Plate Plane Stress Research Articles

A novel method for the vibration optimisation of structures subjected to dynamic loading

The optimum design of structures with frequency constraints is of great importance in the aeronautical industry. In order to avoid severe vibration, it is necessary to shift the fundamental frequency of the structure away from the frequency range of the dynamic loading. This paper develops a novel topology optimisation method for optimising the fundamental frequencies of structures. The finite element dynamic eigenvalue problem is solved to derive the sensitivity function used for the optimisation criteria. An alternative material interpolation scheme is developed and applied to the optimisation problem. A novel level-set criteria and updating routine for the weighting factors is presented to determine the optimal topology. The optimisation algorithm is applied to a simple two-dimensional plane stress plate to verify the method. Optimisation for maximising a chosen frequency and maximising the gap between two frequencies are presented. This has the application of stiffness maximisation and flutter suppression. The results of the optimisation algorithm are compared with the state of the art in frequency topology optimisation. Test cases have shown that the algorithm produces similar topologies to the state of the art, verifying that the novel technique is suitable for frequency optimisation.

Symplectic Analysis of the Shear Lag Phenomenon in a T-Beam

AbstractThe principal purpose of this paper is to develop a new analytical method for the shear lag problem of T-beams. The flange slab of a T-beam is simplified into a plane stress plate. A system of Hamilton dual equations of the flange slab using symplectic elasticity theory can be obtained. Analytical solutions of the flange slab under uniform or linear shear force applied on the entire slab length are obtained by solving the equation. Based on the solution, the cause of the positive shear lag and negative shear lag phenomenon are studied in this paper. Theoretical research indicates that the method is superior for analyzing the shear lag phenomenon. The analytical solution has been validated by comparison with the beam solution and FEM solution of a cantilever T-beam subjected to a uniform load.

Effective flange width of simply supported box girder under uniform load

A new method for the determination of effective flange width under uniform load on simply supported box girder bridges considering shear lag effect is proposed in this paper. Based on the Symplectic Elasticity method, the flange slab of the box girder is simplified into a plane stress plate. Using equilibrium conditions of the plates, the Hamilton dual equations for top plate element is established. The analytical formulas of each plate element considering shear lag effect are derived. The closed polynomial effective width expression of flange slab under uniform load on the whole span length has been obtained. Through examples using the finite element method, the results obtained by the proposed method are examined and the accuracy of the proposed method is verified.

Linear buckling analysis of perforated plates subjected to localised symmetrical load

Holes are often unavoidable in webs of steel beams and in plates, due to inspection, maintenance and also aesthetic purposes. In these situations, the presence of holes may cause redistribution of plane stresses in plates with a significant reduction of stability. In this paper, linear buckling analyses of perforated plates subjected to localised symmetrical load, with circular and rectangular holes, are developed. The results show some relevant differences between the behaviour of the plate subjected to localised symmetrical load with respect to that of uniform compressive load, already known in literature, with a significant increase in the elastic critical load value. The buckling coefficient of square and rectangular perforated panels under symmetrical uniform compression assumes the same value of corresponding whole panels, and it strongly increases for localised symmetrical load, reducing its increment on the basis of the diameter of the hole. Lastly, some indications are given on the best orientation of rectangular holes when localised load are applied showing differences with the uniform compression load case. Holes with the longest edge parallel to the transverse direction of the panel subjected to uniform compression loads are preferable for stability purposes. According to present results obtained for localised loads, holes with the longest edge along the longitudinal direction are preferable.

Analysis of symmetric laminated rectangular plates in plane stress

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.

Solution of the moiré hole drilling method using a finite-element-method-based approach

The moiré hole drilling method in a biaxially loaded infinite plate in plane stress is an inverse problem that exhibits a dual nature: the first problem results from first drilling the circular hole and then applying the biaxial loads, while the other problem arises from doing the opposite, i.e., first applying the biaxial load and then drilling the circular hole. The first problem is hardly ever addressed in the literature but implies that either separation of stresses or material property identification may be achieved from interpreting the moiré signature around the hole. The second is the well-known problem of determination of residual stresses from interpreting the moiré fringe orders around the hole. This paper addresses these inverse problem solutions using the finite element method as the means to model the plate with a hole, rather than the typical approach using the Kirsch solution, and a least-squares optimization approach to resolve for the quantities of interest. To test the viability of the proposed method three numerical simulations and one experimental result in a finite width plate are used to illustrate the techniques. The results are found to be in excellent agreement. The simulations employ noisy data to test the robustness of this approach. The finite-element-method-based inverse problem approach employed in this paper has the potential for use in applications where the specimen shape and boundary conditions do not conform to symmetric or well-used shapes. Also, it is a first step in testing similar procedures in three-dimensional samples to assess the residual stresses in materials.

Finite element limit analyses of under-matched tensile specimens

This paper quantifies the effect of under-matching on plastic limit loads and fully plastic stress triaxialities for mismatched flat plate (in plane strain and plane stress) and round bar tensile specimens, via parametric finite element (FE) limit analyses based on elastic–perfectly-plastic materials. It is found that the effect of the strength mismatch ratio (the ratio of the yield strength of the weld zone to that of the base material) on plastic limit loads and fully plastic stress triaxialities can be significant for plane strain plate and round bar specimens, but much less significant for plane stress plate specimens. Its effect is dependent significantly on the slenderness of the weld zone (defined by the ratio of the weld zone width to the specimen width). The effect of the slenderness of the weld zone is apparent only when the weld width is less than the specimen width or diameter. In particular, the stress triaxiality in the softer weld zone can increase significantly with decreasing the ratio of the weld zone width to the specimen width (radius). Based on the present limit load results, a simple method to extract intrinsic tensile properties of the under-matched weld zone from test results of under-matched tensile specimens is proposed.

Localized vibration in elastic structures with slowly varying thickness

High frequency localized vibration modes for linear isotropic elastic plates and rods of slowly varying thickness are shown to occur within the vicinity of maximal or minimal cross-sections. Long wave asymptotic integration close to the eigenmodes of such cross-sections is provided. The cases of a one integer parameter of thickness resonances (for a plate in plane stress) and two integer parameters (for a rod of circular cross-section) are fully investigated. One-dimensional Hermite-type equations for the eigenmodes are derived and associated frequency shifts from those of the thickness resonances of the maximal or minimal cross-section obtained. It is established that the existence of this type of localization phenomena is strongly dependent on the sign of the group velocity of the associated uniform thickness structure this being the same as that of the local group velocity calculated at the maximal or minimal cross-section. It is confirmed that positive group velocity allows localization for elevated structures with negative group velocity allowing localization only for ligaments. In each case intervals of the Poisson's ratio providing localization are given for the first few modes.

Fiber volume ratio effect on the crack bridging mechanism in uniaxial reinforced ceramic composites

Abstract An unidirectional fiber reinforced and centrally cracked infinite composite plate under uniaxial stress is considered. Elastic and frictional bridging relations which express the relation between the stress in the fiber and the crack opening displacement along the crack axis which are consistent with infinite brittle fiber-reinforced composite behaviour are chosen to be utilized in the formulation of the problem. The differential equation of the orthotropic plate in plane stress is solved through Fourier transform techniques and the stresses are obtained in terms of the crack surface derivative function. The singular integral equation system is reduced to the solution of a non-linear system of algebraic equations by introducing Chebyshev polynomials. Stress intensity factor at crack tips, crack opening displacement and the point where the bridging type transition point on the crack axis are calculated for SiC/Al 2 O 3 . The effect of the variations of the fiber volume ratio on the bridging mechanism is presented.

An algorithm for gradient-regularized plasticity coupled to damage based on a dual mixed FE-formulation

A thermodynamically consistent theory of gradient-regularized plasticity with coupling to isotropic damage was presented by Svedberg and Runesson [23]. Its main purpose was to describe the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure. Explicit constitutive relations were established for a model problem based on von Mises plasticity with isotropic hardening of two types: local and gradient-enhanced. The main purpose of this paper is to describe a new numerical algorithm for the gradient-enhanced formulation. Using the Backward Euler rule leads to a gradient-enhanced format of the classical Closest-Point-Projection-Method. The plastic multiplier increment is solved via a dual mixed method, which is a convenient feature in the present context. This leads to an unambiguous representation of the constitutive relations in the equilibrium equation. Moreover, the classical local algebraic Kuhn-Tucker conditions are obtained in a straightforward fashion in the limit case of local theory. The developed algorithm was implemented in the educational finite element package [5]. Numerical results are shown for the extension of a plate in plane stress. These results demonstrate the convergence characteristics with mesh refinement, as well as the salient features of the adopted damage-based constitutive model.

Exact analytical solutions for non-selfadjoint variable-topology shape optimization problems: perforated cantilever plates in plane stress subject to displacement constraints. Part I

Lurie (1994, 1995a, b) proved recently that variabletopology shape optimization of perforated plates in flexure for non-selfadjoint problems leads to rank-2 microstructures which are in general nonorthogonal. An extension of the same optimal microstructures to perforated plates in plane stress will be presented in Part II of this study. Using the above microstructure, the optimal solution is derived in this part for cantilever plates in plane stress, which are subject to two displacement constraints. For low volume fractions the above solutions are shown to converge to the known truss solutions of Birkeret al. (1994). The problem of homogenizing the stiffness of nonorthogonal rank-2 microstructures is also discussed.

The Michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: An asymptotic study

It is shown that Michell's problem of least-weight truss layouts can be obtained by asymptotic expansion of the optimized strain and complementary energies derived for minimum compliance topology of perforated plates in plane stress.

Lithospheric stresses associated with nontransform offsets of the Mid‐Atlantic Ridge: Implications from a finite element analysis

Nontransform offsets are a fundamental aspect of the offset geometry exhibited along the mid‐oceanic ridge system, independent of spreading rate. Along the slow/intermediate opening (<40 mm/y full rate) Mid‐Atlantic Ridge these offsets of the ridge axis range in length from less than 10 km to approximately 30 km and vary in age offset from 0.5 to 2.0 m.y. The variable morphotectonic geometries associated with these discontinuities indicate that horizontal shear strains are accommodated by both extensional and strike‐slip tectonism and that the geometries are unstable in time. In many cases, there appears to be an evolutionary relationship between transform fault boundaries and nontransform offsets as the result of prolonged differential asymmetric spreading between adjoining ridge segments. The finite element method is used to study the complex stress field associated with these small‐offset discontinuities of ridges with slow (30 mm/y) and fast (100 mm/y) total opening rates. A plane stress plate model examines the variation in the horizontal tectonic stress field produced by offsets with different lengths and changes in the ratio of a ridge‐normal tensile stress resisting plate separation to a shear stress resisting relative plate motion along the discontinuity. The predicted fault patterns based on the calculated stress field are compared with seafloor observations in terms of the morphotectonic patterns and evolution of nontransform offsets. For a slow spreading rate, the analysis shows that all structural geometries observed can be modeled by a range of offset lengths (5, 10, 20, 30, and 40 km) and by a ridge‐normal stress 3 to 5 times greater than the discontinuity shear stress. These findings suggest that nontransform offsets are zones of mechanical weakness relative to the surrounding lithosphere. An offset length between 10 and 20 km is predicted to be the threshold length for maintaining a transform fault geometry. As inferred from ridge axis morphology, there seems to be a strong link between the magnitude of the stress ratio and the time varying magmatic activity along and between ridge segments. While our models are consistent with a weak discontinuity shear stress relative to the ridge‐normal stress to explain the geometries of nontransform offsets of slow‐spreading centers, a weaker ridge‐normal stress to discontinuity shear stress most closely models the development of an overlapping spreading center geometry, the distinctive geometry of nontransform offsets of spreading centers opening at fast rates. This difference is attributed to magma supply along‐axis, relatively continuous for fast‐spreading centers and intermittent for slow‐spreading centers, and a preexisting zone of mechanical weakness linked to the evolution of nontransform offsets from transform faults on slow‐spreading centers.

Elastic-plastic analysis of cracked plates in plane stress: An experimental study

An experimental method is presented for the complete solution of the elastic-plastic plane stress problem of an edge-cracked plate obeying the Mises yield criterion and the Prandtl-Reuss incremental stress-strain flow rule. The material of the plate is assumed as a strain-hardened one with different degrees of hardening. The elastic and plastic components of strain were determined by using the method of birefringent coatings cemented on the surface of the metallic specimens made of the material under study. Normal incidence of circularly polarized light yielded the isolinics and isochromatics of the coating which provided the principal elastic strain differences and strain-directions at the interface. Evaluation of the stress intensity factor at the crack tip, by using the Griffith-Irwin definition, gave the sum of principal stresses at the crack tip. These data were sufficient to separate the components of strain at the coating-plate interface by using the classical shear-difference method.

局所積分の概念を用いた確率有限要素法

This paper, introducing local integration, formulates a new stochastic finite element method for estimating the response variability of multi-dimensional stochastic systems. Young's modulus is assumed to have a spatial variation and is idealized as a multi-dimensional continuous Gaussian stochastic field. An essential feature of the proposed method is that the continuous stochastic field is rigorously taken care of by means of local integrations to construct element stiffness matrices, as the results, the issue involving the stochastic field is transformed into a problem involving only a few random variables, and the perturbation technique is then utilized with considerable ease. This may lead to substantial improvement in computational efficiency. And the accuracy of the solution from the proposed method appears to be independent of the way in which discretization is performed, whereas the problem associated with the convergence of the solution from conventional methods, which are based on a discretized stochastic field, always remains in any case. In this paper, it is shown that stochastic stiffness matrices can be easily derived from the principle of complementary virtual work, when the spatially varying Young's modulus is idealized as mentioned above. In order to examine the validity of the proposed method, two kinds of stochastic structures subjected to deterministic loads; a portal frame and a plane stress plate, are analyzed. As the results, the proposed method has a great advantage not only in the computational cost but also in the solution accuracy. Finally, it should be emphasized that the formulation of this method is so systematic that it can be easily extended to various engineering problems.

Dynamic line-spring model for the simplified analysis of the dynamic stress-intensity factor for 3-D surface cracks.

A dynamic line-spring model is proposed for the simplified analysis of the dynamic stress-intensity factor for part-through surface cracks. A plate containing two surface cracks is modeled as a 2-D plate in plane stress with a through crack. Surfaces of the through crack are connected by dynamic line-springs distributed continuously along the through crack. In the case of static loading, the present model reduces to the static model proposed by Rice and Levy. As an example of the application of the present model, the dynamic response of a plate containing two semi-elliptical surface cracks is analysed where the plate is subjected to a vibrating stress at its ends. It is found that the dynamic stress-intensity factor takes its maximum value at the deepest penetration point of the surface crack and begins to increase markedly near the first resonance frequency as the frequency of the applied stress increases.

Lithospheric stress near a ridge‐transform intersection

Oceanic lithosphere at a ridge‐transform intersection has distinct structural characteristics. The spreading axis and normal fault scarps, which parallel the ridge axis away from the intersection, locally curve towards the transform fault. This implies that the least compressive stress direction becomes oblique to the spreading direction near a ridge‐transform intersection. This stress field may be the result of a tensile ridge axis stress resisting plate separation and a transform shear stress resisting relative plate motion. A plane stress plate model with prescribed plate boundary stresses is formulated to explore this hypothesis. The orientation of the least compressive horizontal stress near an intersection depends strongly on the ratio of the applied boundary stresses. Thus ridge axis curvature may place an important constraint on plate boundary forces.

On a hybrid method to determine strain fields around propagating cracks

The moving singularity of the crack tip in a plane-stress plate causes a highly dynamic stress field of varying intensity with time, throughout the period of the propagation of the crack. This dynamic stress field results in a considerable change of the mechanical and optical properties of a strain-rate dependent material. An analysis of this varying dynamic stress field was presented in this paper which contradicts assumptions and simplifications introduced in a previous paper [7], referring to the same problem. For the experimental determination of the K d I- factor the optical method of the dynamic caustics was utilized in combination with a high-speed camera and a comparison was sketched between the possibilities of this method and the strain-gauge method used in Ref. [7].

Small-Scale Yielding at the Tip of a Through-Crack in a Shell

Deformation theory is used to model plastic deformation at the tip of a through-crack in a thin shell. In the vicinity of the crack the shell is subjected to both stretching and bending, but stretching is assumed to dominate. Thus the stresses are tensile, but with a nonuniform distribution through the thickness, which depends on the material properties as well as on the geometry. The nonlinear near-tip fields (which are singular) have been analyzed asymptotically. Cracks in shallow shells and spherical shells have been investigated in some detail. It is shown that the angular variations are the same as for generalized plane-stress plate problems. Assuming small-scale yielding a path-independent integral, which is valid in a region close to the crack edge, is used to connect the nonlinear near-tip fields with the corresponding singular parts of the linear fields. It is shown that the nonlinear behavior significantly affects the through-the-thickness variations of the near-tip fields. The singular parts of the membrane stresses tend to become more uniform through the thickness of the shell with a flatter strain-hardening curve.

Improved boundary–integral equation method for time‐dependent inelastic deformation in metals

AbstractEfficient solution of boundary value problems for time‐dependent inelastic deformation in metallic structures is of great practical importance. These problems are generally solved by finite element methods and separate descriptions for time‐independent plasticity and time‐dependent creep are normally used. The boundary–integral equation method was recently applied for the first time to such problems. This paper presents a very efficient numerical implementation of the method with a linear description of the relevant variables over each boundary element and a newly developed Euler‐type time‐integration scheme with automatic time‐step control for time integration. Numerical results for plates in plane stress with and without cutouts, under different loading histories, are presented. A combined creep–plasticity constitutive theory with state variables is used to model material behaviour. The results are more accurate and are obtained with much less computational effort compared to a previous attempt with an uniform description of variables over each boundary element and a predictor–corrector scheme for time integration. The computer program developed is quite general and can handle plane stress problems for plates of arbitrary shapes subjected to arbitrary time‐histories of loadings. The numerical results presented in the paper are for certain illustrative problems.