In-plane Normal Stresses Research Articles

Nonlinear analysis of RC shear walls considering tension-stiffening effect

An analytical model which can simulate the nonlinear behavior of reinforced concrete (RC) structures (such as panels and shear walls) subjected to in-plane shear and normal stresses is introduced. Based on the concept of equivalent uniaxial strain, constitutive relations of concrete are presented in the axes of orthotropy which coincide with the principal axes of total strain and rotate according to the loading history. The proposed model includes the description of biaxial failure criteria which show compressive strength enhancement and tensile resistance reduction effects for the stress states of biaxial compression and tension–compression, respectively. After tensile cracking, concrete compressive strength degradation was implemented and the tensile capacity of concrete maintained by the reinforcing steel (tension-stiffening effect) is considered. Using the concept of average stresses and strains, a criterion is proposed to simulate the tension-stiffening effect based on the force equilibriums, compatibility conditions, and bond stress-slip relationship between reinforcement and the surrounding concrete. The finite element model predictions are validated by comparison with available experimental data. In addition, correlation studies between analytical results and experimental values from idealized shear panel tests were conducted. Load–displacement relations of shear panel beams and walls under various stress conditions are then evaluated to verify the soundness of the proposed model.

Planar elastic laminates and their homogenization

The state of stress and deformation of a planar elastic-homogeneous transversely isotropic thick layer in the case of plane and axisymmetric strain respectively is determined in a systematic and uniform manner using integral transforms and transfer matrices. Next, a laminate with an arbitrary number of different layers is considered without any simplifying assumptions. Then we analyze a periodic structure consisting of many thin and identical layer groups by means of a suitable homogenization, where a layer group contains two or more different transversely isotropic homogeneous basic layers. As an example exact closed form solutions for a periodically layered half space are evaluated. The well known result, that a medium which is composed of alternating thin layers of two different elastic-isotropic substances is elastostatically equivalent to a homogeneous transversely isotropic medium is extended to the above mentioned more general case. Further, the in-plane normal stresses which are discontinuous for a finite layering are evaluated in addition to the smeared ones which are continuous and correct only in the limit of a vanishing thickness of the individual layers. The explicit knowledge of the resultant elastic constants (called effective ones) turns out to be dispensable; rather, the effective material parameters (defined in this paper) which are the weighted sums of the material parameters of the basic layers are proved to be relevant. Nevertheless, for the purpose of comparison with some published results the effective elastic constants, especially for a layer group consisting of two different elastic-isotropic substances, are evaluated additionally.

CMOS stress sensors on [100] silicon

CMOS analog stress sensor circuits based upon the piezoresistive behavior of MOSFET's are presented. On the [100] surface, these circuits provide temperature-compensated outputs that are proportional to the in-plane normal stress difference (/spl sigma/(11)'-/spl sigma//sub 22/') and the in-plane shear stress /spl sigma//sub 22/'. The circuits provide high sensitivity to stress, well-localized stress-state measurement, and direct voltage or current outputs that eliminate the need for tedious /spl Delta/R/R measurements required with more traditional resistor rosettes. The theoretical and experimental results also provide design guidance for calculating and minimizing the sensitivity of traditional analog circuits to packaging-induced die stress.

Use of Cavitating Jet for Introducing Compressive Residual Stress

In an attempt to strengthen the surface of materials, the potential of using a cavitating jet to form compressive residual stress has been investigated. Introducing compressive residual stress to a material surface provides improvement of the fatigue strength and resistance to stress corrosion cracking. In general, cavitation causes damage to hydraulic machinery. However, cavitation impact can be used to form compressive residual stress in the same way as shot peening. In the initial stage, when cavitation erosion progresses, only plastic deformation, without mass loss, takes place on the material surface. Thus, it is possible to form compressive residual stress without any damage by considering the intensity and exposure time of the cavitation attack. Cavitation is also induced by ultrasonic, high-speed water tunnel and high-speed submerged water jet, i.e., a cavitating jet. The great advantage of a cavitating jet is that the jet causes the cavitation wherever the cavitation impact is required. To obtain the optimum condition for the formation of compressive residual stress by using a cavitating jet, the residual stresses on stainless steel (JIS SUS304 and SUS316) and also copper (JIS C1100) have been examined by changing the exposure time of the cavitating jet. The in-plane normal stresses were measured in three different directions on the surface plane using the X-ray diffraction method, allowing for the principal stresses to be calculated. Both of the principal stresses are found changing from tension to compression within a 10 s exposure to the cavitating jet. The compressive residual stress as a result of the cavitating jet was found to be saturated after a certain time, but it starts decreasing, and finally, it approaches zero asymptotically. It could be verified in the present study that it was possible to form compressive residual stress by using a cavitating jet, and the optimum processing time could also be realized. The great difference between the water jet in water and air has also been shown in this regard. [S1087-1357(00)00501-3]

Unified approach to shear analysis and design

The theory of reinforced concrete originated in the late 19th century with the development of the linear theory for bending. As the first rational theory in reinforced concrete behavior, it satisfied Navier's three principles of mechanics of materials: stress equilibrium; strain compatibility; and constitutive law of material. By the 1950s this linear bending theory was generalized to become the Bernoulli compatibility truss model which covered both the linear and non-linear theories for bending, axial load and their combination. For shear and torsion, however, a rational theory that satisfies Navier's principles took the entire 20th century to develop. By the 1990s the theory for shear and torsion contained four truss models: the equilibrium (plasticity) truss model; the Mohr compatibility truss model; the rotating-angle softened-truss model; and the fixed-angle softened-truss model. Together with the strut-and-tie model for local regions and the Bernoulli compatibility truss model for bending and axial load, a unified theory of six rational models for reinforced concrete was established.The two non-linear softened-truss models have capacities for predicting the behavior of reinforced concrete elements subjected to in-plane shear and normal stresses. These two models satisfy the two-dimensional stress equilibrium, Mohr's circular strain compatibility and the softened biaxial constitutive laws of concrete. As a result, they can be used to predict the strength as well as the load-deformation history of a membrane element. The rotating-angle softened-truss model is simple to use and is adequate for situation that does not require the ‘contribution of concrete’ ( V c ), while the fixed-angle softened-truss model is more complex but is capable of predicting V c . The rotating-angle softened-truss model has been used to analyze four types of structures that are subjected predominantly to shear: low-rise shear walls, framed shear walls, deep beams and shear transfer zones. In addition, both softened truss models are used to construct the concrete stiffness matrices used in the finite element analysis. These matrices contain only three moduli in the diagonal elements. The assumption that Poisson ratios are zero in the non-diagonal elements leads to undesirable limitations of applicability. To overcome this weakness, a general model of the stiffness matrix is proposed that includes two additional Poisson ratios. All five material properties in the general matrix are currently being established by new biaxial tests of panels using proportional loading and strain-control procedures.

ZIGZAG AND INTERLAMINAR EQUILIBRIA EFFECTS IN LARGE-DEFLECTION AND POSTBUCKLING ANALYSIS OF MULTILAYERED PLATES

Concerning, higher-order shear deformation theories (HSDT) for composite plates, this article presents a numerical investigation on local characteristics (distribution along the plate thickness of in-plane displacement, in-plane normal stress, and transverse shear stress component;) in the large-deflection and postbuckling fields. The von Kdrmdn theory in conjunction with a recent mixed two-dimensional model, elsewhere denoted by the acronym RMZC (Reissner-Mindlin zigzag continuity) is employed. The RAfZC model assumes two independent fields along the plate thickness for displacements and transverse shear stresses, respectively; the displacement model describes the so-called zigzag form for the in-plane components, while the stress field fulfils interlaminar equilibria. Standard displacement formulation is enforced by employing variationalty consistent constitutive equations between stress and displacement unknowns. Finite-element-type approximations are introduced on the plate domain, leading to governing equations in the form of a nonlinear system of algebraic equations. This is solved through application of the Newton-Raphson method accounting for arc-length strategies. Explicit forms of the linear and nonlinear resulting matrices are quoted in appendixes. The numerical investigation and discussion concern large-deflection and postbuckling analysis of symmetrically laminate square thick/thin plates subjected to simply supported and clamped boundary conditions. Results of the RMZC model are compared with those related to CLT classical lamination theory (CLT) and first shear deformation theory (FSDT). It has been confirmed that the importance of effects related to the RMZC model are more evident when local characteristics are described. Furthermore, it has been found that such effects become more significant when large deflections as well as postbuckled equilibrium states of thick plates are considered, while for the thin-plate analyses, FSDT and RMZC have furnished very close results.

Stress analysis of laminated composite annular disks subjected to a concentrated transverse load using layer-wise zig-zag theory

This paper presents accurate inter-lamina stress distributions of laminated composite annular disks incorporating layer-wise zig-zag theory. This theory is based on the superposition of a global higher order shear deformation displacement field and a local linear zig-zag displacement field (RHOT). RHOT automatically satisfies displacement continuity at the layer interfaces and by further application of stress continuity at the layer interfaces and traction free boundary conditions, the unknown degrees of freedom are reduced to seven regardless of the number of layers. These are two in-plane displacements, two shear rotations, a transverse displacement and two section rotations. A four-node sector finite element in a cylindrical coordinate system is developed using RHOT. Two in-plane displacements and two shear rotations which are C0 continuous are interpolated using bilinear functions and a transverse displacement and two section rotations which are C1 continuous are interpolated using higher order Hermitian functions. In-plane normal stress and inter-lamina transverse shear stress variation through the thickness are calculated for laminated disks using a RHOT sector element and comparison is made with first order shear deformation theory predictions. The present work precedes the application of the zig-zag theory to transient dynamic analysis and is a first step at establishing the accuracy of implementing the zig-zag theory into a sector finite element.

Fixed Angle Softened Truss Model for Reinforced Concrete

A softened truss model has previously been developed for reinforced and prestressed concrete membrane elements subjected to in-plane shear and normal stresses. This existing model satisfies the three principles of mechanics of materials : two-dimensional stress equilibrium, Mohr's circular strain compatibility, and the softened biaxial constitutive laws of concrete. That is, the model can predict the strength of a membrane element as well as its load-deformation history. However, this model cannot predict the contribution of concrete observed in tests, because it is based on the assumption that the direction of the cracks (and thus, the concrete struts) is inclined at the rotating angle following the postcracking principal stresses of the concrete. This paper presents a new and more general softened truss model in which the direction of the cracks is assumed to incline at the fixed angle following the principal stresses of the applied loading. This new model, although more complex, is capable of predicting the contribution of concrete. The fixed angle softened truss model requires four constitutive laws of materials. Three have been established previously for the rotating angle softened truss model (concrete in compression, concrete in tension, and steel embedded in concrete). This paper presents the fourth constitutive law relating the average shear stress of concrete to the average shear strain.

Nonequilibrium molecular dynamics study of shear and shear-free flows in simple fluids

Nonequilibrium molecular dynamics simulations have been performed in order to compare the characteristics of planar Couette, planar elongation, uniaxial stretching, and biaxial stretching flows in simple fluids at different strain rates. After deriving the periodic boundary conditions for general flow fields and introducing some methodological improvements for elongation flow calculations we simulated the combination of shear and shear-free flows as well. We found that even at high strain rates where simple fluids exhibit strong non-Newtonian behavior (shear-thinning) it is a reasonable approximation to consider the two planar flows to be rotationally equivalent. This is because in planar Couette flow the in-plane normal stress difference of simple fluids is approximately zero even far from equilibrium. Similarly to planar Couette flow, the trace of the pressure tensor and the internal energy vary approximately as function of the 3/2 power of the strain rate in shear free flows. However, the individual diagonal elements of elongation flow pressure tensors deviate considerably from this approximation. In the extension direction the pressure seems to have a minimum in terms of the strain rate in every shear-free flow. We have discussed the implications of these results.

Residual stresses and stress-induced cracks in coated components

Based on beam theory, this paper proposes an analytical study of the thermal expansion mismatch-induced tensile residual stresses in brittle coated components. Numerical results show that the shear stress and the peeling stress at the coating-substrate interface are significant near the free ends of the coated components and become negligible in the interiorregion. In contrast, the in-plane normal stress dominantly develops in the interior region and exhibits a mirror near the free ends. In addition to the stress analysis, calculations of the strain energy release rate reveal that the crack initiation and crack spacing of the coating is a function of coating thickness and coating's elastic modulus.

Stress-based elastodynamic discrete laminated plate theory

A static laminated plate theory based on an assumed piecewise linear through-the-thickness in-plane stress distribution has been extended to include inertia effects. Based on this in-plane stress distribution assumption, outof-plane shear and normal stress component distributions were derived from the three-dimensional equations of motion, resulting in six nonzero stress components. Hamilton's variational principle was used to derive the plate equations of motion, the plate constitutive relationships, and the interface continuity equations. The governing equations were written in a form that is self-adjoint with respect to the convolution bilinear mapping. The resulting system of equations for a single lamina consists of 25 field equations in terms of 9 weighted displacement field variables, 10 stress and moment resultant field variables, and 6 out-of-plane shear and normal stress boundary field variables. For the laminated system, the mixed formulation enforces both traction and displacement continuity at lamina interfaces as it satisfies layer equilibrium. A finite element formulation based on a specialized form of the governing functional was developed. The method is illustrated with results of a free vibration analysis of sandwich and homogeneous plates for which exact solutions are available.

Interface shear stresses induced by non-uniform heating of a film on a substrate

When a thin film bonded to a substrate is heated, normal stresses in the film and shear stresses at the interface develop that may lead to debonding or fracture of the film. The thermal stresses in such a situation are analyzed here using a shear lag model. Both the thin film and substrate are modeled as linearly elastic, isotropic, homogeneous materials and are in the plane strain condition. Results from the shear lag model are compared with 2D finite element results in order to validate the approximations inherent in the shear lag model. The shear lag model is found to be valid as long as the film is thin compared with the dimension that characterizes the spatial variation of the temperature field. Studies of the shear stresses along the interface of the film and substrate using the model indicate that the maximum shear stress may be as high as 20% of the maximum in-plane normal stress in the film and hence exceed the shear strength of the interface. The work is motivated by possible applications in optical disc technology, materials processing, etc.

Interlayer shear slip theory for cross-ply laminates with nonrigid interfaces

In the conventional analysis for laminated composite materials, the composite interface is always assumed to be rigidly bonded. However, due to the low shear modulus and poor bonding, the composite interface can be nonrigid. Based on the previously developed inter laminar shear stress continuity theory and a linear shear slip law, this study presents a so-called interlayer shear slip theory to investigate the effect of the interfacial bonding on the behavior of cross-ply laminates. Closed-form solutions for the cases of the cylindrical bending of long composite strips with [0/0] and [0/90] sequences are obtained. Numerical results for the laminates with different length-to-thickness ratios are presented. They reveal that the interlayer shear slip theory is valid for cross-ply laminates with rigid and nonrigid interfaces. It is also concluded from this study that at some special locations, namely, singular points, the transverse shear stress or in-plane normal stress remains insensitive to the condition of interfacial bonding.

Imperfection sensitivity and postbuckling behavior of shear-deformable composite doubly-curved shallow panels

The static postbuckling of simply-supported single and multilayered composite doublecurved shallow panels subjected to a system of in-plane compressive edge loads is studied. The effects caused by transverse shear deformation, lamination, the character of in-plane boundary conditions and transverse normal stress, are considered and a series of pertinent conclusions are outlined. Comparisons are made of the obtained results with their classical counterparts and conclusions related to their range of applicability arc presented. Moreover, by incorporating the initial geometric imperfections, their influence on the load carrying capacity of curved composite panels is discussed and the peculiarities of their effect as compared to the case of flat panels arc emphasized.

Cylindrical bending of arbitrary cross-ply laminated plates

A refined shear deformation theory for cylindrical bending of arbitrary cross-ply laminated plates is presented. A piecewise linear in-plane displacement and constant transverse displacement field is employed. The transverse shear strains across any two different layers are assumed to be proportional to each other. Governing equations and consistent boundary conditions are deduced based upon the Principle of Minimum Potential Energy. The accuracy of the present theory is examined by applying it to a cylindrical bending problem of laminated plates which has been solved exactly. The results indicate that the present theory can accurately predict transverse displacement and in-plane displacement and normal stress at low span-to-thickness ratios.

Micromechanical investigation of the convexity of yield surfaces of metal matrix composites

This paper examines the convexity of initial yield surfaces of metal matrix composites in the presence of stiffening response of the fiber phase using a recently proposed micromechanics model. The loss of convexity of yield surfaces of unidirectional and multidirectional graphite/aluminum laminates is investigated under selected loading conditions for different fiber stiffening rates and temperature changes. In the case of unidirectional laminae, loss of convexity of initial yield surfaces for the chosen set of material parameters and loading conditions occurs for fibers whose limiting longitudinal response is four times stiffer than the initial response. Application of a negative temperature change of a sufficient magnitude eliminates the concavity. In the case of multidirectional laminates, loss of convexity of initial yield surfaces may occur for fibers which exhibits substantially less stiffering. In particular, initial yield surfaces of angle-ply configurations in the low off-axis range become concave under loading by inplane normal and shear stresses for fibers whose limiting longitudinal response is two times stiffer than the initial response.

Analysis of Interlaminar Mode II Bending Specimens Using a Higher Order Beam Theory

A higher order beam theory is used to analyze two Mode II interlaminar bending specimen geometries. The theory is based on second order displacements in the thickness coordinate and is derived in conjunction with Reissner's variational principle. Homogeneous orthotropic materials are considered. Both the inplane normal stress and the interlaminar shear stress distributions exactly satisfy the equilibrium equations of clas sical theory of elasticity. The resulting field equations are applied to an analysis of the end notch flexure (ENF) specimen and end notch cantilever (ENC) specimen. Numerical results from the present theory are compared to those obtained from finite elements for the end notch flexure specimen. Excellent agreement is obtained.

Experimental Determination of the Two-Dimensional State of Stress and the Orthotropic Elasticity Coefficients for Coatings

ABSTRACTThe state of stress for a uniform coating away from the edges reduces to that of plane stress, two in-plane normal stresses, and an in-plane shear stress. For this state, the interface between the coating and the substrate is totally stress free. Since the substrate and the coating are not interacting mechanically, an internal section of the substrate can be removed creating a tensioned drum-like membrane without altering the stress state. Holographic interferometry of vibrationally excited membranes is used to evaluate the stress. Using this technique, up to thirty vibrational modes can be obtained. This high degree of redundancy enables one to determine the one shear and two normal stresses that act in the plane of the coating. The only physical property requires is the coating density. The density is obtained from commonly reported literature values. Simple variations on the membrane vibration scheme, e.g., cutting the membrane to create a uniaxially tensioned ribbon, enables one to determine the in-plane Poisson's ratio and shearmodulus.In separate but related experiments on commercially made free-standing films with residual orientation, the above techniques, combined with special free and axially constrainedcompressibility experiments should enable all of the Poisson's ratios and elasticmoduli for an orthotropic material (nine elastic constants) to be determined. Methods for measuring the state of stress and the elastic constants are required to predict the state of stress in complex coating geometries.

Nonlinear Analysis of Cracked Reinforced Concrete

A nonlinear material model for cracked reinforced concrete subjected to inplane shear and normal stresses has been developed. As a result, a set of constitutive equations suitable for incremental finite element analysis is derived. Features of the present model include the smeared crack representation, rotating crack approach, tension stiffening, stress degrading effect for concrete parallel to the crack direction, and shear retention of concrete on the crack surface. This material model has been tested against the experimental data of Vecchio and Collins and it has been demonstrated that this material model is adequate in describing the post-cracking behavior of reinforced concrete.

Numerical/Experimental Correlation of Three-Dimensional Thermal Stress Distributions in Graphite/Epoxy Laminates

Three-dimensional finite element analysis was used to predict thermal stress distribu tions in thick graphite/epoxy laminates. The laminates analyzed were [On/90n]s and [+45n/-45n]s. For the [On/90 n]s laminates, high thermal in-plane tensile normal stresses were predicted in the transverse (2-) directions in both the 0° and 90 ° plies. High inter laminar normal (z- or 3-direction) tensile stresses were predicted to exist in the 90° plies near the center of two opposite free edges, with large compressive interlaminar normal stresses along the other two edges. Stresses were predicted to be low in all corners. In the [+45n/-45n]s laminates, high in-plane transverse (2-) direction normal tensile stresses were also predicted. High tensile interlaminar normal stresses were predicted to exist at two diagonally opposite corners, with large compressive interlaminar normal stresses at the other corners. Subsequent experiments using thick laminates with similar stacking sequences slowly cooled to -150°C confirmed the predicted stress distribu tions and the existence of a "free corner" effect in angle-ply laminates, an effect which could not have been predicted without performing a fully three-dimensional analysis. Absence of inplane symmetry in mid-plane symmetric angle-ply laminates was predicted analytically and verified experimentally. Failure mode and location were shown, analytically (based on a maximum stress failure criterion) and experimentally, to depend on stacking sequence.