The following syntax is used to describe a typical irreducible continuum element:

Similar documents
Analysis of Planar Truss

Stretching of a Prismatic Bar by its Own Weight

Two-Dimensional Steady State Heat Conduction

MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command.

Modeling a Composite Slot Cross-Section for Torsional Analysis

MATERIAL ELASTIC ANISOTROPIC command

Finite Element Method in Geotechnical Engineering

Post Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method

ME FINITE ELEMENT ANALYSIS FORMULAS

Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,

1 Exercise: Linear, incompressible Stokes flow with FE

Poly3d Input File Format

Module 2: Thermal Stresses in a 1D Beam Fixed at Both Ends

NUMERICAL ANALYSIS OF A PILE SUBJECTED TO LATERAL LOADS

Numerical Properties of Spherical and Cubical Representative Volume Elements with Different Boundary Conditions

DISPENSA FEM in MSC. Nastran

Lecture 4 Implementing material models: using usermat.f. Implementing User-Programmable Features (UPFs) in ANSYS ANSYS, Inc.

Theoretical Manual Theoretical background to the Strand7 finite element analysis system

1 Slope Stability for a Cohesive and Frictional Soil

ENGN 2340 Final Project Report. Optimization of Mechanical Isotropy of Soft Network Material

Semiloof Curved Thin Shell Elements

JEPPIAAR ENGINEERING COLLEGE

Computational Materials Modeling FHLN05 Computer lab

1 Slope Stability for a Cohesive and Frictional Soil

CAEFEM v9.5 Information

UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES

An Assessment of the LS-DYNA Hourglass Formulations via the 3D Patch Test

The Finite Element Method for Solid and Structural Mechanics

Thick Shell Element Form 5 in LS-DYNA

Contents as of 12/8/2017. Preface. 1. Overview...1

Application of pseudo-symmetric technique in dynamic analysis of concrete gravity dams

Non-linear and time-dependent material models in Mentat & MARC. Tutorial with Background and Exercises

MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM

Linear Static Analysis of a Simply-Supported Truss (SI)

Content. Department of Mathematics University of Oslo

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Interpolation Functions for General Element Formulation


Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS

3. Overview of MSC/NASTRAN

Chapter 2 Finite Element Formulations

Stress analysis of deflection analysis flexure and obif Vertical Load orientation

Computational Inelasticity FHLN05. Assignment A non-linear elasto-plastic problem

Settlement and Bearing Capacity of a Strip Footing. Nonlinear Analyses

Tutorial 2. SSMA Cee in Compression: 600S F y = 50ksi Objective. A the end of the tutorial you should be able to

Computational Inelasticity FHLN05. Assignment A non-linear elasto-plastic problem

BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I

Leaf Spring (Material, Contact, geometric nonlinearity)

Constitutive models. Constitutive model: determines P in terms of deformation

MATERIAL MECHANICS, SE2126 COMPUTER LAB 4 MICRO MECHANICS. E E v E E E E E v E E + + = m f f. f f

MATERIAL MECHANICS, SE2126 COMPUTER LAB 2 PLASTICITY

MAE 323: Chapter 6. Structural Models

A Basic Primer on the Finite Element Method

Effect of Mass Matrix Formulation Schemes on Dynamics of Structures

Converting Plane Stress Statics to 2D Natural Frequencies, changes in red

Permafrost Thawing and Deformations

Alternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE. University of Liège Aerospace & Mechanical Engineering

Nonlinear analysis in ADINA Structures

As an example, the two-bar truss structure, shown in the figure below will be modelled, loaded and analyzed.

Investigation of Utilizing a Secant Stiffness Matrix for 2D Nonlinear Shape Optimization and Sensitivity Analysis

Finite Element Method

MATERIAL MECHANICS, SE2126 COMPUTER LAB 3 VISCOELASTICITY. k a. N t

Computation Time Assessment of a Galerkin Finite Volume Method (GFVM) for Solving Time Solid Mechanics Problems under Dynamic Loads

Using Thermal Boundary Conditions in SOLIDWORKS Simulation to Simulate a Press Fit Connection

Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method

ENGN 2290: Plasticity Computational plasticity in Abaqus

Basic Equations of Elasticity

Software Verification

MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4

2D Kirchhoff Thin Beam Elements

FINITE ELEMENT APPROACHES TO MESOSCOPIC MATERIALS MODELING

A two-dimensional FE truss program

Card Variable MID RO E PR ECC QH0 FT FC. Type A8 F F F F F F F. Default none none none 0.2 AUTO 0.3 none none

Size Effects In the Crushing of Honeycomb Structures

Software Verification

Finite Element Solutions for Geotechnical Engineering

1 FLUID-MECHANICAL INTERACTION SINGLE FLUID PHASE

Finite Element Modeling and Analysis. CE 595: Course Part 2 Amit H. Varma

Project. First Saved Monday, June 27, 2011 Last Saved Wednesday, June 29, 2011 Product Version 13.0 Release

AN ALGORITHM FOR TOPOLOGY OPTIMIZATION

Advantages, Limitations and Error Estimation of Mixed Solid Axisymmetric Modeling

INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA

2D Embankment and Slope Analysis (Numerical)

Module 10: Free Vibration of an Undampened 1D Cantilever Beam

The Finite Element Method for Mechonics of Solids with ANSYS Applicotions

CRITERIA FOR SELECTION OF FEM MODELS.

ME 1401 FINITE ELEMENT ANALYSIS UNIT I PART -A. 2. Why polynomial type of interpolation functions is mostly used in FEM?

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

AN ALTERNATIVE TECHNIQUE FOR TANGENTIAL STRESS CALCULATION IN DISCONTINUOUS BOUNDARY ELEMENTS

Conservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body

SIMULATION OF FLUID-STRUCTURAL INTERACTION

Schur decomposition in the scaled boundary finite element method in elastostatics

4 Finite Element Method for Trusses

Effect of Plasticity on Residual Stresses Obtained by the Incremental Hole-drilling Method with 3D FEM Modelling

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004

2D Liquefaction Analysis for Bridge Abutment

Back Matter Index The McGraw Hill Companies, 2004

SIMULATION OF PLANE STRAIN FIBER COMPOSITE PLATES IN BENDING THROUGH A BEM/ACA/HM FORMULATION

Internet Parallel Structure Analysis Program (IPSAP) User's Guide

Chapter 11 Three-Dimensional Stress Analysis. Chapter 11 Three-Dimensional Stress Analysis

Transcription:

ELEMENT IRREDUCIBLE T7P0 command.. Synopsis The ELEMENT IRREDUCIBLE T7P0 command is used to describe all irreducible 7-node enhanced quadratic triangular continuum elements that are to be used in mechanical analyses. Syntax The following syntax is used to describe a typical irreducible continuum element: Explanatory Notes ELEment IRReducible TYPe T7P0 NODes #:#:# (MATerial #) (INItial #) (THIckness #.#) (INTcode #) (CONstruction #) (EXCavation #) (DONT PRINT Results) (DONT PRINT STRAins) (DONT PRINT STREsses) (PRINT PRIN STRAins) (PRINT PRIN STREsses) (PRINT VOLUMETRIC STRAIN).. The T7P0 is an irreducible, enhanced quadratic, isoparametric triangular continuum element [1]. The element Contains three (3) vertex nodes. Contains three (3) mid-side nodes. Contains one (1) mid-side node. Has two (2) displacements degrees of freedom at each node. Possesses a total of fourteen (14) displacement degrees of freedom. The numbering order of NODES associated with T6P0 elements, which must be specified sequentially from 1 to 7, is shown in Figure 1. NOTE: Presently APES does not possess the ability to generate T7P0 elements. It is assumed that the analyst will thus use some stand-alone pre-processing software to accomplish this task. The resulting element and node data will then be translated to the format expected by APES. 1 V. N. Kaliakin

x 2 1 x 1 4 6 7 2 3 5 Figure 1: Node Numbering Associated with a Typical Irreducible 7-Node (T7P0) Triangular Element The MATERIAL keyword is used to specify the number of the material idealization associated with the element. The default values for the MATERIAL number is one (1). The INITIAL keyword is used to specify the initial state number associated with the element. The default value for the INITIAL is zero (0). The THICKNESS keyword is used to specify the material thickness assumed for the element. Over a given element, the thickness is assumed to be constant. The default THICKNESS value is equal to one (1.0). For AXISYMMETRIC and PLANE STRAIN idealizations (see discussion of the ANALYSIS IDEALIZATION command), the THICKNESS must be equal to 1.0. For such idealizations, specified values different from 1.0 are ignored and the proper value is used. The value specified in conjunction with the INTCODE keyword describes the order of numerical integration scheme to be used in developing the element equations for the element. The commonly used numerical integration rule for T7P0 elements corresponds to a 6- point numerical integration scheme (degree of precision equal to 4) for the primary dependent variables (i.e., nodal displacements) and a 4-point scheme (degree of precision equal to 3) for the secondary dependent variables (i.e., strains and stresses). This is the default condition and requires no input using the INTCODE keyword. If a quadrature order different from the default condition is desired, the following integer values are associated with this keyword: INTCODE = 31: a 3-point numerical integration scheme (degree of precision equal to 2) is used to compute the primary dependent variables (i.e., nodal displacements) and a 1- point scheme (degree of precision equal to 1) is used to compute the secondary dependent variables (i.e., strains and stresses). INTCODE = 63: a 6-point numerical integration scheme (degree of precision equal to 4) is used to compute the primary dependent variables (i.e., nodal displacements). A 3- point scheme (degree of precision equal to 2) is used is used to compute the secondary dependent variables (i.e., strains and stresses). 2 V. N. Kaliakin

INTCODE = 64: a 6-point numerical integration scheme (degree of precision equal to 4) is used to compute the primary dependent variables (i.e., nodal displacements). A 4- point scheme (degree of precision equal to 3) is used is used to compute the secondary dependent variables (i.e., strains and stresses). This is equivalent to the aforementioned default setting. The incremental CONSTRUCTION and EXCAVATION numbers represent the time increment in which the material in this element(s) is added to or removed from the model. A CONSTRUCTION number equal to zero corresponds to a material in existence at the beginning of the analysis. Since this is the default condition, no input is required in such a case. The condition of no excavation is likewise the default. The purpose of the PRINT commands is to eliminate unnecessary output generated by APES. More precisely, if the time history of strains and/or stresses is desired only for a select few elements, this option greatly speeds program output and facilitates inspection of results by the user. Information associated with the elements specified in this section will be printed for every solution (time) step. If generation is performed using this ELEMENT IRREDUCIBLE command, then all the elements generated will be affected in a like manner by the above print control commands. Specification of the keyword DONT PRINT Results indicates that the analyst does not desire to see output of secondary dependent variables (i.e., strains and stresses) for this element. Specification of the DONT PRINT STRAINS keyword indicates that element strains are not to be printed. Under the default condition both strains are printed. Specification of the keyword DONT PRINT STRESSES indicates that stresses are not to be printed. Under the default condition stresses are printed. The PRINT PRIN STRAINS keyword indicates that principal strains are to be computed and printed for the element. Under the default condition these quantities are not computed and printed. The PRINT PRIN STRESSES keyword indicates that principal stresses are to be computed and printed for the element. Under the default condition these quantities are not computed and printed. The keyword PRINT VOLUMETRIC STRAIN causes the volumetric strain to be computed and printed for the element. In addition, the ratio of the absolute value of the volumetric strain to the absolute value of the minimum non-zero normal strain in the element is printed. That is, ε vol min (ε 11, ε 22, ε 33 ) ; min (ε 11, ε 22, ε 33 ) 0 This ratio is instructive in the assessment of mixed and mixed/penalty elements used to simulate material response in the incompressible limit. As such, this keyword would likely not 3 V. N. Kaliakin

be used in conjunction with the T7P0 element. Under the default condition the volumetric strain and the aforementioned ratio are not computed and printed. 4 V. N. Kaliakin

Example of Command Usage Element Performance in Simple Patch Test Consider a simple eight-element mesh of T7P0 elements. Although the solution domain is square with a side dimension of 2.0, the middle node is purposely not placed at the centroid of the solution domain Ω. As such, the elements are mildly distorted. A distributed traction of 20.0, acting in the positive x 2 -coordinate direction, is applied along the top boundary of Ω. Along the right boundary, a distributed traction of 10.0, acting in the positive x 1 -coordinate direction, is applied. The material is characterized using the isotropic elastic constitutive model with elastic modulus equal to 1000 and a Poisson s ratio equal to 0.30. The input data associated with this problem is given next. ana tit "patch test C involving 8 quadratic T7P0 triangles" ana tit " sig_11 = 20.0 ; sig_22 = 10.0 ; sig_33 = sig_12 = 0.0" analysis type mech analysis idealization plane_stress analysis temp transient echo init off echo memory off echo grav off echo warn off integration time parameter 0.50 dim max material isotropic elastic 1 dim max nodes 33 dim max t7p0 8 finished settings mat elastic isotropic number 1 desc " test 1 " mod = 1000.0 poisson 0.30 nodes line number 1 x1 0.0 x2 0.0 nodes line number 5 x1 10.0 x2 0.0 incr 1 nodes line number 6 x1 0.0 x2 2.5 nodes line number 10 x1 10.0 x2 2.5 incr 1 nodes line number 11 x1 0.0 x2 4.8 nodes line number 15 x1 10.0 x2 5.2 incr 1 nodes line number 16 x1 0.0 x2 7.5 nodes line number 20 x1 10.0 x2 7.5 incr 1 5 V. N. Kaliakin

nodes line number 21 x1 0.0 x2 10.0 nodes line number 25 x1 10.0 x2 10.0 incr 1 nodes line number 26 x1 1.667E+00 x2 3.267E+00 nodes line number 27 x1 3.333E+00 x2 1.667E+00 nodes line number 28 x1 6.667E+00 x2 1.667E+00 nodes line number 29 x1 8.333E+00 x2 3.400E+00 nodes line number 30 x1 1.667E+00 x2 6.600E+00 nodes line number 31 x1 3.333E+00 x2 8.333E+00 nodes line number 32 x1 6.667E+00 x2 8.333E+00 nodes line number 33 x1 8.333E+00 x2 6.733E+00 element irreducible type "t7p0" nodes 1 13 11 7 12 6 26 mat 1 element irreducible type "t7p0" nodes 1 3 13 2 8 7 27 mat 1 element irreducible type "t7p0" nodes 3 5 13 4 9 8 28 mat 1 element irreducible type "t7p0" nodes 5 15 13 10 14 9 29 mat 1 element irreducible type "t7p0" nodes 11 13 21 12 17 16 30 mat 1 element irreducible type "t7p0" nodes 13 23 21 18 22 17 31 mat 1 element irreducible type "t7p0" nodes 13 25 23 19 24 18 32 mat 1 element irreducible type "t7p0" nodes 13 15 25 14 20 19 33 mat 1 spec line quad mech node_b 21 node_end 25 1_incr 2 2_incr 1 1_hist 0 2_hist 0 & np_begin 10.0 np_end 10.0 spec line quad mech node_b 25 node_end 5 1_incr -10 2_incr -5 1_hist 0 2_hist 0 & np_begin 20.0 np_end 20.0 specification conc mech nodes 1:5 1_for 1_val 0.0 1_hist 0 2_dis 2_val 0.0 specification conc mech nodes 1:21:5 1_dis 1_val 0.0 2_for 2_val 0.0 2_hist 0 finished data solution time final 1.0 increments 1 output 1:5:1 finished load Using the above data in conjunction with the APES computer program, the results shown below are obtained. For clarity, the header that is printed at the top of the file is omitted from this file. patch test C involving 8 quadratic T7P0 triangles sig_11 = 20.0 ; sig_22 = 10.0 ; sig_33 = sig_12 = 0.0 6 V. N. Kaliakin

D Y N A M I C S T O R A G E A L L O C A T I O N Largest NODE number which can used in the mesh = 33 Max. no. of ISOTROPIC, LINEAR ELASTIC materials = 1 Max. no. of 7-node triangular (T7P0) elements = 8 = G E N E R A L A N A L Y S I S I N F O R M A T I O N = --> MECHANICAL analysis shall be performed --> Fluid flow is NOT accounted for in the analysis --> Thermal effects are NOT accounted for in analysis --> TWO-DIMENSIONAL solution domain assumed (PLANE STRESS idealization) --> Nodal coordinates will NOT be updated --> solver type used: SKYLINE --> storage type: SYMMETRIC --> "Isoparametric" mesh generation scheme used = I N T E G R A T I O N O P T I O N S = In approximating time derivatives, the value of "THETA" = 5.000E-01 7 V. N. Kaliakin

= N O N L I N E A R A N A L Y S I S I N F O R M A T I O N = --> LINEAR analysis = H I S T O R Y F U N C T I O N I N F O R M A T I O N = <<< NONE >>> = M A T E R I A L I D E A L I Z A T I O N S = --> Material number: 1 ~~~~~~~~~~~~~~~ type : isotropic linear elastic info. : test 1 Modulus of Elasticity = 1.000E+03 Poisson s ratio = 3.000E-01 Elastic bulk modulus of the solid phase = 0.000E+00 Material density of the solid phase = 0.000E+00 Combined bulk modulus for solid/fluid = 0.000E+00 = N O D A L C O O R D I N A T E S = node : 1 x1 = 0.000E+00 x2 = 0.000E+00 node : 2 x1 = 2.500E+00 x2 = 0.000E+00 node : 3 x1 = 5.000E+00 x2 = 0.000E+00 node : 4 x1 = 7.500E+00 x2 = 0.000E+00 node : 5 x1 = 1.000E+01 x2 = 0.000E+00 node : 6 x1 = 0.000E+00 x2 = 2.500E+00 8 V. N. Kaliakin

node : 7 x1 = 2.500E+00 x2 = 2.500E+00 node : 8 x1 = 5.000E+00 x2 = 2.500E+00 node : 9 x1 = 7.500E+00 x2 = 2.500E+00 node : 10 x1 = 1.000E+01 x2 = 2.500E+00 node : 11 x1 = 0.000E+00 x2 = 4.800E+00 node : 12 x1 = 2.500E+00 x2 = 4.900E+00 node : 13 x1 = 5.000E+00 x2 = 5.000E+00 node : 14 x1 = 7.500E+00 x2 = 5.100E+00 node : 15 x1 = 1.000E+01 x2 = 5.200E+00 node : 16 x1 = 0.000E+00 x2 = 7.500E+00 node : 17 x1 = 2.500E+00 x2 = 7.500E+00 node : 18 x1 = 5.000E+00 x2 = 7.500E+00 node : 19 x1 = 7.500E+00 x2 = 7.500E+00 node : 20 x1 = 1.000E+01 x2 = 7.500E+00 node : 21 x1 = 0.000E+00 x2 = 1.000E+01 node : 22 x1 = 2.500E+00 x2 = 1.000E+01 node : 23 x1 = 5.000E+00 x2 = 1.000E+01 node : 24 x1 = 7.500E+00 x2 = 1.000E+01 node : 25 x1 = 1.000E+01 x2 = 1.000E+01 node : 26 x1 = 1.667E+00 x2 = 3.267E+00 node : 27 x1 = 3.333E+00 x2 = 1.667E+00 node : 28 x1 = 6.667E+00 x2 = 1.667E+00 node : 29 x1 = 8.333E+00 x2 = 3.400E+00 node : 30 x1 = 1.667E+00 x2 = 6.600E+00 node : 31 x1 = 3.333E+00 x2 = 8.333E+00 node : 32 x1 = 6.667E+00 x2 = 8.333E+00 node : 33 x1 = 8.333E+00 x2 = 6.733E+00 = E L E M E N T I N F O R M A T I O N = --> number: 1 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 1 13 11 7 12 6 26 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 2 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 1 3 13 2 8 7 27 9 V. N. Kaliakin

integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 3 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 3 5 13 4 9 8 28 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 4 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 5 15 13 10 14 9 29 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 5 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 11 13 21 12 17 16 30 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 6 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 13 23 21 18 22 17 31 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 7 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 13 25 23 19 24 18 32 10 V. N. Kaliakin

integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... --> number: 8 (type: T7P0 ) (kind: IRREDUCIBLE ) nodes: 13 15 25 14 20 19 33 integration rule for primary variables: 6-point formula for triangles integration rule for secondary variables: 4-point formula for triangles material no. = 1 material type: isotropic linear elastic thickness = 1.000E+00... = N O D E P O I N T S P E C I F I C A T I O N S = Node ( c o o r d i n a t e s ) Number s p e c i f i c a t i o n: ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ 1 : ( x1 = 0.000E+00, x2 = 0.000E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 displacement-2 = 0.000E+00 ; history no. = -2 2 : ( x1 = 2.500E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 3 : ( x1 = 5.000E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 4 : ( x1 = 7.500E+00, x2 = 0.000E+00 ) force-1 = 0.000E+00 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 5 : ( x1 = 1.000E+01, x2 = 0.000E+00 ) force-1 = 1.600E+01 ; history no. = 0 displacement-2 = 0.000E+00 ; history no. = -2 11 V. N. Kaliakin

6 : ( x1 = 0.000E+00, x2 = 2.500E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 10 : ( x1 = 1.000E+01, x2 = 2.500E+00 ) force-1 = 6.933E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 11 : ( x1 = 0.000E+00, x2 = 4.800E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 15 : ( x1 = 1.000E+01, x2 = 5.200E+00 ) force-1 = 3.333E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 16 : ( x1 = 0.000E+00, x2 = 7.500E+00 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 0.000E+00 ; history no. = 0 20 : ( x1 = 1.000E+01, x2 = 7.500E+00 ) force-1 = 6.400E+01 ; history no. = 0 force-2 = 0.000E+00 ; history no. = 0 21 : ( x1 = 0.000E+00, x2 = 1.000E+01 ) displacement-1 = 0.000E+00 ; history no. = -2 force-2 = 8.333E+00 ; history no. = 0 22 : ( x1 = 2.500E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 3.333E+01 ; history no. = 0 23 : ( x1 = 5.000E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 1.667E+01 ; history no. = 0 24 : ( x1 = 7.500E+00, x2 = 1.000E+01 ) force-1 = 0.000E+00 ; history no. = 0 force-2 = 3.333E+01 ; history no. = 0 25 : ( x1 = 1.000E+01, x2 = 1.000E+01 ) force-1 = 1.733E+01 ; history no. = 0 force-2 = 8.333E+00 ; history no. = 0 12 V. N. Kaliakin

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ end of mathematical model data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ At time 1.000E+00 (step no. 1) NO iteration was required = E L E M E N T S T R A I N S & S T R E S S E S = --> element 1 ( type = T7P0 ):... @(x1 = 1.667E+00, x2 = 3.267E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 1.885E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 7.250E-08 @(x1 = 1.000E+00, x2 = 1.979E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -6.169E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -2.373E-09 @(x1 = 3.000E+00, x2 = 3.947E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 2.740E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 1.054E-07 @(x1 = 1.000E+00, x2 = 3.899E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 3.026E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 1.164E-07 --> element 2 ( type = T7P0 ):... @(x1 = 3.333E+00, x2 = 1.667E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -2.426E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -9.331E-08 13 V. N. Kaliakin

@(x1 = 2.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -3.205E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.233E-07 @(x1 = 4.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -1.049E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -4.036E-08 @(x1 = 4.000E+00, x2 = 3.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -3.025E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.163E-07 --> element 3 ( type = T7P0 ):... @(x1 = 6.667E+00, x2 = 1.667E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 2.106E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 8.100E-09 @(x1 = 6.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 1.530E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 5.884E-08 @(x1 = 8.000E+00, x2 = 1.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 8.722E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 3.355E-09 @(x1 = 6.000E+00, x2 = 3.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -9.859E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -3.792E-08 --> element 4 ( type = T7P0 ):... @(x1 = 8.333E+00, x2 = 3.400E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -2.300E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -8.848E-08 @(x1 = 9.000E+00, x2 = 2.021E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -3.325E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.279E-07 14 V. N. Kaliakin

@(x1 = 9.000E+00, x2 = 4.101E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -7.141E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -2.746E-08 @(x1 = 7.000E+00, x2 = 4.053E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -2.884E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.109E-07 --> element 5 ( type = T7P0 ):... @(x1 = 1.667E+00, x2 = 6.600E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -1.981E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -7.617E-08 @(x1 = 1.000E+00, x2 = 5.899E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 3.822E-12 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 1.470E-09 @(x1 = 3.000E+00, x2 = 5.947E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -3.005E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.156E-07 @(x1 = 1.000E+00, x2 = 7.979E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -3.002E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -1.155E-07 --> element 6 ( type = T7P0 ):... @(x1 = 3.333E+00, x2 = 8.333E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 9.784E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 3.763E-08 @(x1 = 4.000E+00, x2 = 7.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -9.418E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -3.622E-08 @(x1 = 4.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 1.525E-10 15 V. N. Kaliakin

sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 5.866E-08 @(x1 = 2.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 2.352E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 9.045E-08 --> element 7 ( type = T7P0 ):... @(x1 = 6.667E+00, x2 = 8.333E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -1.269E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -4.881E-08 @(x1 = 6.000E+00, x2 = 7.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -1.152E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -4.433E-08 @(x1 = 8.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -2.053E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -7.898E-08 @(x1 = 6.000E+00, x2 = 9.000E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -6.017E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -2.314E-08 --> element 8 ( type = T7P0 ):... @(x1 = 8.333E+00, x2 = 6.733E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 1.308E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 5.031E-08 @(x1 = 7.000E+00, x2 = 6.053E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 1.669E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 6.418E-08 @(x1 = 9.000E+00, x2 = 6.101E+00): eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = -1.917E-11 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = -7.374E-09 @(x1 = 9.000E+00, x2 = 8.021E+00): 16 V. N. Kaliakin

eps_11 = 1.700E-02 ; eps_22 = 4.000E-03 ; eps_33 = -9.000E-03 ; gam_12 = 2.422E-10 sig_11 = 2.000E+01 ; sig_22 = 1.000E+01 ; sig_33 = 0.000E+00 ; sig_12 = 9.317E-08 = N O D A L Q U A N T I T I E S = node: 1 ( x1 = 0.000E+00, x2 = 0.000E+00 ) u_1 = 9.340E-22, u_2 = 2.047E-22 node: 2 ( x1 = 2.500E+00, x2 = 0.000E+00 ) u_1 = 4.250E-02, u_2 = 1.308E-22 node: 3 ( x1 = 5.000E+00, x2 = 0.000E+00 ) u_1 = 8.500E-02, u_2 = 3.210E-22 node: 4 ( x1 = 7.500E+00, x2 = 0.000E+00 ) u_1 = 1.275E-01, u_2 = 1.308E-22 node: 5 ( x1 = 1.000E+01, x2 = 0.000E+00 ) u_1 = 1.700E-01, u_2 = 4.397E-22 node: 6 ( x1 = 0.000E+00, x2 = 2.500E+00 ) u_1 = 2.597E-22, u_2 = 1.000E-02 node: 7 ( x1 = 2.500E+00, x2 = 2.500E+00 ) u_1 = 4.250E-02, u_2 = 1.000E-02 node: 8 ( x1 = 5.000E+00, x2 = 2.500E+00 ) u_1 = 8.500E-02, u_2 = 1.000E-02 node: 9 ( x1 = 7.500E+00, x2 = 2.500E+00 ) u_1 = 1.275E-01, u_2 = 1.000E-02 node: 10 ( x1 = 1.000E+01, x2 = 2.500E+00 ) u_1 = 1.700E-01, u_2 = 1.000E-02 node: 11 ( x1 = 0.000E+00, x2 = 4.800E+00 ) u_1 = 6.483E-22, u_2 = 1.920E-02 node: 12 ( x1 = 2.500E+00, x2 = 4.900E+00 ) 17 V. N. Kaliakin

u_1 = 4.250E-02, u_2 = 1.960E-02 node: 13 ( x1 = 5.000E+00, x2 = 5.000E+00 ) u_1 = 8.500E-02, u_2 = 2.000E-02 node: 14 ( x1 = 7.500E+00, x2 = 5.100E+00 ) u_1 = 1.275E-01, u_2 = 2.040E-02 node: 15 ( x1 = 1.000E+01, x2 = 5.200E+00 ) u_1 = 1.700E-01, u_2 = 2.080E-02 node: 16 ( x1 = 0.000E+00, x2 = 7.500E+00 ) u_1 = 2.630E-22, u_2 = 3.000E-02 node: 17 ( x1 = 2.500E+00, x2 = 7.500E+00 ) u_1 = 4.250E-02, u_2 = 3.000E-02 node: 18 ( x1 = 5.000E+00, x2 = 7.500E+00 ) u_1 = 8.500E-02, u_2 = 3.000E-02 node: 19 ( x1 = 7.500E+00, x2 = 7.500E+00 ) u_1 = 1.275E-01, u_2 = 3.000E-02 node: 20 ( x1 = 1.000E+01, x2 = 7.500E+00 ) u_1 = 1.700E-01, u_2 = 3.000E-02 node: 21 ( x1 = 0.000E+00, x2 = 1.000E+01 ) u_1 = 8.230E-22, u_2 = 4.000E-02 node: 22 ( x1 = 2.500E+00, x2 = 1.000E+01 ) u_1 = 4.250E-02, u_2 = 4.000E-02 node: 23 ( x1 = 5.000E+00, x2 = 1.000E+01 ) u_1 = 8.500E-02, u_2 = 4.000E-02 node: 24 ( x1 = 7.500E+00, x2 = 1.000E+01 ) u_1 = 1.275E-01, u_2 = 4.000E-02 node: 25 ( x1 = 1.000E+01, x2 = 1.000E+01 ) u_1 = 1.700E-01, u_2 = 4.000E-02 node: 26 ( x1 = 1.667E+00, x2 = 3.267E+00 ) u_1 = 2.834E-02, u_2 = 1.307E-02 node: 27 ( x1 = 3.333E+00, x2 = 1.667E+00 ) 18 V. N. Kaliakin

u_1 = 5.666E-02, u_2 = 6.668E-03 node: 28 ( x1 = 6.667E+00, x2 = 1.667E+00 ) u_1 = 1.133E-01, u_2 = 6.668E-03 node: 29 ( x1 = 8.333E+00, x2 = 3.400E+00 ) u_1 = 1.417E-01, u_2 = 1.360E-02 node: 30 ( x1 = 1.667E+00, x2 = 6.600E+00 ) u_1 = 2.834E-02, u_2 = 2.640E-02 node: 31 ( x1 = 3.333E+00, x2 = 8.333E+00 ) u_1 = 5.666E-02, u_2 = 3.333E-02 node: 32 ( x1 = 6.667E+00, x2 = 8.333E+00 ) u_1 = 1.133E-01, u_2 = 3.333E-02 node: 33 ( x1 = 8.333E+00, x2 = 6.733E+00 ) u_1 = 1.417E-01, u_2 = 2.693E-02 apes -> end of analysis........ 19 V. N. Kaliakin

Bibliography [1] Kaliakin, V. N., Approximate Solution Techniques, Numerical Modeling and Finite Element Methods. New York: Marcel Dekker, Inc. (2001). 20

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback