elfug

Elfini Solver Verification

Version 5 Release 14

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Overview Conventions What's New? User Tasks Static Analysis Cylindrical Roof Under its Own Weight Morley's Problem Twisted Beam Thick Cylinder Space Structure on Elastic Supports Bending of a Beam Frequency/Modal Analysis Free Vibrations of a Compressor Blade Free Thin Square Plate Free Vibrations of a Simply-supported Thin Square Plate Plane Vibrations of a Simply-supported Double Cross Deep Simply-supported Solid Beam Modal Analysis of a Beam with Axial Load Buckling Analysis Buckling of a Straight Beam (Out-of-Plane Buckling) Buckling of a Straight Beam (In-Plane Buckling) Lateral Buckling of Narrow Rectangular Beam Dynamic Response Analysis Harmonic Forced Vibration of a Simply-supported Thin Square Plate Transient Forced Vibration of a Simply-supported Thin Square Plate Transient Dynamic Response of a Clamp Beam with Different Inertia Transient Response of a Spring-mass System with Imposed Acceleration Harmonic Forced Vibration of a Plane Grid Thermo Mechanical Analysis Thermal Expansion of a Beam Analysis of an Assembly Beams with Fastened Connection Composites Bonded Blades in Compression Index



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OverviewWelcome to the ELFINI Solver Verification Guide. This guide provides a series of mechanical problems to test the accuracy of the ELFINI Finite Element Solver. This overview provides the following information:q

Before reading this manual Getting the most out of this guide Accessing sample documents Conventions used in this guide

q

q

q

Before Reading this GuideBefore reading this guide, we recommend that you read the Generative Structural Analysis User's Guide and Advanced Meshing Tools User's Guide. You may also like to read the following product guides, for which the appropriate license is required:q

Infrastructure User's Guide Version 5 Part Design User's Guide Assembly Design User's Guide

q

q

Getting the Most Out of this GuideTo get the most out of this guide, each mechanical problem has an analytical or reference solution and an associated CATPart or CATAnalysis document. This guide is divided in the following distinctive chapters:q

Static Analysis: description of several linear static analysis problems and comparison of results. Modal Analysis: free vibration analysis of several problems and comparison of results. Buckling Analysis: computation of critical loads creating structure instability and comparison of results. Dynamic Response Analysis: dynamic analysis of several problems and comparison of results. Thermo Mechanical Analysis: thermal expansion of an element and comparison of results. Analysis Assembly: analysis of several elements assembled with a fastened connection and comparison of results. Composites: analysis of composites created with isotropic materials and comparison of results.

q

q

q

q

q

q



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Accessing sample documentsTo perform the scenarios, you will be using sample documents contained in the online/elfug/samples folder. For more information about this, please refer to Accessing Sample Documents in the Infrastructure User's Guide.



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ConventionsCertain conventions are used in CATIA, ENOVIA & DELMIA documentation to help you recognize and understand important concepts and specifications.

Graphic ConventionsThe three categories of graphic conventions used are as follows:q

Graphic conventions structuring the tasks Graphic conventions indicating the configuration required Graphic conventions used in the table of contents

q

q

Graphic Conventions Structuring the TasksGraphic conventions structuring the tasks are denoted as follows: This icon... Identifies... estimated time to accomplish a task a target of a task the prerequisites the start of the scenario a tip a warning information basic concepts methodology reference information information regarding settings, customization, etc. the end of a task



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functionalities that are new or enhanced with this release allows you to switch back to the full-window viewing mode

Graphic Conventions Indicating the Configuration RequiredGraphic conventions indicating the configuration required are denoted as follows: This icon... Indicates functions that are... specific to the P1 configuration specific to the P2 configuration specific to the P3 configuration

Graphic Conventions Used in the Table of ContentsGraphic conventions used in the table of contents are denoted as follows: This icon... Gives access to... Site Map Split View mode What's New? Overview Getting Started Basic Tasks User Tasks or the Advanced Tasks Workbench Description Customizing Reference Methodology Glossary



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Index

Text ConventionsThe following text conventions are used:q

The titles of CATIA, ENOVIA and DELMIA documents appear in this manner throughout the text. File -> New identifies the commands to be used. Enhancements are identified by a blue-colored background on the text.

q

q

How to Use the MouseThe use of the mouse differs according to the type of action you need to perform. Use this mouse button... Whenever you read...

q

Select (menus, commands, geometry in graphics area, ...) Click (icons, dialog box buttons, tabs, selection of a location in the document window, ...) Double-click Shift-click Ctrl-click Check (check boxes) Drag Drag and drop (icons onto objects, objects onto objects)

q

q

q

q

q

q

q

q

Drag Move

q

q

Right-click (to select contextual menu)



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What's New?New FunctionalitiesAnalysis AssemblyBeams with Fastened Connections You can test an assembly of two beams and compare it with an entire beam.

CompositesBonded Blades in compression You can test surface mesh on composites created with isotropic materials.



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User TasksStatic Analysis Frequency/Modal Analysis Buckling Analysis Dynamic Response Analysis Thermo Mechanical Analysis Analysis of an Assembly Composites



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Static AnalysisThis chapter contains static linear analysis problems which illustrate some of the features and capabilities of CATIA-ELFINI. Static linear analysis consists in finding the deformed shape and the internal strains and stresses of an elastic structure subject to prescribed boundary conditions (displacement and traction types). This chapter contains the following models and tasks: Cylindrical Roof Under its Own Weight Morley's Problem Twisted Beam Thick Cylinder Space Structure on Elastic Supports Bending of a Beam



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Cylindrical Roof Under Its Own WeightThis test, also known as the Scordelis-Lo roof, lets you check analysis results for a cylindrical roof under its own weight, in the context of a static case. You will use 2D meshes.

Before you begin: The Cylindrical_Roof_QD8_15nodes.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic quadrangle elements (QD8). To compute the case with linear quadrangle (QD4), linear triangle (TR3) and parabolic triangle (TR6) elements, proceed as follow: 1. Open the CATAnalysis document. 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below. 3. In the Generative Structural Analysis workbench, compute the case.

SpecificationsGeometry Specifications

Length: L = 3000 mm

Radius: R = 3000 mm

Thickness: th = 30 mm

Angle: = 40 deg



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Analysis Specifications

Young Modulus (material): E = 30000 MPa

Poisson's Ratio (material): =0

Load: Surface Force Density (weight of the roof): q = 6250 N_m 2 on y

Mesh Specifications:q

Number of nodes: 5 x 5 Number of nodes: 9 x 9 Number of nodes: 15 x 15

q

q

Restraints: On B-C: Tx = Ty = Rz = 0 On C-D: Tz = Rx =Ry = 0 On D-A: Tz = Rx = Rz = 0



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ResultsThe results (in mm) correspond to the normalized vertical displacement at Point A.

Computed Values Linear Triangle Number of Nodes Shallow Shell Deep Shell (TR3) Parabolic Triangle (TR6) Parabolic Quadrangle (QD8) Theoretical Values

Linear Quadrangle (QD4)

5x5

29.56

40.83

40.18

37.05

37.03

36.10

9x9

34.57

37.88

37.84

36.39

37.03

36.10

15 x 15

37.04

38.54

37.12

36.74

37.03

36.10

Reference: Computer analysis of Cylindrical Shells. SCORDELIS A.C., LO K.S., Journal of the American Concrete Institute VOL 61 pp 539-561 1969



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Morley's ProblemThe purpose of this test is to evaluate the sensitivity of a plate element to the direction (angular orientation) of the mesh on a plate in bending, in a context of a static case. You will use 2D meshes.

Before you begin: The Morley_60degree_TR6_7nodes.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of 7 nodes per edge and parabolic triangle elements (TR6). The Morley_80degree_QD8_13nodes.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of 13 nodes per edge and parabolic quadrangle elements (QD8). To compute cases with other element types, angular orientation, and number of nodes, proceed as follow: 1. Open the corresponding CATAnalysis document. 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below. 3. In the Generative Structural Analysis workbench, compute the case.


Length: L = 1 mm

Thickness: th = 0.001 mm



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Angle: = 80, 60, 40, 30 deg

Analysis SpecificationsYoung Modulus (material): E = 3.107 MPa

Poisson's ratio (material): = 0.3

Boundary conditions: Tx = Ty = Tz = 0 on all edges

Pressure load: 1 MPa on the plate



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ResultsThe vertical displacements at the center [mm] for various mesh directions are given in the following tables.

Output Results (7x7 nodes)Computed Value Linear Angle Parabolic Linear Quadrangle (QD4) Parabolic Quadrangle (QD8) Theoretical value

Triangle (TR3) Triangle (TR6)

80 deg

1.359 10-3

1.414 10-3

1.42 10-3

1.425 10-3

1.409 10-3

60 deg

0.925 10-3

0.9334 10-3

0.935 10-3

0.901 10-3

0.9318 10-3

40 deg

0.3455 10-3

0.3592 10-3

0.3538 10-3

0.2923 10-3

0.3487 10-3

30 deg

0.1474 10-3

0.1483 10-3

0.1538 10-3

0.1106 10-3

0.1485 10-3

Output results (13x13 nodes)Computed Value Linear Angle Parabolic Linear Quadrangle (QD4) Parabolic Quadrangle (QD8) Theoretical value

Triangle (TR3) Triangle (TR6)



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80 deg

1.409 10-3

1.431 10-3

1.422 10-3

1.422 10-3

1.409 10-3

60 deg

0.9403 10-3

0.9412 10-3

0.937 10-3

0.9306 10-3

0.9318 10-3

40 deg

0.3566 10-3

0.3558 10-3

0.3493 10-3

0.3271 10-3

0.3487 10-3

30 deg

0.153 10-3

0.1537 10-3

0.1497 10-3

0.1307 10-3

0.1485 10-3

Reference:q

MORLEY L.S.D., Skew plates and Structures, Pergamon Oxford 1963 RAZZAQUE A., Program for Triangular Bending Elements with Derivative Smoothing, IJNME Vol 6, pp333-343, 1973

q



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Twisted BeamThis test lets you check analysis results for a twisted beam, in the context of a static case. You can use either 2D and 3D meshes.

Before you begin: The Twisted_beam_TE10.CATAnalysis document presents a complete analysis, computed with a mesh formed of parabolic tetrahedron elements (TE10). The Twisted_beam_TR6.CATAnalysis document presents a complete analysis, computed with a mesh formed of parabolic triangle elements (TR6). To compute the case with linear triangle (TR3), linear quadrangle (QD4), parabolic quadrangle (QD8), and linear tetrahedron (TE4) elements, proceed as follow: 1. Open the corresponding document:r

Twisted_beam_TE10.CATAnalysis to compute a volume mesh, Twisted_beam_TR6.CATAnalysis to compute a surface mesh.

r

2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below. 3. In the Generative Structural Analysis workbench, compute the case.

SpecificationsGeometry specificationsLength: L = 12 m

Width: B = 1.1 m




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Rotation: 90 degrees between O and A

Analysis specificationsYoung Modulus (material): E = 29 MPa

Poisson's ratio (material): = 0.22

Size of Elements: 275 mm

Restraint: Clamp on section O

Loads: Two loads (surface force density) are applied on the beam section at A. Pz = 1 N or Py = 1 N



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ResultsThe results correspond to the displacement in the direction of the load at Edge A. Normalized results (computed results divided by analytic solution) are listed.

Linear Triangle (TR3)

Parabolic Triangle (TR6)

Linear Quadrangle (QD4)

Parabolic Quadrangle (QD8)

Linear Tetrahedron (TE4)

Parabolic Tetrahedron (TE10)

Uz [mm] Pz = 1

2.842

1.751

1.752

1.75

0.83

1.749

Normalized Results

1.62

1.00

1.00

1.00

0.47

1.00

Uy [mm]

9.806

5.416

5.416

5.418

2.352

5.422

Py = 1 Normalized Results 1.81 1.00 1.00 1.00 0.43 1.00

Reference: MAC NEAL,R.,HARDER,R.L., A proposed standard set of problems to test finite element accuracy, Finite Element Analysis Design, Vol. 1, P.3-20, 1985.



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Thick Cylinder Under Internal PressureThis test lets you check analysis results for a thick cylinder under internal pressure, in the context of a static case. You will use 3D meshes.

Before you begin: The Thick_cylinder_TE10.CATAnalysis document presents a complete analysis, computed with a mesh formed of parabolic tetrahedron elements (TE10). The Thick_cylinder_HE20.CATAnalysis document presents a complete analysis, computed with a mesh formed of parabolic hexahedron elements (HE20). To compute the case with linear tetrahedron elements (TE4) and linear hexahedron elements (HE8), proceed as follow:

1. Open the document:r

Thick_cylinder_TE10.CATAnalysis for linear tetrahedron elements (TE4), Thick_cylinder_HE20.CATAnalysis for linear hexahedron elements (HE8).

r

2. Enter the Advanced Meshing Tools workbench. 3. In the specification tree, double-click on the mesh:r

Volume mesh for linear tetrahedron elements (TE4), Surface mesh for linear hexahedron elements (HE8).

r

The Global Parameters dialog box is open. 4. Select the Linear element type. 5. Compute the case in the Generative Structural Analysis workbench.



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SpecificationsGeometry SpecificationsInner radius: a = 0.1 m

Outer radius: b= 0.2 m

Height: H = 20 mm

The structure and the load being cylindrically symmetrical, only one quarter of the cylinder is modeled.

Analysis SpecificationsYoung Modulus (material): E = 2. 105 MPa

Poisson's Ratio (material): = 0.3

Mesh Specifications: Size = 10mm

Restraints: The normal direction is restraint for upper, downer and side faces

Load: Uniform radial pressure: P = 60 MPa on the inner face



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The analytical solution of this problem is q For both components of the stress tensor within the part cylindrical axis:

q

For the component of a displacement using the same axis:

ResultsComputed Value Linear Tetrahedron Position in Space Image Type Reference Value (TE4) Parabolic Tetrahedron (TE10) Linear Hexahedron (HE8) Parabolic Hexahedron (HE20)

Inner Quarter

Stress principal tensor component (with C3 component filter) -60 MPa -56.8 MPa -60.1 MPa -52.7 MPa -59.8 MPa



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Stress principal tensor component (with C1 component filter) 100 MPa 104 MPa 100 MPa 104 MPa 101 MPa

Translational displacement vector Outer Quarter 59x10-6m 57.2x10-6m 57.2x10-6m 57.1x10-6m 57.2x10-6m

Stress principal tensor component (with C3 component filter) 0 MPa -0.413 MPa 0.068 MPa -1.02 MPa 0.0432 MPa

Stress principal tensor component (with C1 component filter) 40 MPa 39 MPa 40 MPa 39.3 MPa 39.7 MPa

Translational displacement vector 40x10-6m 36.2x10-6m 36.4x10-6m 36.3x10-6m 36.4x10-6m

Reference: New developments in the finite element analysis of shells. LINDBERG G.M., OLSON M.D., COWPER G.R. - Q. Bull div. Mech. Eng. and Nat. Aeronautical Establishment, National Research Council of Canada.



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Space Structure on Elastic SupportThis test lets you check space structure on elastic support, in the context of a static case. You will use 1D mesh. This test proposed by SFM is used to validate the following attribute: discrete elastic coupling.

The SpaceStructure.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of beam elements. Proceed as follow:

1. Open the CATAnalysis document, 2. Compute the case and generate an image called Deformed mesh.

SpecificationsGeometry SpecificationsLength: L = 2000 mm

Inertia: Ix/2 = Iy = Iz = 10-6 m4

Area: a = 10-3 m2

Analysis SpecificationsYoung Modulus (material): E = 210 GPa



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Load (traction at D): Fz = -10000 N

Boundary Conditions:q

Displacement Type:r

at A: four springs (one in translation, three in rotation, one of which is infinitely rigid)

r

at B: four springs (one in translation, three in rotation, one of which is infinitely rigid)

r

at H: ball joint (articulation with all three rotations released)

q

Traction Type:r

at D: Fz = -10000 N

ResultsThe results correspond to the quantity at each point and according to x, y, or z.

Beam Output ResultsPoint Quantity Reference Computed Result



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A

Mx [N_m]

-8437.5

-8438.14

A

My [N_m]

-1562.5

-1562.3

A

Mz [N_m]

3125.0

3124.6

B

Mx [N_m]

1562.5

1562.5

B

My [N_m]

8437.5

8437.05

B

Mz [N_m]

3125.0

3125.0

A

v [mm]

-29.76

-29.76

A

[deg]

9.208

9.209

D

w [mm]

370.04

370.07

Reference: Guide de validation des progiciels de calcul de structures, SSLL 04/89, pp.26-27, AFNOR technique 1990, SFM 10 Avenue Hoche 75008 PARIS



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Bending of a BeamThis test lets you check analysis results for the bending of a beam, in the context of a static case. You will use 3D meshes. Before you begin: The Bending_of_a_beamTET10.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic tetrahedron elements (TE10). The Bending_of_a_beamHE20.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic hexahedron elements (HE20). To compute the case with linear tetrahedron elements (TE4) and linear hexahedron elements (HE8), proceed as follow:


Bending_of_a_beamTET10.CATAnalysis for linear tetrahedron elements (TE4), Bending_of_a_beamHE20.CATAnalysis for linear hexahedron elements (HE8).

r


Volume mesh for linear tetrahedron mesh (TE4), Surface mesh for linear hexahedron mesh (HE8).

r

The Global Parameters dialog box is now open. 4. Select the Linear element type. 5. Compute the case in the Generative Structural Analysis workbench.

SpecificationsGeometry SpecificationsCH = L = 6m

AH = L' = (2/3) L

DE = GF = 2a = 2m

EF = DG = 2b = 2m


Elfini Solver VerificationYoung Modulus (material): E = 2e + 005 GPa


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Mesh Specifications: 3D elements of 500mm

Load: Moment on C: My = 4/3e + 007 Nm

Restraint: U = V = W = Rz = 0 on H

Resultsq

The analytic solution is:

q

The table below presents the analysis results. The results correspond to the means of the stress and the displacements in the section A. Normalized results (computed results divided by analytical solution) are listed.

Elfini Solver VerificationLocalization Type of Values Analytical Solution Linear

Version 5 Release 14Parabolic Tetrahedron (TE10) Linear Hexahedron (HE8)

Page 29Parabolic Hexahedron HE20

Tetrahedron (TE4)

Computed Normalized Computed Normalized Computed Normalized Computed Normalized Results Results Results Results Results Results Results Results

Section A

zz[Mpa]

10

7.6

0.76

10

1

9.97

0.997

9.99

0.99

A

UA [mm]

-0.4

-0.329

0.823

-0.388

0.97

-0.389

0.973

-0.39

0.975

B

WB [mm]

0.2

0.17

0.85

0.197

0.985

0.197

0.985

0.197

0.985

F or G

VF=-VG [mm]

0.015

0.012

0.8

0.01485

0.99

0.0149

0.993

0.0149

0.993

D or E Reference:

VD=-VE [mm]

-0.015

-0.012

0.8

-0.01485

0.99

-0.0149

0.993

-0.0149

0.993

Guide de validation des prologiciels de calcul de structures, SFM, Afnor Technique pp124-125



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Modal AnalysisThis chapter contains modal analysis problems. A modal analysis consists in finding the natural vibrations frequencies and mode shapes of an elastic structure. This chapter contains the following models and tasks: Free Vibrations of a Compressor Blade Free Thin Square Plate Free Vibrations of a Simply-supported Thin Square Plate Plane Vibration of Simply-supported Double Cross Deep Simply-supported Solid Beam Modal Analysis of a Beam with Axial Load



Page 31

Free Vibrations of a Compressor BladeThis test lets you check analysis free vibrations results for a compressor blade in the context of a modal (or frequency) case. You will use 2D meshes.

Before you begin: The Free_vibrations_compressor_blade_QD8.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic and quadrangular elements (QD8). To compute the case with linear quadrangle (QD4), linear triangle (TR3) and parabolic triangle (TR6) elements, proceed as follow: 1. Open the CATAnalysis document, 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below, 3. In the Generative Structural Analysis workbench, compute the case.

SpecificationsGeometry SpecificationsLength: L = 304.8 mm

Radius: R = 609.6 mm

Thickness: th = 3.048 mm



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Angle: A = 28.6479 deg (or 0.5 rad)

Analysis SpecificationsYoung Modulus (material): E = 2.0685e+011 N_m2


Density: r = 7857.2 Kg/m3

Mesh Specifications: 11 x 11 nodes

Restraint: BC straight edge is free AD straight edge is clamped



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ResultsThe table below presents the frequencies of the first six modes. Computed Value Linear Triangle 11 x 11 nodes (TR3) Parabolic Triangle (TR6) Linear Quadrangle (QD4) Parabolic Quadrangle (QD8) Reference value [Hz]

Mode 1

84.2461

82.8426

83.6888

85.7079

85.6

Mode 2

141.824

136.908

136.16

138.555

134.5

Mode 3

248.017

242.163

252.195

245.927

259.0

Mode 4

341.879

333.202

337.331

340.94

351.0

Mode 5

380.636

373.174

381.633

382.989

395.0

Mode 6

538.508

517.818

546.994

527.982

531.0

Reference:q

OLSON,M.D.,LINDBERG,G.M., Vibration analysis of cantilevered curved plates using a new cylindrical shell finite element, 2nd Conf. Matrix Methods in Structural Mechanics, WPAFB, Ohio 1968 OLSON,M.D.,LINDBERG,G.M., Dynamic analysis of shallow shells with a doubly curved triangular finite element, JSV, Vol. 19, No 3, pp 299-318, 1971

q



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Free Thin Square PlateThis test lets you check analysis results for a free thin square plate, in the context of a Free Frequency Case. You will use 2D meshes. This test proposed by NAFEMS is used to validate the following attributes:q

Rigid body modes (3 modes) Repeated eigen values Kinematically incomplete suppressions

q

q

Before you begin: The Free_Thin_Square_Plate_QD4.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of linear quadrangle elements (QD4). To compute the case with other elements (QD8, TR3 and TR6), proceed as follow: 1. Open the CATAnalysis document, 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below, 3. In the Generative Structural Analysis workbench, compute the case.


Length: L = 10000 mm



Elfini Solver VerificationYoung Modulus (material): E = 200 GPa


Page 35


Density: =8000 kg/m3


8x8 elements mesh for linear element (TR3 and QD4) 4x4 elements mesh for parabolic element (TR6 and QD8)

q

The structure is free, hence the boundary conditions are: Ty = Tz = Rx = 0

The Frequency case must be computed in Lanczos method with Shift Auto.

ResultsThe computed results present the frequency of the six modes after the three rigid body modes. The normalized results are the computed results divided by the reference solution.

Values Linear Triangle (TR3) Mode Reference Value Linear Quadrangle (QD4) Parabolic Triangle (TR6) Parabolic Quadrangle (QD8)

Computed Normalized Computed Normalized Computed Normalized Computed Normalized Results Results Results Results Results Results Results Results 4 1.622 1.614 0.995 1.624 1.001 1.584 0.977 1.622 1.000

Elfini Solver Verification5 2.360 2.379 1.008

Version 5 Release 142.389 1.012 2.376 1.007 2.367

Page 361.003

6

2.922

2.961

1.013

2.980

1.020

2.950

1.010

2.935

1.004

7

4.233

4.229

0.999

4.252

1.004

4.135

0.977

4.174

0.986

8

4.233

4.231

1.000

4.252

1.004

4.149

0.980

4.174

0.986

9

7.416

7.653

1.032

7.793

1.051

7.439

1.003

7.550

1.018

Reference: BENCHMARK newsletter, April 1989, p.17, NAFEMS - Glasgow



Page 37

Free Vibrations of a Simply-supported Thin Square PlateThis test lets you check analysis results for a simply-supported thin square plate, in the context of a free frequency case. You will use 2D meshes. This test proposed by NAFEMS is used to validate the following attributes:q

2D shell elements (i.e. the elementary stiffness and mass matrices) Free frequency solve algorithms (mainly repeated eigen values) Convergence of 2D shell elements (via several meshes)

q

q

Before you begin: The Free_vibrations_simply_Thin_Square_Plate.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of 16 elements per edge, and parabolic quadrangle elements (QD8). To compute the case with other mesh sizes and element types, proceed as follow: 1. Open the CATAnalysis document. 2. In the Advanced Meshing Tools workbench, replace the mesh size and element type. 3. In the Generative Structural Analysis workbench, compute the case, and activate the image called Deformed mesh.





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Density: = 8000 kg/m3

The boundary conditions are: Tx = Ty = Rz = 0 at all nodes Tz = 0 along all four edges Rx = 0 along edges x = 0 and x = 10000 mm Ry = 0 along edges y = 0 and y = 10000 mm



Page 39

ResultsYou will find here the results for different finite elements:q

parabolic triangle shell linear triangle shell parabolic quadrangle shell linear quadrangle shell

q

q

q

Parabolic Triangle Shell (32x32 elements)Parabolic triangle shell (32x32 elements)

Mode Reference Frequency [Hz]

Frequency [Hz]

Error [%]

1

2.377

2.377

0.013

2

5.942

5.943

0.019

3

5.942

5.943

0.021

4

9.507

9.516

0.095

5

11.884

11.882

0.013



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6

11.884

11.883

0.010

7

15.449

15.467

0.118

8

15.449

15.469

0.126

To visualize the convergence of parabolic triangle shell elements with other mesh refinements, please click here.

Linear Triangle Shell (64x64 elements)Linear triangle shell (64x64 elements)


Frequency [Hz]

Error [%]

1

2.377

2.378

0.022

2

5.942

5.947

0.082

3

5.942

5.947

0.084

4

9.507

9.520

0.133

5

11.884

11.904

0.167

6

11.884

11.904

0.171

7

15.449

15.483

0.218

8

15.449

15.483

0.221



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To visualize the convergence of linear triangle shell elements with other mesh refinements, please click here.

Parabolic Quadrangle Shell (16x16 elements)Parabolic quadrangle shell (16x16 elements)


Frequency [Hz]

Error [%]

1

2.377

2.376

0.022

2

5.942

5.940

0.026

3

5.942

5.940

0.026

4

9.507

9.503

0.038

5

11.884

11.880

0.034

6

11.884

11.880

0.034

7

15.449

15.442

0.049

8

15.449

15.442

0.049

To visualize the convergence of parabolic quadrangle shell elements with other mesh refinements, please click here.

Linear Quadrangle Shell (32x32 elements)Linear quadrangle shell (32x32 elements)



Page 42


Frequency [Hz]

Error [%]

1

2.377

2.378

0.030

2

5.942

5.954

0.203

3

5.942

5.954

0.203

4

9.507

9.523

0.171

5

11.884

11.952

0.568

6

11.884

11.952

0.568

7

15.449

15.510

0.392

8

15.449

15.510

0.392

To visualize the convergence of linear quadrangle shell elements with other mesh refinements, please click here.

Reference: BENCHMARK newsletter, Report No. E1261/R002, February 1989, p.21, NAFEMS - Glasgow



Page 43

Plane Vibrations of a Simply-supported Double CrossThis test lets you check the plane vibration of a simply-supported double cross in the context of a modal (or frequency) case. You will use a 1D mesh. This test proposed by NAFEMS is used to validate the following attributes:q

bending-extension coupling multiple and close eigen values.

q

Before you begin: The DoubleCross.CATAnalysis document presents a complete analysis of this case. Proceed as follow:


SpecificationsGeometry SpecificationsThe recommended Finite Element Model uses four beam elements per arm.

Length: L = 5000 mm



Page 44

Height: h = 125 mm



Density: d = 8000 Kg/m3

The structure is planar (all out-of plane motion is blocked). The eight ends of the double cross are simply-supported, hence the boundary conditions are: u = v = 0 at A, B, ... H. The output consists of the 16 lowest natural frequencies of this structure.



Page 45

ResultsThe table below presents the results (frequencies in Hz):

Mode

Reference Value [Hz]

Computed Result [Hz]

Error [%]

1

11.336

11.336

0

2-3

17.687

17.687

0

4-8

17.715

17.715

0

9

45.477

45.477

0

10-11

57.364

57.364

0

12-16

57.683

57.683

0

Reference: BENCHMARK newsletter, April 1989, p.14, NAFEMS - Glasgow



Page 46

Deep Simply-supported Solid BeamThis test lets you check modal analysis results for a beam with 3D meshes formed of linear or parabolic hexahedron elements. Before you begin: The Deep_simply_supported_solid_beam_HE8.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of linear hexahedron elements (HE8). To compute the analysis with parabolic hexahedron elements (HE20), open the Deep_simply_supported_solid_beam.CATPart document, and proceed as follow: 1. Define the mesh for the 2D surface presented in the CATPart document, using the Surface Meshing workshop in the Advanced Meshing Tools workbench. 2. Extrude the mesh to obtain a 3D mesh. 3. Define analysis specifications as indicated below. 4. Compute the CATAnalysis and generate an image called Deformed mesh.

For a 3D hexahedron mesh, you need to create a mesh on a 2D part, then to extrude it with the Extrude Mesher in Translation icon, in the Advanced Meshing Tools workbench.


Length: L = 10 m



Page 47

Width and Height: h = 3000 mm

Analysis SpecificationsYoung Modulus (material): E = 200000 MPa


Density: = 8000 kg_m3 Mesh specifications: On x, y and z axis:q

For HE8: 1 x 3 x 10 elements For HE20: 1 x 1 x 5 elements

q



Page 48

Boundary conditions: In the skew coordinate system (applied to the box group): Tx = 0 on face ABCD Ty = 0 on edge CC' Ty = Tz = 0 on edge DD'

ResultsThe results of the test are presented in the table below:

Parabolic Hexahedron (HE20) Reference Mode Solution [Hz]

Linear Hexahedron (HE8)

Frequencies Normalized Frequencies Normalized Results Results Results Results

1

38.2

38.27

1.002

38.28

1.002

2

85.21

87.67

1.029

83.95

0.985

3

152.23

157.5

1.035

157.57

1.035

4

245.53

259.01

1.055

264.92

1.079

5

297.05

306.06

1.030

298.33

1.004



Page 49

Reference: Free vibrations benchmarks, F. Abbassian, D.J. Dawswell, N.C. Knowles, NAFEMS, Glasgow, November 1987, pp438-452.



Page 50

Modal Analysis of a Beam with Axial LoadThis test lets you check analysis results for a beam with a static preload. You will use a 1D mesh.

Before you begin: The Modal_analysis_beam_with_axial_load.CATAnalysis document presents a complete analysis of this case, computed with a load of 105 N. Proceed as follow:


To compute the unloaded case, change in this CATAnalysis the value of the distributed force.

SpecificationsGeometry SpecificationsLength: L = 2m

Width and height: a = 0.05m

Area: 2.5e-00.3m2

Inertial: Iz = 5.2083e007m4



Page 51

Analysis SpecificationsYoung Modulus (material): E = 2e + 011 N_m2



Restraints: Point A: x = y = 0 Point B: y = 0

Mesh Specifications: 10 elements

Distributed Force: The load Fx is a static pressing preload in the axis of the beam. There is two cases of preloading:q

|Fx| = 0 |Fx| = 105 N

q



Page 52

Resultsq


The table below presents the analysis results. Relative errors are listed:

Fundamental natural mode of vibration

Frequencies [Hz] Relative Errors [%] Analytical Results Computed Results

Bending mode 1 |Fx|=0 Bending mode 2

28.702

28.694

0.03

114.807

114.701

0.09

Bending mode 1 |Fx|=105N Bending mode 2

22.434

22.428

0.03

109.08

108.981

0.09

Reference: Guide de validation des prologiciels de calcul de structures, SFM, Afnor Technique pp192-193.



Page 53

Buckling AnalysisThis chapter contains buckling analysis problems. A buckling analysis consists in finding the buckling mode shapes and the buckling critical factors corresponding to a specified load case applied to an elastic structure. This chapter contains the following model and task: Buckling of a Straight Beam (Out-of-Plane Buckling) Buckling of a Straight Beam (In-Plane Buckling) Lateral Buckling of Narrow Rectangular Beam



Page 54

Buckling of a Straight Beam (Out-of-Plane Buckling)This test lets you compare analysis results for the buckling of a straight beam in CATIA with the Beam Theory. In other words, you will study the quality of distorted shell elements in a buckling analysis. You will use 2D meshes. Before you begin: The Straight_Beam.CATAnalysis document presents the complete analysis of the buckling case, computed with an irregular mesh formed of parabolic triangle elements (TR6).

To compute the case with linear triangle (TR3), linear quadrangle (QD4) and parabolic quadrangle (QD8) elements, open the Straight_Beam.CATPart document and proceed as follow:

1. Define the mesh using the Surface Meshing workshop in the Advanced Meshing Tools workbench. 2. In Generative Structural Analysis workbench, define analysis specifications as shown below. 3. Insert and compute buckling case solution in Generative Structural Analysis workbench.

To create an irregular mesh, use the Add/Remove Constraints icon in the Local Specifications toolbar, then select the points on the part.


Length: L = 100 mm

Width: W = 12 mm



Elfini Solver VerificationYoung Modulus (material): E = 1e+09 N_m = 1000 MPa


Page 55


Boundary conditions: On point B: Tx = 0 At edge AB: Ty = Tz = Rx = 0

Line forced density: At edge CD: Ty = -83.333 N_m

Resultsq

The critical load is given by column Buckling theory:

In this particular case, the result is Nc = 0.2467 N

q

The table below presents the results of the solver's computation and the percentage of error compared with the results of the Beam Theory.

Linear triangle shell (TR3)

Parabolic triangle shell (TR6)

Linear quadrangle shell (QD4)

Parabolic quadrangle shell (QD8)

Mesh

Regular

Irregular

Regular

Irregular

Regular

Irregular

Regular

Irregular

Computed buckling factor [N] 0.2488 0.2491 0.2463 0.2470 0.2467 0.2479 0.2467 0.2471

Elfini Solver VerificationError [%]


Page 56

0.9

1

0.2

0.1

0

0.5

0

0.2

Reference: Beam Theory



Page 57

Buckling of a Straight Beam (In-Plane Buckling)This test lets you compare analysis results for the buckling of a straight beam in CATIA with the Beam theory. In other words, you will study the quality of distorted membrane elements in a buckling analysis. You will use 2D meshes, with membrane elements. Before you begin: The Buckling_StraightBeam_in_plane.CATAnalysis document presents the complete analysis of the buckling case, computed with an irregular mesh formed of parabolic triangle elements (TR6).

To compute the case with linear triangle (TR3), linear quadrangle (QD4) and parabolic quadrangle (QD8) elements, open the Straight_Beam.CATPart document and proceed as follow:

1. Define the mesh using the Surface Meshing workshop in the Advanced Meshing Tools workbench. 2. In Generative Structural Analysis workbench, define analysis specifications as shown below. 3. Insert and compute buckling case solution in Generative Structural Analysis workbench.

To create an irregular mesh, use the Add/Remove Constraints icon in the Local Specifications toolbar, then select the points on the part.


Length: L = 100 mm

Width: W = 12 mm



Elfini Solver VerificationYoung Modulus (material): E = 1000 MPa


Page 58


Boundary conditions: On point B: Tx = 0 At edge AB: Ty = Tz = Rx = 0

Mesh Specifications: Membrane Elements

Line forced density: At edge CD: Ty = -83.333 N_m

Resultsq

The critical load is given by column buckling theory:

In this particular case, the result is Nc = 35.53 N

q

The results of the solver's computation and the normalized results (computed results divided by analytic solution) are given in the table below.

Linear triangle shell (TR3)


Linear quadrangle shell (QD4)

Parabolic quadrangle shell (QD8)

Mesh

Regular Irregular

Regular

Irregular

Regular

Irregular

Regular

Irregular

Elfini Solver VerificationComputed buckling factor [N] 317.435 287.831 35.483


Page 59

35.537

36.135

47.478

35.221

35.272

Normalized Results

8.934

8.101

0.999

1.000

1.017

1.336

0.991

0.993

Reference: Beam Theory



Page 60

Lateral Buckling of Narrow Rectangular BeamThis test lets you check analysis results for rectangular beams, in the context of a buckling case. You will use 1D meshes. Before you begin: The Lateral_buckling_rectangular_beam.CATAnalysis document presents the complete lateral buckling case, and includes the tests 1 and 2. Proceed as follow:

1. Open the CATAnalysis document, 2. Compute the analysis and generate an image called Deformed Mesh.

SpecificationsGeometry SpecificationsLength: L1 = 240mm

Height: b = 30 mm

Thickness: t = 0.6mm




Page 61

Young Modulus (material): E = 71240 GPa


Inertia: Ix = 2.16 mm4 (torsionnal) Iy = 0.54 mm4 (minor) Iz = 1350 mm4 (major)

Mesh Specifications: 21 nodes, 20 beam elements.


Test 1: At End 1, all DOF for nodes are fixed Test 2: At End 1, Tx = Ty = Tz = Rx = Ry = 0 At End 2, Ty = Tz = Rx = Ry = 0

q



Page 62

Load:q

Test 1: At End 2, unit force Fx = 1 N Test 2: At End 1, moment Mz = -1 N x mm At End 2, moment Mz = 1 N x mm

q

ResultsThe following table presents the results for buckling eigen values.

Argyris Theory

CATIA Beam

Test 1

1.088

1.088

Test 2

624.77

624.77

Reference: Handbook of Engineering Mechanics, W. Flugge, McGraw-Hill, New York, 1962 Finite Element MethodThe Natural Approach, J.H. Argyris



Page 63

Dynamic Response AnalysisThis chapter contains the following model:

Harmonic Forced Vibration of a Simply-supported Thin Square Plate Transient Forced Vibration of a Simply-supported Thin Square Plate Transient Dynamic Response of a Clamp Beam with Different Inertia Transient Response of a Spring-mass System with Imposed Acceleration Harmonic Forced Vibration of a Plane Grid



Page 64

Harmonic Forced Vibration of a Simply-supported Thin Square PlateThis test lets you check analysis results for a simply-supported thin square plate, in the context of an harmonic dynamic response case. You will use 2D meshes. This test proposed by NAFEMS is used to validate the following attributes:q

2D shell elements (i.e. the elementary stiffness and mass matrices) Harmonic dynamic response solve algorithms

q

Before you begin: The Harmonic_forced_vibration_of_a_simply_supported_thin_square_plate.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of linear quadrangle elements (QD4). To compute the case with parabolic quadrangle (QD8), linear triangle (TR3) and parabolic triangle (TR6) elements, proceed as follow: 1. Open the CATAnalysis document, 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below, 3. In the Generative Structural Analysis workbench, compute the case.





Page 65


Analysis SpecificationsYoung Modulus (material) E = 200 GPa

Poisson's Ratio (material) = 0.3


Boundary conditions: Tx = Ty = Rz = 0 at all nodes Tz = 0 along all four edges Rx = 0 along edges x = 0 and x = 10000 mm Ry = 0 along edges y = 0 and y = 10000 mm



Page 66

Load: F = F0 sin wt where:q

F0 = 100 N/m2 w=2 f

q

q

f = 0 to 4.16 Hz

Damping: d = 2% in all 16 modes used

ResultsThe results for different finite elements are presented in the table below. The peak is the value at undamped natural frequency. In this particular test, the undamped natural frequency is 2.377 Hz.

Values

Linear triangle shell (TR3) Type of Values Reference Solution 64x64 elements

Parabolic triangle shell (TR6) 32x32 elements

Linear quadrangle shell (QD4) 32x32 elements

Parabolic quadrangle shell (QD8) 16x16 elements

Computed Error Computed Error Computed Error Computed Error Results [%] Results [%] Results [%] Results [%]



Page 67

Peak displacement [mm] at 2.377 Hz 45.420 45.430 0.023 45.430 0.022 45.477 0.125 45.429 0.020

Peak stress [MPa]

30.030

32.005

6.58

32.082

6.83

31.976

6.48

32.227

7.32




Page 68

Transient Forced Vibration of a Simply-supported Thin Square PlateThis test lets you check analysis results for a simply-supported thin square plate, in the context of a transient dynamic response case. You will use 2D meshes. This test proposed by NAFEMS is used to validate the following attributes:q

2D shell elements (i.e. the elementary stiffness and mass matrices) Transient dynamic response solve algorithms

q

Before you begin: The Transient_forced_vibration_of_a_simply_supported_thin_square_plate.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of linear quadrangle elements (QD4). To compute the case with parabolic quadrangle (QD8), linear triangle (TR3) and parabolic triangle (TR6) elements, proceed as follow: 1. Open the CATAnalysis document, 2. In the Advanced Meshing Tools workbench, replace the mesh specifications as indicated below, 3. In the Generative Structural Analysis workbench, compute the case.





Page 69





Boundary conditions: Tx = Ty = Rz = 0 at all nodes Tz = 0 along all four edges Rx = 0 along edges x = 0 and x = 10000 mm Ry = 0 along edges y = 0 and y = 10000 mm



Page 70

Load: F0 = 100 N/m2 over whole plate Damping: d = 2 % in all 16 modes used Time step: t = 0.002 s

ResultsYou will find here the results for different finite elements:

Values

Linear triangle shell (TR3) Type of Values


Linear quadrangle shell (QD4) 32x32 elements

Parabolic quadrangle shell (QD8) 16x16 elements

Reference 64x64 elements 32x32 elements Solution

Values

Error [%]

Values

Error [%]

Values

Error [%]

Values

Error [%]



Page 71

Peak displacement [mm] at t=0.210s 3.523 3.444 2.24 3.445 2.21 3.451 2.04 3.446 2.19

Peak stress [MPa]

2.484

2.221

10.59

2.217

10.75

2.251

9.38

2.234

10.06

Static displacement [mm] 1.817 1.774 2.37 1.775 2.31 1.776 2.26 1.775 2.31




Page 72

Transient Dynamic Response of a Clamp Beam with Different InertiaThis test lets you check analysis results for a beam with different inertia, in the context of a transient dynamic response case. You will use a 1D mesh. This test proposed by AFNOR is used to validate the transient dynamic response solve algorithms.

Before you begin: The Transient_dynamic_response_beam_inertia.CATAnalysis document presents the complete analysis of the transient dynamic response case. Proceed as follow:

1. Open the CATAnalysis document, 2. Compute the Frequency case, and generate an image called Deformed mesh.

SpecificationsGeometry SpecificationsBeams AC and DB:q

l = AC = DB = 1.25 m area = 2.872e3 m2 Iy = 1.943e5 m4

q

q



Page 73

Beams CD:q

L = CD = 2.5 m area = 5.356e3 m2 Iy = 8.356e5 m4

q

q

The structure and the load being symmetrical, only one half of the beam is modeled.



Clamp in A Edges AC and CO: Rx = 0 Point O: Tx = Ty = Rx = Ry = Rz = 0

q

q

Mesh Specifications: Beam mesh with 62.5 elements.



Page 74

Masses for Frequency Case:q

Edge AC: Line mass density = 22.402 kg/m Edge OC: Line mass density = 42.398 kg/m

q

Damping: Global damping = 0 %

Load:q

q1 = 10000 N/m all beam long t1 = 2.38e-2 s t2 = 5.95e-2 s

q

q

ResultsThe tables below present the analysis results:

q

Frequency Response: Frequency [Hz] Bending Mode Analytic Solution Computed Results Error [%]

1

63.009

62.8796

0.21

Elfini Solver Verificationq


Page 75

Transient Dynamic Response: Displacement V [mm] Time [s] Point Localization [mm] Analytical Solution Computed Results Error [%]

0.0595

2437.5

2.0469

2.467

0.09

Reference: Guide de validation des prologiciels de calcul de structures, AFNOR Technique, SFM pp188-189.


Transient Response of a Spring-mass System with Imposed Acceleration


Page 76

This test lets you check analysis results for a spring-mass system, in the context of a transient dynamic response case. You will use a 1D mesh. This test proposed by NAFEMS is used to validate the following attributes:q

1D beam element Transient dynamic response solve algorithms

q

Before you begin: The Transient_response_spring_mass_system.CATAnalysis document presents a complete analysis of this case. Proceed as follow:



Mass: m = 1 kg

Stiffness: k = 1000 N_m

Length: L=1 m

Section: area = 1.10-5 m2

Analysis SpecificationsMesh Specification Size = 187.5 mm

Young Modulus (material): E = 100 MPa

Elfini Solver VerificationBoundary conditions:q


Page 77

Clamp in A Ty = Tz = Rx = Ry = Rz = 0 for each beam

q

Damping: d=0%

Time Step:

q

q

0 s < t < 0.1 s

Resultsq

Analytical Solution:

The natural frequencies are the roots of the following equation:

The response at point D is given by:

with imposed acceleration coefficient:

and

modal participated factor for a translational motion with u = 1

q

Modal Analysis:

Frequencies [Hz] Mode Analytic Solution Values Error [%]

1

2.239

2.2399

0.038

Elfini Solver Verification2 6.275

Version 5 Release 146.2760 0.015

Page 78

q

Transient Dynamic Analysis:

Displacement [mm] Time [s] Analytic Solution Values Error [%]

0.02

2.7

2.69

0.25

0.04

42.6

42.77

0.40

0.05

104.1

104.32

0.21

0.06

215.8

216.18

0.18

0.08

681.3

682.16

0.13

0.1

1658

1659.71

0.10

Reference: Guide de validation des prologiciels de calcul de structures, AFNOR Technique, SFM, pp182-183



Page 79

Harmonic Forced Vibration of a Plane GridThis test lets you check analysis results for a plane grid, in the context of an harmonic dynamic response case. You will use a 1D mesh. This test proposed by AFNOR is used to validate the following attributes:q

1D beam elements Harmonic dynamic response solve algorithms

q

Before you begin: The Harmonic_forced_vibrations_plane_grid.CATAnalysis document presents a complete analysis of this case. Proceed as follow:


SpecificationsGeometry SpecificationsLength: L1 = L2 = 5 m

Section IPE 200 :q

area = 2.872e-3 m2 Iy = 1.943e-5 m4 Iz = 1.424e-6 m4

q

q



Page 80

The geometry is built with two Parts, inserted in a Product. The first Part includes the beam AC and the beam DF. The second Part includes the beam BE.

Analysis SpecificationsYoung Modulus (material) E = 200 GPa

Poisson's Ratio (material) = 0.3



500 mm elements for the 3 edges No automatic mesh capture for all For ball joints B and E, rigid connection property with transmitted degrees of freedom Tx = Ty = Tz = 0

q

q

Boundary conditions:q

At points A, C, D and F : Tx = Ty = Tz = 0 Edges AC and DF : Ry = 0 Edge BE : Rx = 0

q

q



Page 81

Load:q

F = F0 sin t F0 = -100 000 N = 80 rad/s, so f = 12.7324 Hz

q

q

Resultsq

Frequency Response:

Frequency [Hz] Fundamental Mode Error [%]

Analytical Solution

Computed Results

1

16.456

16.410

0.28

2

38.165

37.941

0.59

q

Harmonic Dynamic Response:

Displacement on z axis at f = 12.7324 Hz

Point

Type of Values

Analytical solution

Computed Results

Error [%]

B and E

WBmax [mm]

-98.00

-100.35

2.4



Page 82

G

WG+WGmax [mm]

-227.00

-226.50

0.2

Reference: Guide de validation des prologiciels de calcul de structures, AFNOR Technique, SFM, pp198-199.



Page 83

Thermo Mechanical AnalysisThis chapter contains the following model: Thermal Expansion of a Beam



Page 84

Thermal Expansion of a BeamThis test lets you check analysis results for the thermal expansion of a beam, in the context of a static case. You will use 3D mesh with tetrahedron and hexahedron elements.

Before you begin: The Thermal_expansion_of_a_beamTE10.CATAnalysis document presents the complete analysis of this case, computed with a mesh formed of parabolic tetrahedron elements (TE10). The Thermal_expansion_of_a_beamHE20.CATAnalysis document presents the complete analysis of this caes, computed with parabolic hexahedron elements (HE20). To compute the case with linear tetrahedron elements (TE4) and linear hexahedron elements (HE8), proceed as follow:


Thermal_expansion_of_a_beamTE10.CATAnalysis for linear tetrahedron elements (TE4), Thermal_expansion_of_a_beamHE20.CATAnalysis for linear hexahedron elements (HE8).

r


Volume mesh for linear tetrahedron elements (TE4), Surface mesh for linear hexahedron elements (HE8).

r

The Global Parameters dialog box is open. 4. Select the Linear element type. 5. Compute the case in the Generative Structural Analysis workbench.



Page 85

SpecificationsGeometry SpecificationsLength: L = 1000 mm

Section: square a = 100 mm


Poisson's Ration (material): = 0.266

Thermal Expansion Coefficient: = 11,7.10-6/

Load:q

Case 1: Thermal Load T1 = 325 K Case 2: Thermal Load T1 = 325 K Case 3: Thermal Load T1 = 325 K and a surface force density Fy = -5.85e + 007N_m2

q

q

The environment temperature is T0 = 300 K



Page 86

Restraint:q

Case 1: Ty = 0 on End 1, Tz = 0 on Face 1 and Tx = 0 on Face 2 Case 2: Ty = 0 on End 1 and End 2, Tz = 0 on Face 1 and Tx = 0 on Face 2 Case 3: Ty = 0 on End 1, Tz = 0 on Face 1 and Tx = 0 on Face 2

q

q

Results

q


Case 1:

Case 2:

Case 3:



Page 87

q

Transient Dynamic Response: Computed Results Linear Case Quantity Analytical solution Tetrahedron (TE4) Parabolic Tetrahedron (TE10) Linear Hexahedron (HE8) Parabolic Hexahedron (HE20)

Case 1

Uth[mm]

0.2925

0.2925

0.2925

0.2925

0.2925

Case 2 22 [MPa]

-58.5

-58.5

-58.5

-58.5

-58.5

Case 3

Uth- UFy [mm]

0

0

0

0

0

Reference: Thermal Expansion Theory



Page 88

Analysis of an AssemblyThis chapter contains the following model:

Beams with Fastened Connection



Page 89

Beams with Fastened ConnectionThis test lets you check analysis results for an assembly of two beams. The aim of this test is to compare the analysis of two beams assembled with a fastened connection with an entire beam. You will use a 3D mesh.

Before you begin: The Assembly_with_fastened_connection.CATAnalysis document presents two complete static analysis and a modal analysis. Proceed as follow:

1. Open the CATAnalysis document. 2. Compute the case and generate an image called Deformed Mesh.


Length: L = AB = 500 mm

Width: a = 50 mm

Thickness: h = 50 mm




Page 90

Poisson's Ration (material): = 0.346

Case 1:q

Load: Pressure Pz = -250 000 N_m2 on the top face of the beam. Restraint: Tx = Ty = Tz = 0 on section A.

q

Case 2:q

Load: Force Fy = -40e + 006 N_m2 on the section B of the beam Restraint: Tx = Ty = Tz = 0 on section A.

q

Resultsq

Case 1: Results for bending Pz = -250 000 N_m2 Entire Beam Assembled Beams Error [%]

Maximum Displacement of the Beam [mm]

-2.67

-2.66

0.37

Energy [J]

3.347

3.333

0.42



Page 91

Von Mises in the Assembled Zone [Mpa] 3.2->22 3.08->22.9 maximum 4%

q

Case 2: Results for compression Fy = -40e + 006 N

Entire Beam Assembled Beams

Error [%]

Maximum Displacement of the Beam [mm]

-2.67

-2.66

0.37

Energy [J]

3.347

3.333

0.42

Von Mises in the Assembled Zone [Mpa] 3.2->22 3.08->22.9 maximum 4%

q

Modal Analysis: Entire Beam Assembled Beams Error [%]

Mode 1

164.207

164.453

0.15

Mode 2

164.242

164.489

0.15

Mode 3

984.085

985.227

0.12

Mode 4

984.257

985.554

0.13



Page 92

Mode 5

1450.44

1451.32

0.06

Mode 6

2552.59

2553.51

0.04

Mode 7

2589.29

2591.54

0.09

Mode 8

2582.79

2593.41

0.14

Mode 9

4353.26

4355.59

0.05

Mode 10

4700.51

4703.55

0.06

Reference: The results of the entire beam are computed with CATIA.



Page 93

CompositesThis chapter contains the following model:

Bonded Blades in Compression



Page 94

Bonded Blades in CompressionThis test lets you check the membrane bending coupling in non-symmetric stack composite plates and the traverse shear behavior in a thick plate. For this test, the reference is the mesh formed of linear hexahedron elements (HE8). You will use 2D meshes.

Before you begin: The Bonded_blades_compression_QD8.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic quadrangle elements (QD8). The Bonded_blades_compression_HE8.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of linear hexahedron elements (HE8).

To compute the mesh with linear quadrangle elements (QD4), proceed as follow:

1. Open the Bonded_blades_compression_QD8.CATAnalysis document. 2. Enter the Advanced Meshing Tools workbench. 3. In the specification tree, double-click on the mesh. The Global Parameters dialog box is open. 4. Select the Linear element type. 5. Compute the case in the Generative Structural Analysis workbench.

SpecificationsGeometry SpecificationsLength: L = 10 000 mm

Width: b = 1 000 mm



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Thickness: h = 400 mm divided in two layers

Analysis SpecificationsMaterial 1:q

Young Modulus (material): E = 100 MPa Poisson's Ration: = 0.3

q

Material 2:q

Young Modulus (material): E = 10 MPa Poisson's Ration: = 0.2

q


10 x 2 elements for QD4 and QD8 10 x 2 x 1 elements for the reference HE8

q

Restraint: Clamp on End 1

Load: Fx = -30 N on End 2



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ResultsThe static analysis results correspond to the displacements U in x and z at End 2. The buckling analysis is performed for this 30 N load case. The buckling coefficients for the lowest two modes, computed with CATIA, are presented in the table below. Computed Results

Reference

Linear Hexahedron Linear Quadrangle Parabolic Quadrangle Analysis Case Quantity (HE8) (QD4) (QD8)

Ux max [mm] Static Case Uz max [mm]

-0.027

-0.0277

-0.0281

-0.834

-0.834

-0.835

Coefficient 1 Buckling Case Coefficient 2

121.634

121.197

120.743

1126.56

1095.34

1079.94

Reference: 3D Model with HE8 element, computed with CATIA.



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IndexBbeams with fastened connection bending of a beam bonded blades in compression buckling of a straight Beam (in-plane buckling) buckling of a straight Beam (out-of-plane buckling)

Ccylindrical roof under its own weight

Ddeep simply-supported solid beam

Ffree thin square plate free vibration of a simply-supported thin square plate free vibrations of a compressor blade

Hharmonic forced vibration of a plane grid harmonic forced vibration of a simply-supported thin square plate peak



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Llateral buckling of narrow rectangular beam

Mmodal analysis of a beam with axial load Morley's Problem

Pplane vibrations of a simply supported double cross

Sspace structure on elastic support

Tthermal expansion of a beam thick cylinder under internal pressure transient dynamic response of a clamp beam with different inertia transient forced vibration of a simply-supported thin square plate transient response of a spring-mass system with imposed acceleration twisted beam