On Systematic Errors of Two-Dimensional Finite Element Modeling of Right Circular Planar Flexure Hinges

2004 ◽  
Vol 127 (4) ◽  
pp. 782-787 ◽  
Author(s):  
B. Zettl ◽  
W. Szyszkowski ◽  
W. J. Zhang
Keyword(s):  
Finite Element ◽  
Plane Strain ◽  
Plane Stress ◽  
Compliant Mechanisms ◽  
Three Dimensional ◽  
Systematic Errors ◽  
Plane Stress State ◽  
3D Analysis ◽  
Two Dimensional ◽  
Flexure Hinges

This paper discusses the finite element method (FEM) based modeling of the behavior of typical right circular flexure hinges used in planar compliant mechanisms. Such hinges have traditionally been approximated either by simple beams in the analytical approach or very often by two-dimensional (2D) plane stress elements when using the FEM. The three-dimensional (3D) analysis presented examines these approximations, focusing on systematic errors due to 2D modeling. It is shown that the 2D models provide only the lower (assuming the plane stress state) or the upper (assuming the plane strain state) limits of the hinge’s stiffness. The error of modeling a particular hinge by 2D elements (with either the plane stress or the plane strain assumptions) depends mainly on its depth-to-height ratio and may reach up to about 12%. However, this error becomes negligible for hinges with sufficiently high or sufficiently low depth-to-height ratios, in which either the plane strain or stress states dominate respectively. It is also shown that the computationally intensive 3D elements can be replaced, without sacrificing accuracy, by numerically efficient 2D elements if the material properties are appropriately manipulated.

scholarly journals Buckling of regular, chiral and hierarchical honeycombs under a general macroscopic stress state

2014 ◽  
Vol 470 (2167) ◽  
pp. 20130856 ◽  
Author(s):  
Babak Haghpanah ◽  
Jim Papadopoulos ◽  
Davood Mousanezhad ◽  
Hamid Nayeb-Hashemi ◽  
Ashkan Vaziri
Keyword(s):  
Finite Element ◽  
Stress State ◽  
Closed Form ◽  
Plane Stress ◽  
Plane Stress State ◽  
Cellular Structures ◽  
Two Dimensional ◽  
Buckling Strength ◽  
Macroscopic Stress ◽  
Unit Cells

An approach to obtain analytical closed-form expressions for the macroscopic ‘buckling strength’ of various two-dimensional cellular structures is presented. The method is based on classical beam-column end-moment behaviour expressed in a matrix form. It is applied to sample honeycombs with square, triangular and hexagonal unit cells to determine their buckling strength under a general macroscopic in-plane stress state. The results were verified using finite-element Eigenvalue analysis.

Evaluation of Residual Stresses in Small Diameter Stainless Steel Pipe Welds by Two-Dimensional and Three-Dimensional Finite Element Analyses

2014 ◽  
Author(s):  
Francis H. Ku ◽  
Trevor G. Hicks ◽  
William R. Mabe ◽  
Jason R. Miller
Keyword(s):  
Residual Stress ◽  
Finite Element ◽  
Stainless Steel ◽  
Residual Stresses ◽  
Three Dimensional ◽  
Steel Pipe ◽  
3D Analysis ◽  
Two Dimensional ◽  
Finite Element Analyses ◽  
Stainless Steel Pipe

Two-dimensional (2D) and three-dimensional (3D) weld-induced residual stress finite element analyses have been performed for 2-inch Schedule 80 Type-304 stainless steel pipe sections joined by a multi-layer segmented-bead pipe weld. The analyses investigate the similarities and differences between the two modeling approaches in terms of residual stresses and axial shrinkage induced by the pipe weld. The 2D analyses are of axisymmetric behavior and evaluate two different pipe end constraints, namely fixed-fixed and fixed-free, while the 3D analysis approximates the non-axisymmetric segmented welding expected in production, with fixed-free pipe end constraints. Based on the results presented, the following conclusions can be drawn. The welding temperature contour results between the 2D and 3D analyses are very similar. Only the 3D analysis is capable of simulating the non-axisymmetric behavior of the segmented welding technique. The 2D analyses yield similar hoop residual stresses to the 3D analysis, and closely capture the maximum and minimum ID surface hoop residual stresses from the 3D analysis. The primary difference in ID surface residual stresses between the 2D fixed-fixed and 2D fixed-free constraints cases is the higher tensile axial stresses in the pipe outside of the weld region. The 2D analyses under-predict the maximum axial residual stress compared to the 3D analysis. The 2D ID surface residual stress results tend to bound the averaged 3D results. 2D axisymmetric modeling tends to significantly under-predict weld shrinkage. Axial weld shrinkage from 3D modeling is of the same magnitude as values measured in the laboratory on a prototypic mockup.

Finite element analysis of peak stress and stress gradient at an elliptical through-hole

2017 ◽  
Vol 52 (5) ◽  
pp. 277-287
Author(s):  
Kristine Klungerbo ◽  
Gunnar Härkegård
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Plane Strain ◽  
Stress Gradient ◽  
Three Dimensional ◽  
Peak Stress ◽  
Two Dimensional ◽  
Element Analysis ◽  
Dimensional Finite Element ◽  
Through Hole

The peak stress and stress gradient (parameters required for fatigue strength assessment) at an elliptical through-hole in a wide plate under uniaxial tension have been studied by means of three-dimensional finite element analysis with high mesh density. Dimensionless variables have been used throughout the investigation. The accuracy of two-dimensional finite element analysis has been assessed by extrapolating peak stress at an elliptical hole to infinite plate width and mesh density and comparing the extrapolated value with the closed-form Kolosov–Inglis solution (deviation < 0.2%). First- and second-order elements with full and reduced integration have been employed. Methods for determining stress gradients, using a varying number of nodal stresses, have been investigated. The accuracy of three-dimensional finite element analysis has been assessed by comparing the plane-strain peak stress for an elliptical through-hole with the corresponding plane-strain value from two-dimensional analysis (deviation < 0.1%). Peak stresses at the apex of the elliptical through-hole have also been determined for this three-dimensional mesh assuming a free plate surface. In particular, beside the maximum peak stress and its location, peak stresses have been determined at the surface and at the mid-plane of the plate for thicknesses ranging from 0.2 to 10 times the axis of the elliptical hole. The stress gradients at these locations have been determined, too. The minimum stress gradient is observed at the location of maximum stress. For sufficiently thin and thick plates, the mid-plane stresses approach two-dimensional plane-stress and generalised plane-strain solutions, respectively.

Three-dimensional thermo-mechanical analysis of continuous casting and comparison with two-dimensional models

2018 ◽  
Vol 53 (6) ◽  
pp. 421-434
Author(s):  
Reza Vaghefi ◽  
MR Hematiyan ◽  
Ali Nayebi
Keyword(s):  
Continuous Casting ◽  
Phase Change ◽  
Galerkin Method ◽  
Plane Strain ◽  
Plane Stress ◽  
Three Dimensional ◽  
Casting Process ◽  
Mechanical Analysis ◽  
Two Dimensional ◽  
Generalized Plane Strain

In this study, a three-dimensional thermo-elasto-plastic model is developed for simulating a continuous casting process. The obtained results are compared with those from different two-dimensional analyses, which are based on plane stress, plane strain, and generalized plane strain assumptions. All analyses are carried out using the meshless local Petrov–Galerkin method. The effective heat capacity method is employed to simulate the phase change process. The von Mises yield criterion and elastic–perfectly-plastic model are used to simulate the stress state during the casting process; while, material parameters are assumed to be temperature-dependent. Based on the three-dimensional and two-dimensional models, numerical results are provided to determine the stress, displacement, and temperature fields induced in the cast material. It is observed that the present meshless local Petrov–Galerkin method is accurate in three-dimensional thermo-mechanical analysis of highly nonlinear phase change problems. Reasonable agreements are observed between the results obtained from the three-dimensional analysis with those retrieved by the generalized plane strain assumption. However, it is observed that the results obtained under plane stress/strain conditions have some significant differences with the results obtained from three-dimensional modeling of continuous casting.

scholarly journals Analysis of Excitation and Dead Vibration Modes of Quartz Resonators

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zi-Gui Huang ◽  
Zheng-Yu Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Resonance Frequency ◽  
Plane Strain ◽  
Excitation Frequency ◽  
Three Dimensional ◽  
Vibration Modes ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Natural Vibrations

This study uses the finite element method (FEM) to analyze the excitation and dead vibration modes of two-dimensional quartz plates. We first simplify three-dimensional quartz plates with plane strain simplification and then compare the modes of the simplified three-dimensional plates to those of two-dimensional plates. We then analyze quartz vibrating elements of AT-cut plates and SC-cut plates. To understand the regularity of the resonance frequency of plates that are excitable by voltage loading, we compare the natural vibrations of quartz plates with the excitation frequency generated after the plates are excited by voltage loading.

A Method to Approximate the Steady-State Creep Response of Three-Dimensional Pipe Bend Finite Element Models Under Internal Pressure Loading Using Two-Dimensional Axisymmetric Models

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
J. P. Rouse ◽  
W. Sun ◽  
T. H. Hyde ◽  
A. Morris ◽  
W. Montgomery
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Cross Section ◽  
Three Dimensional ◽  
Pressure Vessels ◽  
Steady State Creep ◽  
3D Analysis ◽  
Two Dimensional ◽  
Rupture Stress ◽  
Pipe Bend

Pipe bends are regions of geometric discontinuities in the pipe systems used in power plants and most industry recorded failures have been located around similar regions. Understanding these potential locations of weakness is therefore of great interest for the safe and economic operation of piping components. Increased predictive accuracy would assist in component design, condition monitoring, and retirement strategy decisions. Modeling of piping components for finite element analysis (FEA) is complicated by the variation of the cross section dimensions (changes in wall thicknesses or cross section ovality) around the pipe bend due to the manufacturing procedure implemented. Quantities such as peak rupture stress and creep rupture life can be greatly affected by these geometric variation (Rouse, J. P., Leom, M. Z., Sun, W., Hyde, T. H., Morris, A., “Steady-state Creep Peak Rupture Stresses in 90 Pipe Bends with Manufacture Induced Cross Section Dimension Variations”International Journal of Pressure Vessels and Piping, Volumes 105–106, May–June 2013, pp. 1–11). Three dimensional (3D) models can be used to approximate to the realistic level of detail found in pipe bends. These simulations may however be computationally expensive and could take a considerable amount of time to complete. Two dimensional (2D) axisymmetric models are relatively straight forward to produce and quick to run, but of course cannot represent the full geometric complexity around the pipe bend. A method is proposed that utilises multiple 2D axisymmetric pipe bend models to approximate the result of a 3D analysis through interpolation, thus exploiting the greatly reduced computing time observed for the 2D models. The prediction of peak rupture stress (both magnitude and location) is assessed using a simple power law material model. Comments are made on the applicability of the proposed procedure to a range of bends angles (90 deg, 60 deg, and 30 deg), as well as the effect of the stress exponent (n) and tri-axial (α) material constants. Provided that peak stresses do not occur at the bend/straight interface, the magnitude and location of the peak rupture stress can be predicted by the 2D axisymmetric interpolation method with a typical percentage difference of less than 1%.

Analysis of the Three-Dimensional Zone around the Interfacial Crack Tip: The K-Influence Domain Range in the Plane Stress State

2017 ◽  
Vol 754 ◽  
pp. 119-122
Author(s):  
Jelena M. Djoković ◽  
Snežana D. Vulović ◽  
Ružica R. Nikolić ◽  
Miroslav M. Živković ◽  
Branislav Hadzima
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Stress State ◽  
Intensity Factor ◽  
Plane Strain ◽  
Plane Stress ◽  
Three Dimensional ◽  
Crack Tip ◽  
Plane Stress State ◽  
Relevant Parameter

The bimaterial sample was analyzed to determine the three-dimensional zone at the interface crack tip and the field in the plane stress state. The solutions for the complete 3D field in different zones around the crack tip were approximated by the plane stress and plane strain states' asymptotic solutions. The difference between the solutions for the plane stress and plane strain states is defined by the three variables. The established relationship between the far field and the field around the crack tip in the plane strain conditions, enables relating the measured experimental results of the stress intensity factor to results for the stress intensity factor for the field around the crack tip, which represent the more relevant parameter for formulating the fracture criterion.

scholarly journals Analytical Solution of the Problem of Symmetric Thermally Stressed State of Thick Plates Based on the 3D Elasticity Theory

2021 ◽  
Vol 24 (1) ◽  
pp. 36-41
Author(s):  
Viktor P. Revenko ◽  
Keyword(s):  
Stress State ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Thermoelastic Stress ◽  
Plane Stress State ◽  
Median Surface ◽  
Two Dimensional ◽  
Particular Solution

An important place among thermoelasticity problems is occupied by the plane elasticity problem obtained from the general three-dimensional problem after using plane stress state hypotheses for thin plates. In the two-dimensional formulation, this problem has become widespread in the study of the effect of temperature loads on the stress state of thin thermosensitive plates. The article proposes a general three-dimensional solution of the static problem of thermoelasticity in a form convenient for practical application. To construct it, a particular solution of the inhomogeneous equation, the thermoelastic displacement potential, was added by us to the general solution of Lamé's equations, the latter solution having been previously found by us in terms of three harmonic functions. It is shown that the use of the proposed solution allows one to satisfy the relation between the static three-dimensional theory of thermoelasticity and boundary conditions, and also to construct a closed system of partial differential equations for the introduced two-dimensional functions without using hypotheses about the plane stress state of a plate. The thermoelastic stress state of a thick or thin plate is divided into two parts. The first part takes into account the thermal effects caused by external heating and internal heat sources, while the second one is determined by a symmetrical force load. The thermoelastic stresses are expressed in terms of deformations and known temperature. A three-dimensional thermoelastic stress-strain state representation is used and the zero boundary conditions on the outer flat surfaces of the plate are precisely satisfied. This allows us to show that the introduced two-dimensional functions will be harmonic. After integrating along the thickness of the plate along the normal to the median surface, normal and shear efforts are expressed in terms of three unknown two-dimensional functions. The three-dimensional stress state of a symmetrically loaded thermosensitive plate was simplified to the two-dimensional state. For this purpose, we used only the hypothesis that the normal stresses perpendicular to the median surface are insignificant in comparison with the longitudinal and transverse ones. Displacements and stresses in the plate are expressed in terms of two two-dimensional harmonic functions and a particular solution, which is determined by a given temperature on the surfaces of the plate. The introduced harmonic functions are determined from the boundary conditions on the side surface of the thick plate. The proposed technique allows the solution of three-dimensional boundary value problems for thick thermosensitive plates to be reduced to a two-dimensional case.

A Numerical Study on CMOD Compliance for Single-Edge Notched Bending Specimens

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Enyang Wang ◽  
Wenxing Zhou ◽  
Guowu Shen ◽  
Daming Duan
Keyword(s):  
Crack Length ◽  
Plane Strain ◽  
Plane Stress ◽  
Stress Condition ◽  
Three Dimensional ◽  
Plane Strain Condition ◽  
Strain Condition ◽  
Two Dimensional ◽  
Single Edge ◽  
Plane Stress Condition

Several well-known equations for estimating the crack length in the single-edge notched bending (SE(B)) specimens from the normalized crack mouth opening displacement (CMOD) compliance are evaluated based on two-dimensional (2D) and three-dimensional (3D) finite element analyses (FEAs). Two-dimensional FEAs are first carried out to verify the reported accuracy and applicable ranges for each equation based on the plane strain models with six different crack lengths. Three-dimensional FEAs are then carried out to estimate the errors of prediction of equations that evaluate the crack length from the plane stress- and plane strain-based CMOD compliances. Both plane-sided and side-grooved models are included in 3D FEAs and have seven different thickness-to-width ratios. The error of prediction of a given equation is largely impacted by the thickness-to-width ratio, the crack length, the presence of side grooves, and the use of the plane stress- or plane strain-normalized CMOD compliance. Based on the errors of prediction, the relevance of the actual state of stress in the ligament of the SE(B) specimens to the plane strain condition or the plane stress condition is inferred. Knowledge of the relevance of the plane stress condition or the plane strain condition can be used to select the corresponding CMOD compliance in crack length-CMOD equations, and, therefore, the corresponding predictive accuracy can be improved.

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