Internal Stress Resultants Research Articles

Variable order secant matrix technique applied to the elastic postbuckling of arches

AbstractThis paper presents the implementation of Reissner's (J. Appl. Math. Phys. 1972; 23:795–804) strain–displacement relations for a planar beam using a variable order secant matrix (VOSM) technique. The secant matrices are formed using a Taylor expansion of the strain energy and hence the order of the secant and tangent matrices to be included in the nonlinear analysis can be varied depending upon the degree of nonlinearity involved in the problem. Nevertheless, a maximum of 24 matrices is sufficient to obtain predicted internal stress resultants to within 0.2% of the converged solution in all the problems tested. Using a three‐noded isoparametric beam element, the accuracy of the VOSM method is demonstrated for the elastic postbuckling of arches. Copyright © 2007 John Wiley & Sons, Ltd.

Erosion Resistance of Thick Nitride and Carbonitride Coatings Deposited using Plasma Enhanced Magnetron Sputtering

In this paper, we report the microstructural and erosion test results of thick coatings deposited using a PEMS technique. Nitrides (TiN, CrN, and ZrN) and nanocomposite carbonitride (TiSiCN) were deposited on Ti-6Al-4V disk coupon and turbine blade samples. The samples were analyzed using SEM with energy dispersive EDS, nanoindentation, and XRD. Selected samples were subjected to alumina erosion tests separately using two incident angles (30° and 90°, respectively). The nanohardness of the nitride coatings was 30 GPa for carbonitride. It was observed that at 30° TiSiCN renders the best erosion resistance. At 90° incidence, the erosion damage becomes more severe. However, one TiSiCN sample still showed an order of magnitude improved erosion resistance over the commercial TiN while delamination occurred on other samples. The delamination is believed to be the result of high internal stress of the TiSiCN coatings. A multilayered TiSiCN/Ti coating is proposed to maintain the high erosion resistance while reducing the internal stress. It is believed that this technology, once fully developed, may be applied to protect turbine engine blades and vanes from both solid particle erosion (SPE) and liquid droplet erosion (LDE) as well as piston rings of heavy-duty diesel engines from wear.

Soft-core Sandwich Panels with Embedded Rigid Inclusions

The structural response of sandwich panels hosting rigid inclusions (inserts) embedded in a soft core is analytically investigated. Three approaches for the mathematical modeling of the structural behavior of the sandwich panel hosting the rigid inclusion are formulated, examined, and compared. In the first approach, the rigid body displacement field of the inclusion is introduced through a unique constitutive law. The second approach uses a multilayered high-order model in which the rigid insert is modeled using significantly increased but finite mechanical properties. In the third approach, the embedded inclusion is considered infinitely rigid and the compatibility and interaction between the rigid insert and the adjacent structural components are imposed using Lagrange multipliers. In all the cases, the field equations and the boundary condition are derived using variational principles. The continuity conditions that connect between the region hosting the rigid insert and the adjacent ordinary sandwich regions are also derived using the variational principle. The mathematical models resulting from the first and second approaches take the form of a set of coupled differential equations. On the other hand, the third approach, which takes advantage of the rigid body displacement field of the embedded inclusion, yields a reduced order set of coupled differential-integral equations. An analytical procedure for the solution of this set of equations is derived. Numerical results that focus on the deformed shape of the panel under load, the distributions of the internal stress resultants, the shear and vertical normal stresses in the core, and the interaction between the rigid insert and the adjacent components are presented. The numerical results compare between the various modeling approaches, demonstrate their capabilities, and reveal some of the characteristics of such sandwich panels. A summary and conclusions closes the paper.

Formation of poorly crystalline iron monosulfides: Surface redox reactions on high purity iron, spectroelectrochemical studies

In the use of iron for reductive dehalogenation of chlorinated solvents in ground water, due to presence of sulfate-reducing bacteria the formation of hydrogen sulfide is expected. To simulate those processes the interface between 99.99% pure iron and 0.1 M NaHCO 3 deoxygenated solution with 3.1 × 10 −5–7.8 × 10 −3 M Na 2S · 9H 2O added was studied. The surface processes were characterised by the in situ normal Raman spectroscopy (NRS) and ex situ techniques; X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), scanning electron microscopy (SEM), energy dispersive X-ray (EDX). The open circuit potential (OCP) was monitored during in situ NRS measurements, and potentiodynamic anodic polarization measurements were carried out to reveal electrochemical behaviour of iron electrode. Open circuit potential–time transients indicated that the native oxide is unstable in deaerated bicarbonate solution and undergoes reductive dissolution (i.e. autoreduction) leaving the metallic Fe covered by Fe(OH) 2, adsorbed OH −, and patches of ‘magnetite-like’ oxide. Immediately upon injection of the Na 2S-solution the iron interface undergoes complex redox surface processes and a poorly crystalline FeS film forms. Potentiodynamic anodic polarization measurements indicated a mechanical breakdown of the FeS film. The origin and initiation of this breakdown process is not clear but is probably a result of internal stress developed during film growth. Based on surface studies supported by electrochemical measurements, a conceptual model for the complex redox processes occurring at the iron interface is proposed. This model describes the structural development of a poorly crystalline FeS, which breaks down, allowing further dissolution of the Fe and formation of FeOOH at the interface. Simultaneously and despite the existence of thick layer of FeS the entrance of hydrogen was evident as the typical hydrogen cracks in bulk of the iron material. This work shed the light on the complexity of the iron/sulfide solution interface, this knowledge is important to understand the kinetic of reduction of organic groundwater contaminants.

High-Order Bending Analysis of Unidirectional Curved ‘‘Soft’’ Sandwich Panels with Disbonds and Slipping Layers

The bending behavior of unidirectional singly curved sandwich panels with a ‘‘soft’’ core and delaminated (debonded) zones or slipping layers is described herein. The delaminated zone is assumed to exist through the width of the panels, and is in the form of an interfacial debonded zone at one of the face–core interfaces, thus subdividing the length of the panel into three zones, i.e. two fully bonded zones and one that is debonded (delaminated). The formulation uses an interfacial spring description, i.e. distributed shear and radial normal springs at the face–core interfaces, to model the effects of the slipping layers and the various delaminated (debonded) zones. The formulation is based on a closed-form high-order approach (HSAPT) that incorporates the high-order effects of the ‘‘soft’’ core. The variational approach is used to derive the governing equations along with the general explicit boundary and continuity conditions. A typical curved sandwich panel with an inner or an edge delamination, with or without contact, subjected to a distributed load and pinned supports is discussed. A comparison between the various delamination types and the fully bonded case is presented in terms of the displacements, stresses and internal stress resultants of the faces and in the vicinity of the edges of the delaminated zones. Finally, the effect of slipping interfacial layers, with or without contact when subjected to a distributed load, is considered and compared with the behavior of a fully bonded sandwich panel. From the numerical studies it is found that the membrane action of the face sheets plays a major role in the overall structural resistance of the sandwich panel compared to that of flat panels. However, in the near vicinity of the edges of the debonded zone, the membrane action does not prevent the interfacial stresses to reach very large values in the case of delamination without contact. For all of the delamination types, contact exerts a favorable effect, in the sense that the induced interfacial stresses are much smaller when compared to cases of delamination without contact.

Exact static solutions to piezoelectric smart beams including peel stresses. II. Numerical results, comparison and discussion

This part presents the numerical results, comparisons and discussion for the exact static solutions of smart beams with piezoelectric (PZT) actuators and sensors including peel stresses presented in Part I. (International Journal of Solids and Structures, 39, 4677–4695) The actuated stress distributions in the adhesive and the adhesive edge stresses varying with the thickness ratios are firstly obtained and presented. The actuated internal stress resultants and displacements in the host beam are then calculated and compared with those predicted by using the shear lag model. The stresses in the adhesive caused by an applied axial force, bending moment and shear force are calculated, and then used to compute the sensing electric charges for comparison with those predicted using the shear lag model. The numerical results are given for the smart beam with (a) one bonded PZT and (b) two symmetrically bonded PZTs, with a comparison to those predicted using the shear lag model. Novel, simple and more accurate formulas for the equivalent force and bending moment induced by applied electric field are also derived for the host beam with one PZT or two symmetrically bonded PZTs. The symmetric shear stress and the anti-symmetric peel stress components caused by a shear force are discussed. In addition, in the case of PZT edge debonding, the stress redistribution in the adhesive and the self-arresting mechanism are also investigated.

On a bending problem of thick laminated composite plates

A high-order displacement field in quadrilateral element with nine nodes and twelve-degrees-of-freedom per node is developed for bending analysis of thick arbitrary layered composite plates under transverse loads. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form and other finite element solutions.

Derivation of particular solutions for linear loading in the thick plate theory

In this paper, the particular solutions in the theory of thick plates are derived. These new particular solutions are the analytical solutions of the governing differential equations of equilibrium due to linear domain loading. Uniform domain loading can be considered as a special case of the present formulation. The expressions for the displacement and the traction kernels are derived and given in explicit form. Also the necessary kernels for the internal stress resultants are also derived and given. The derived particular solutions are verified analytically. A new set of boundary integral equation formulation for thick plates using the new particular solutions are outlined.

Diffusion bonded Mokumé Gane decorative metal laminates

Vacuum diffusion bonding has been used successfully to fabricate a range of new metal combinations suitable for Mokume Gane use. New metal systems of mutual solubility, mutual insolubility, and highly reactive couples can be used, greatly extending the range of colours available. Kirkendall porosity occurs during the fabrication of many material couples, but this is normally closed during subsequent rolling. If brittle intermetallics form at the laminate interfaces, delamination may occur during forging and rolling. However, with care, usable laminates can be rolled to sheet. Laminates which contain metals of very different mechanical properties undergo internal fracture during rolling, and this is shown to be the result of internal tensile stresses generated during the rolling operation.

Diffusion bonded Mokumé Gane decorative metal laminates

Vacuum diffusion bonding has been used successfully to fabricate a range of new metal combinations suitable for Mokume Gane use. New metal systems of mutual solubility, mutual insolubility, and highly reactive couples can be used, greatly extending the range of colours available. Kirkendall porosity occurs during the fabrication of many material couples, but this is normally closed during subsequent rolling. If brittle intermetallics form at the laminate interfaces, delamination may occur during forging and rolling. However, with care, usable laminates can be rolled to sheet. Laminates which contain metals of very different mechanical properties undergo internal fracture during rolling, and this is shown to be the result of internal tensile stresses generated during the rolling operation.

Wagner term in flexural-torsional buckling of thin-walled open-profile columns

This paper considers the Wagner term in flexural-torsional buckling of thin-walled open-profile columns, which has drawn much attention following its rejection by Ojalvo in 1981. During the ensuing debate on the subject, further issues of flexural-torsional buckling were raised highlighting the need for a comprehensive clarification. This paper collates various questions that have not been convincingly answered. In tackling these questions, two simple bar models are first used to clarify important differences between alternative representations of internal stress resultants, and their relationship with the Wagner term. With this background, a statically admissible free-body diagram is described, establishing the Wagner term without resorting to disputed means such as the use of a fictitious lateral load. The Wagner term is shown to be an internal stress resultant of the longitudinal prebuckling forces distributed across the member section; this stress resultant is of the same order of magnitude as the St Venant and warping torque. The spatial effect of stresses contributing to the Wagner term cannot be captured by the single line representation as proposed by Ojalvo.

Magnetostriction of heat-treated Fe 73.5Cu 1Nb 3Si 13.5B 9

Amorphous alloy Fe 73.5Cu 1Nb 3Si 13.5B 9 after heat treatment crystallizes into nanocrystalline phase containing supersaturated solid solution α-Fe(Si) and Fe 3Si. The magnetostriction of the alloy strongly decreases after annealing even for a few minutes from the value of 24 × 10 −6 almost by one order of magnitude. With spontaneous volume magnetostriction, the invar effect disappears. The quantity ∂ω ∂H follows the crystallization process, exhibiting, however, pronounced maxima. The magnetic anisotropy is a result of internal stresses.

Finite element analysis of laminated composite plates using a higher-order displacement model

A C° continuous displacement finite element formulation of a higher-order theory for flexure of thick arbitrary laminated composite plates under transverse loads is presented. The displacement model accounts for non-linear and constant variation of in-plane and transverse displacement model eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with nine degrees-of-freedom per node. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form, the theory of elasticity and the finite element solutions with another higher-order displacement model by the same authors. A computer program has been developed which incorporates the realistic prediction of interlaminar stresses from equilibrium equations.

Variation of stress resultants in concrete structures due to time-dependent creep and shrinkage strains of concrete

A general procedure for calculating the variation with time of the internal stress resultants and hence the stresses, in concrete structures is discussed. In particular, a study is made of the changes in the stress resultants due to time-dependent creep and shrinkage strains of concrete. A general procedure of calculating the variation in the stress resultants due to differential creep strains in concrete structures has been proposed by the author[1]. A similar procedure is followed in this paper to study these variations when creep and shrinkage strains take place simultaneously. The method leads to a system of n-linear differential equations of the form: X = AXt + B the solution of which is performed by a computer using Runge-Kutta numerical procedures. A reinforced concrete portal frame exhibiting creep and shrinkage strains is solved by the proposed method and the results are given in tabular and graphical form.

Optimally Reinforced Axisymmetric Plates

A sandwich plate is considered in which the reinforcement fibers are located entirely within the outer faces of the plate. The layout of the fibers is determined within the faces of the plate to minimize the total fiber consumption. The variation of fiber material within the optimal plate to resist the internal stress resultants due to a given loading may be obtained once the fiber layout has been determined. Only one loading case is considered for a simply supported plate, clamped plate, and simply supported annular plate.

The Analysis of Fuselages of Arbitrary Cross‐section and Taper

Continuing Chapter V, internal stress resultants are established for the open rings in (he fuselage due to the various statically equivalent and self‐equilibrating load systems. The determination of the frame redundancies leads immediately to the corresponding results for the closed rings. The procedure is developed for both single‐ and double‐cell cross‐sections with cither singly‐or doubly‐connected rings. A brief account is also given of the flexibility of a diaphragm or bulkhead, describing how cither the force or displacement method may be used. The analysis of the fuselage basic system concludes with an investigation of thermal stresses in the closed frames.