetic energy. This latter solution may be extended to cases involving non-rigid boundaries and two-way stiffening and this is done in the second part of this paper. Here, the two solution methods are applied to several examples of one-way stiffened plates subjected to various blast-type pulses. Good agreement is obtained between the present results and those from elastic-plastic beam finite element and finite strip solutions.     

Second order strains and instability of thin walled bars of open cross-section

Complete expressions for the strains, including second order terms, occurring in thin walled bars of open cross-section are derived. Shear deformation due to non-uniform bending and distortion of the cross-section are not considered in the present theory. The correctness of the expressions is verified by incorporating them in a general instability analysis based on energy considerations.     

Instability of cantilever beams with non-linear elements: Butterfly catastrophe

The instability of an elastic cantilever beam combined with a non-linear element, which is subjected to axial and normal concentrated and distributed loads was studied.It was shown that the total potential energy u of the system was equivalent to a universal unfolding of a butterfly catastrophe and, therefore, its approximate locus of failure may be studied by means of the bifurcation set of this type of elementary catastrophe. The problem is a generalization of the case of cantilever beams connected with elastic elements, whose solution is in agreement with the general results. established in the paper.     

A two-level optimization procedure for material characterization of composites using two symmetric angle-ply beams

A two-level optimization procedure for determining elastic constants E₁, E₂, G₁₂, and ν₁₂ of laminated composite materials using measured axial and lateral strains of two symmetric angle-ply beams with different fiber angles subjected to three-point-bending testing is presented. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/−45°)₆]_s beam are used to construct the strain discrepancy function which is a measure of the sum of the squared differences between the experimental and theoretical predictions of the axial and lateral strains. The identification of the material constants is then formulated as a constrained minimization problem in which the best estimates of shear modulus and Poisson's ratio of the beam are determined to make the strain discrepancy function a global minimum. In the second-level optimization process, shear modulus and Poisson's ratio determined in the first level of optimization are kept constant and Young's moduli of the second angle-ply beam with fiber angles different from 45° are identified by minimizing the strain discrepancy function established at this level of optimization. The suitability of the proposed procedure for material characterization of composite materials has been demonstrated by means of a number of examples.     

A four variable refined plate theory for nonlinear cylindrical bending analysis of functionally graded plates under thermomechanical loadings

In this paper we present a new application for a four variable refined plate theory to analyse the nonlinear cylindrical bending behavior of functionally graded plates subjected to thermomechanical loadings. This recent theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The non-linear strain-displacement relations in the von Karman sense are used to study the effect of geometric non-linearity. The solutions are achieved by minimizing the total potential energy and the results are compared to the classical and the first-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the nonlinear cylindrical bending behavior of functionally graded plates.     

Theoretical prediction of tensile instability conditions in rotating disks

An analysis of conditions at instability in rotating disks of uniform thickness is presented. Two plasticity theories are applied. The first analysis based on the Tresca yield criterion and its associated flow rule is an extension of the work of Prager and Weiss. The results of this analysis are correlated with results based on the Hencky deformation theory of plasticity using a finite difference method of solution. Both theories show that if fracture is preceded by a tensile instability condition then large strains occur at the bore of a hollow disk.     

Large deflections of metallic sandwich and monolithic beams under locally impulsive loading

A rigid-perfectly plastic model is adopted to predict the dynamic response of fully clamped sandwich and monolithic beams subjected to localized impulse. Large deflection effect is incorporated in analysis by considering interaction between plastic bending and stretching. Based on the principle of energy equilibrium, a membrane factor for metallic sandwich beams with nonuniform cross-section thickness is derived to consider the effect of axial force induced by large deflection. Then, the dynamic response solution is obtained for the large deflection of metallic sandwich beams subjected to localized impulse. In addition, tighter ‘bounds’ of the solutions for sandwich beams are derived by using the inscribed and circumscribed squares of a new yield criterion including the core effect. As a degenerated limit case, solution for the large deflection response of solid monolithic beams is also obtained. The present solutions are in good agreement with finite element (FE) results and lie in the ‘bounds’ of the solutions. It is demonstrated that the axial (membrane) force associated with stretching plays an important role in the dynamic response of large deflections; in comparison with small deflection solutions, the axial (membrane) forces substantially stiffen the metallic sandwich and monolithic beams.     

Framework instability analysis using Blaszkowiak's stiffness functions

Many years ago Blaszkowiak showed how basic and modified stifinesses could be derived directly from governing differential equations for the uniform beam, with the effects of axial loading, continuous lateral elastic supports, etc. separately included.This paper presents his derivations for the uniform beam-column, allowing for various end conditions. The stiffness coefficients are tabulated here against the parameter = (P/P_e), where P_e = π²EI/l², and some examples of their use are given.These functions comprise a straightforward and comprehensive system for the analysis of framework instability. In the authors' opinion they are the ultimate improvement on previous incomplete formulations.     

Plastic buckling of circular tubes under axial compression—part II: Analysis

In the second part of this study, the evolution of uniform axisymmetric wrinkling in axially compressed cylinders is modeled using the principle of virtual work. A version of this formulation also allows localization of wrinkling. The model domain is assigned an initial axisymmetric imperfection of a chosen amplitude and the wavelength yielded by the first bifurcation check. The solution correctly simulates the growth of wrinkles and results in a limit load instability. The limit strain is influenced by the amplitude of the imperfection. Beyond the limit load, wrinkling tends to localize, eventually leading to local folding.The possibility of bifurcation of the axisymmetric solution to non-axisymmetric buckling modes is examined by using a dedicated bifurcation check. The bifurcation check was found to yield such buckling modes correctly. The evolution of such buckling modes is simulated by a separate non-axisymmetric model assigned imperfections with axisymmetric and nonaxisymmetric components. The domain analyzed is one characteristic wavelength long (2λ_C). Initially, compression activates mainly axisymmetric deformation. In the neighborhood of the bifurcation point, non-axisymmetric deformation starts to develop, eventually leading to a limit load instability. Experimental responses were simulated with accuracy by assigning appropriate values to the two imperfection amplitudes. Prediction of the limit strains for the whole range of diameter-to-thickness ratios (D/t) considered in the experiments was achieved by making the amplitude of the non-axisymmetric imperfection proportional to (D/t)²/m³ (m is the circumferential wavenumber). Matching all aspects of the experiments required inclusion of the anisotropy measured in the tubes tested through Hill's yield criterion in all models.     

Finite element analysis for friction noise of simplified hip joint and its experimental validation

In a hip joint system, squeak noise often occurs due to friction between the ball and hemispherical cup. To analyze the dynamic instability induced by friction in the hip joint system, the dynamic ball joint model was constructed by using the finite element method. The results from stability analysis revealed that the mode-coupling type instability occurred for one bending mode and its adjacent composite mode with the axial and transverse displacements with the increase of friction coefficient. The vitro squeak test and vibration modal tests confirmed that squeak arose near the frequency of the mode pair.     

An experimental study of the burst strength of rotating disks

Details are given of a spinning rig for burst tests on metal disks in vacuo at speeds in excess of 100,000 rpm. The permanent strain distributions and the instability and fracture conditions are observed in the spinning of disks of vacuum remelted alloy steel. It is shown for hollow disks of uniform thickness that at instability two “necks” form at either side of the bore and the bore becomes oval with the direction of the minimum diameter passing through the two “necked” regions.Good correlation between theoretical and experimental strain distributions at instability are obtained provided the ratio of outside to inside radius of the disk is greater than 10. The theory is based on a rigid-plastic material and it is thought that better correlation for radius ratios less than 10 would be achieved if elastic strains were taken into account in the theory.