Displacement Formula Research Articles

Analisis Nonlinier Geometri Struktur Frame-Truss 3D dengan Metode Elemen Hingga

Abstract: Non-linear analysis program for 3D structural geometry has not yet been used for analyzing the frame-truss structure. It needs more accurate and efficient program in analyzing the 3D structural geometry. This research is set out through this analysis procedure. The procedure is based on the displa-cement formula from finite element generating the matrix of geometrical stiffness (k g ) and the matrix of induce moment stiffness (k j ). The matrix of element stiffness is differentiated from the principle of virtual process with high order attachment. Elemen geometry updating is conducted through finite rotation concept and non-linear problems formulated as increment, while iteration is conducted by using Generalized Stiffness Parameter method, through critical points. The matrix of global stiffness is calculated firstly by summarizing the matrix of local stiffness (ke), (kg), and (kj), and then transformed into the global system. Augmented matrix is used in simultaneous linier function in each iteration process. It can hopefully reduce the running time.

Analytical solution for bending beam subject to lateral force with different modulus

A bending beam, subjected to state of plane stress, was chosen to investigate. The determination of the neutral surface of the structure was made, and the calculating formulas of neutral axis, normal stress, shear stress and displacement were derived. It is concluded that, for the elastic bending beam with different tension-compression modulus in the condition of complex stress, the position of the neutral axis is not related with the shear stress, and the analytical solution can be derived by normal stress used as a criterion, improving the multiple cyclic method which determines the position of neutral point by the principal stress. Meanwhile, a comparison is made between the results of the analytical solution and those calculated from the classic mechanics theory, assuming the tension modulus is equal to the compression modulus, and those from the finite element method (FEM) numerical solution. The comparison shows that the analytical solution considers well the effects caused by the condition of different tension and compression modulus. Finally, a calculation correction of the structure with different modulus is proposed to optimize the structure.

Comment on “Thickness dependence of exchange bias and coercivity in a ferromagnetic layer coupled with an antiferromagnetic layer” [J. Appl. Phys. 94, 2529 (2003)

Hu, Jin, and Ma have proposed a theoretical investigation on the influence of the antiferromagnetic layer thickness on the magnetic properties of ferromagnetic∕antiferromagnetic bilayers [J. Appl. Phys. 94, 2529 (2003)], considering both the bilinear and biquadratic exchange couplings, and have claimed that from their formulas for the hysteresis loop displacement and coercivity many interesting conclusions can be extracted. Unfortunately, the mathematical procedure used to find the equilibrium of the system is based on inadequate stability conditions and has led to nonphysical results. More importantly, the simple phenomenological model, employed by the authors, is intrinsically not capable to give the antiferromagnetic layer thickness dependence of the magnetic properties of such exchange-coupled bilayers.

Assessment of the current US seismic displacement requirements for bridges in a low–moderate seismic zone

Current US seismic guidelines indicate that ground acceleration of 0.10 g or greater is likely to produce plastic structural response, yet they require that an elastic seismic analysis be performed for all multi-span bridges located in regions deemed likely to sustain such ground acceleration. The relative superstructure–substructure displacements obtained from an elastic analysis of typical bridges located in a low–moderate seismic zone are usually in the range of 2.5–10 cm. The design code presents an empirical formula which approximates the upper limit of displacements due to plastic response. This formula typically gives displacements in the range of 37.5–60 cm. Since the requirement of the use of the empirical formula is predicted on the assumption of plastic response, an assessment of its validity is warranted. The primary objectives of this paper are to determine if plastic response is a real possibility for earthquakes of low to moderate magnitude (i.e. acceleration coefficient, A=0.10–0.20 g) and further to examine if the empirical displacement formula is suitable for bridges in those seismic zones.

Dynamic crack analysis under thermal shock considering Lord–Shulman theory

A boundary element method using Laplace transform in time domain is developed for the analysis of fracture mechanics considering transient coupled thermoelasticity problems with relaxation time in two-dimensional finite domain. The dynamic thermoelastic model of Lord and Shulman are selected for showing finite thermal propagation speed. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using boundary element method. Actual physical quantities in time domain is obtained, using the numerical inversion of the Laplace transform method. The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. Thermal dynamic stress intensity factor for mode I is evaluated from computed nodal values, using the well-known displacement and traction formulas. The accuracy of the method is investigated through comparison of the results with the available data in literature. Conditions where the inertia term plays an important role are discussed and variations of dynamic stress intensity factor are investigated. Different relaxation times are chosen for briefly showing the effects on stress intensity factor considering Lord and Shulman (LS) theory.

Surface Green's Functions for an Incompressible, Transversely Isotropic Elastic Half-Space

The surface displacements produced by normal and tangential point loads applied to the surface of an incompressible, transversely isotropic material are considered when anisotropy is produced by a single family of fibers oriented perpendicular to the surface normal. Three elastic constants (two shear moduli and a fiber modulus) characterize the linear elasticity of such a material. The problems are solved analytically in two-dimensional Fourier transform space, and explicit surface displacement formulae are given for the inverses in physical space. Simple relations are given as asymptotic expansions for weak anisotropy. Computed surface displacement patterns are illustrated, and the application of the results to atomic force microscopy is discussed.

On pionic hydrogen. Quantum field theoretic, relativistic covariant and model-independent approach

We consider pionic hydrogen $A_{\pi p}$ , the bound $\pi^- p$ state. Within the quantum field theoretic and relativistic covariant approach we calculate the shift and width of the energy level of the ground state of pionic hydrogen caused by strong low-energy interactions treated perturbatively. The generalization of the Deser-Goldberger-Baumann-Thirring (DGBT) formulas (S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96, 774 (1954)) is given. The generalized DGBT formulas for the energy level displacement of the ground state of pionic hydrogen contain the non-perturbative and model-independent correction of about $1\%$ , caused by the relativistic covariant smearing of the wave function of the ground state around the origin. This non-perturbative correction is very important for the precise extraction of the S-wave scattering lengths of the $\pi N$ scattering from the experimental data on the energy level displacements in pionic hydrogen by the PSI Collaboration. In addition, the shift of the energy level of the ground state of pionic hydrogen is improved by the second-order correction of strong low-energy interactions which is about 0.1%. This testifies the applicability of the perturbative treatment of strong low-energy interactions to the analysis of pionic hydrogen. We show that the width of the energy level of the ground state of pionic hydrogen is valid to all orders of the perturbation theory in strong low-energy interactions.

Deformation Analysis of 3D Braided Structural Composites

A nonlinear finite element technique is used to simulate and analyze the deformation behavior of 3D braided composites. The finite element displacement formula is derived from the geometric nonlinear behavior of braided composites. The integral relevant to the element stiffness matrix is simplified by selecting a 20-node hexahedral solid element. The Gauss-Legendre quadrature is used to compute the element matrix. A program referred to as SNLFEDA (Simulation of Nonlinear Finite Element Deformation Analysis) is developed. Tension, compression, and bending of 3D braided composites is simulated, and the results' of these simulations are supported by the experimental data and the values predicted by previously published elastic modulus models.

Hartley-type algebras in displacement and optimization strategies

The Hartley-type ( Ht) algebras are used to face efficiently the solution of structured linear systems and to define low complexity methods for solving general (nonstructured) nonlinear problems. Displacement formulas for the inverse of a symmetric Toeplitz matrix in terms of Ht transforms are compared with the well known Ammar–Gader formula. The LQN unconstrained optimization methods, which define Hessian approximations by updating n× n matrices from an algebra L , can be implemented for L=Ht with an O( n) amount of memory allocations and O( nlog n) arithmetic operations per step. The LQN methods with the lowest experimental rate of convergence are shown to be linearly convergent.

A transmission problem for fluid-structure interaction in the exterior of a thin domain

We study the behavior of weak solutions to a time-harmonic fluid-structure interaction problem in the exterior of a thin, elastic domain, in the limit of vanishing thickness. Boundary integral operators are used to reduce the exterior problem. Formal asymptotic expansions are developed and justified. The asymptotic procedure is simple, and involves the solution of a reduced exterior problem. Exact representation formulae for the elastic displacements in the thin domain are also provided.

On free vibration of a functionally graded piezoelectric rectangular plate

On the basis of three-dimensional theory equations of transversely isotropic piezoelasticity, two independent state equations with variable coefficients are derived. To this end, separation formulae for displacements and shear stresses are employed. A laminated approximation is used to transform the state equations to the ones with constant coefficients in each layer. The free vibration problem of a piezoelectric rectangular plate with a functionally graded property is then investigated. Discussion on the boundary conditions is presented.

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.

Finite element modeling of spur gearing fractures

A new numerical procedure of solution has been developed for the study of stress singularities in gear teeth fractures incorporating kinematic rotations and varying contact conditions. The parametric description of both the geometry and the operation by means of global design variables, and the approximation of elastic deformations by the finite element method enables the development of a computational design procedure which simulates the operation of spur gear drives. Considering the members of the drive as individual subdomains which can elastically deform and rotate, the engagement of operating surfaces at successive points enables the simulation of conjugate action. In the presence of cracks, the behavior of stress singularities is approximated by quarter point elements and the stress intensity factors are evaluated through displacement formulas. This generalized formulation efficiently approximates the load sharing and thus permits evaluation of the dynamic stress intensity factors along the path of contact, for any crack appearing in spur gear drives. Numerical results are presented and discussed for some important gear fractures. This method seems to be a powerful tool for computational procedures in the design of gear drives incorporating fracture defects.

NEW FORMULAE FOR DYNAMICAL THERMAL STRESSES

A generalization of the function of influence of a unit heat source to the displacements is suggested for the boundary value problems in the dynamical uncoupled thermoelasticity. This generalization is a convolution over time and bulk of two influence functions. One of them is a Green's function for the heat conduction problem. The other is a function of influence of unit concentrated forces onto bulk dilatation. Broad possibilities are shown in constructing these influence functions. In particular, the theorem on dilatation constructing is proved. To calculate the convolutions successfully the following properties of the introduced function are found to be useful. (1) In coordinates of the point of observation, the function satisfies the equations used to find the Green's functions in the problem of heat conduction, with the unit heat source being replaced by the influence function of concentrated force onto dilatation; and (2) in coordinates of the point of heat source application, it satisfies the boundary value problem used to find Green's matrix, with the unit concentrated forces being replaced by derivatives of Green's function in the problem of heat conduction. Based on the introduced influence function, some new integral formulae for displacements and stresses are obtained, which are a generalization of Mysel's formula in the theory of dynamical thermal stresses. The proposed formulae have certain advantages allowing us to unite the two-staged process of finding the solutions for boundary value problems in thermoelasticity in a single stage. It is established that, based on the obtained results it becomes possible to compile a whole handbook on the influence functions and integral solutions for boundary value problems in dynamical thermoelasticity. As examples, the solutions for two boundary value problems in the theory of dynamical thermal stresses for the half-space and quarter-space are presented.

NON-STICKING OSCILLATION FORMULAE FOR COULOMB FRICTION UNDER HARMONIC LOADING

In this paper, a new estimate for periodic non-sticking (i.e., zero stop per cycle) solutions is presented for the steady state responses of the Coulomb friction oscillator subjected to harmonic loading. Compared with the Den Hartog (1931 Transactions of the American Society of Mechanical Engineers53, 107–115 [1]) estimate, the new estimate leads to the same formulae for the maximum displacement and its time lag, but only the new estimate offers the closed-form formulae for the maximum velocity and its time lag. More importantly, a simple formula is derived for estimating the minimum driving force amplitude needed to prevent an oscillating object from sticking to the friction surface on which it slides. The validity of the assumptions made for the new estimate and the accuracy of the formulae developed are confirmed by comparing with the exact solutions (Hong and Liu 2000 Journal of Sound and Vibration229, 1171–1192 [2]). It is also found that there exists the best driving force amplitude for maximum dissipation efficiency.

CRITERIA FOR ASSUMPTIONS ON THE STRESS STATE IN THE UNCOUPLED LINEAR THERMOELASTICITY IN THE QUASISTATIONARY DISPLACEMENT FIELD APPROXIMATION IN THE CASE OF CYLINDRICAL SYMMETRY

The equations of the uncoupled linear thermoelasticity in the quasistationary displacement field approximation in case of cylindrical symmetry are transformed to the form, in which: the separated equations for σrz, σzz, et σzφ coordinates of the stress tensor (in the cylindrical coordinate system r,z,φ) are obtained; the explicit formulae for displacement coordinates ur and uz in terms of σrz, σzz volume force coordinate Fr and temperature T are given; also an explicit formula for displacement coordinate uφ in terms of σzφ volume force Fφ is given. If equations for σrz, σzz, and σzφ are replaced by assumptions on the stress state, then these equations represent criteria (necessary conditions) of validity of these assumptions. In particular, for the plane state of stress (σrz = 0 = σzz= σzφ) with no volume force the criterion for σrz reads: (∇2 stands for the Laplacean), and the criteria for σzz and σzφ are satisfied automatically.

Boundary element analysis of fatigue crack propagation micromechanisms in austempered ductile iron

The Dual Boundary Element Method (DBEM) is used in this work to model the micro mechanics of fatigue crack propagation in austempered ductile iron (ADI). Emphasis is put in devising accurate procedures for the evaluation of the interaction effects between very close crack–microcrack arrays. Fracture parameters are computed via the so-called one-point displacement formula using special crack-tip elements. Crack propagation is modelled using an incremental crack extension analysis; with crack extensions calculated using a propagation law that accounts for the near-threshold regime. Obtained results are in agreement with experimental observations, providing evidence to fracture mechanics models proposed in the literature.

Moving Cracks in Composite Materials with Initial Stresses

The dynamic problems of fracture mechanics for composite materials with initial stresses are considered in the case of cracks moving at a constant rate along a straight line. In the continuum approximation, composite materials are modeled by orthotropic nonlinearly elastic bodies with an arbitrary form of the elastic potential. A three-dimensional linearized theory of elasticity is used. The complex potentials of plane and antiplane problems of the linearized theory are used for dynamic problems. Exact solutions for Modes I, II, and III in the case of moving cracks are obtained using the Keldysh-Sedov methods. Asymptotic formulas for stresses and displacements near the crack tip for Modes I, II, and III are presented. The basic mechanical effects are analyzed with respect to the problems considered.

A Method for the Evaluation of the Durability of Reinforced-Concrete Structures from the Viewpoint of Fracture Mechanics

We develop a method for the evaluation of the durability of reinforced-concrete beams based on the concepts of fracture mechanics. A model of fracture of cracked structures taking into account the process of creep of concrete is constructed. We deduce dependences describing the influence of crack opening displacement on the growth of its length under long-term loading and present formulas for the crack-mouth opening displacement under short-term and long-term loads. The development of a periodic system of cracks is described for different types of concrete subjected to long-term loading.

Thermal Fatigue Reliability Assessment for Solder Bump Joint of BGA.

The authors have proposed the estimating method of thermal fatigue reliability for solder joints of electronic package, using the total equivalent inelastic strain range Δein and Manson-Coffin's law. However, as it is difficult to calculate it in the case of huge models, it is needed to establish a simple calculation method of Δein. In this study, the authors proposed a simple approach of FEM analysis for calculation of Δein. Approximate formula for relative displacements of solder joint using the results of the elastic analysis was examined in order to simplify huge inelastic analysis. In addition, using Statistical Design Support System (SDSS), they made the equation of inelastic strain range in the solder joints. And proposed simple estimation method of thermal fatigue life.