OptiStruct Verification Problems - Altair University...This manual presents solved verification models including NAFEMS problems. This chapter covers the following: • Accessing the

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ContentsIntellectual Property Rights Notice.............................................................................iiiTechnical Support............................................................................................................vii

Verification Problems...................................................................................................... 9

Accessing the Model Files.................................................................................................10NAFEMS Linear Elastic......................................................................................................11

OS-V: 0010 Elliptic Membrane.................................................................................. 11OS-V: 0020 Cylinder Shell Patch............................................................................... 14OS-V: 0030 Radial Point Load on a Hemisphere.......................................................... 16OS-V: 0040 Z-Section Cantilever...............................................................................18OS-V: 0050 Skew Plate Normal Pressure....................................................................20OS-V: 0060 Thick Plate Pressure...............................................................................22OS-V: 0070 Solid Cylinder/Taper/Sphere - Temperature............................................... 24OS-V: 0080 Buckling of Shells and Composites with Offset........................................... 26OS-V: 0085 Plane Strain: Analysis of Pressure Vessel.................................................. 31

NAFEMS Thermo-elastic....................................................................................................34OS-V: 0100 Membrane with Hot-Spot........................................................................ 34

NAFEMS Heat Transfer..................................................................................................... 36OS-V: 0110 One Dimensional Transient Heat Transfer.................................................. 36OS-V: 0120 Two-Dimensional Heat Transfer with Convection.........................................38

NAFEMS Nonlinear Quasi-static Analysis............................................................................. 40OS-V: 0200 Simply-Supported Thin Square Plate........................................................ 40OS-V: 0210 Simply-Supported Thick Square Plate....................................................... 43OS-V: 0220 3D Punch (Rounded Edges).....................................................................46OS-V: 0230 3D Loaded Pin.......................................................................................51OS-V: 0240 3D Steel Roller on Rubber...................................................................... 54OS-V: 0250 3D Sheet Metal Forming.........................................................................57OS-V: 0260 Shell Bending under a Tip Load...............................................................59

NAFEMS Frequency Response Analysis............................................................................... 61OS-V: 0300 Deep Simply-Supported Beam Harmonic Forced Vibration Response.............. 61OS-V: 0310 Deep Simply-Supported Beam Periodic Forced Vibration Response................ 63OS-V: 0320 Deep Simply-Supported Beam Random Forced Vibration Response................65OS-V: 0330 Deep Simply-Supported Beam Transient Forced Vibration Response...............67OS-V: 0340 Simply-Supported Thin Square Plate Harmonic ForcedVibration Response.................................................................................................. 69OS-V: 0350 Simply-Supported Thin Square Plate Periodic Forced Vibration Response........ 71OS-V: 0360 Simply-Supported Thin Square Plate Harmonic ForcedVibration Response.................................................................................................. 73OS-V: 0365 Simply-Supported Thin Square Plate Transient ForcedVibration Response.................................................................................................. 75OS-V: 0370 Simply-Supported Thick Square Plate Harmonic ForcedVibration Response.................................................................................................. 77

1

OS-V: 0375 Simply-Supported Thick Square Plate Periodic Forced Vibration Response.......79OS-V: 0380 Simply-Supported Thick Square Plate Random ForcedVibration Response.................................................................................................. 81OS-V: 0385 Simply-Supported Thick Square Plate Transient ForcedVibration Response.................................................................................................. 83

NAFEMS Normal Modes Analysis........................................................................................85OS-V: 0400 Pin-ended Double Cross..........................................................................85OS-V: 0410 Cantilever with Off-Center Point Masses....................................................88OS-V: 0415 Deep Simply-Supported Beam................................................................. 90OS-V: 0420 Free Thin Square Plate...........................................................................92OS-V: 0425 Clamped Thin Rhombic Plate...................................................................94OS-V: 0430 Cantilevered Thin Square Plate................................................................96OS-V: 0435 Clamped Thick Rhombic Plate................................................................100OS-V: 0440 Cantilevered Tapered Membrane............................................................ 102OS-V: 0450 Free Cylinder: Axi-symmetric Vibration................................................... 104OS-V: 0455 Simply-Supported Solid Square Plate......................................................107OS-V: 0460 Dynamic Behavior of a Fluid-containing Structure using MFLUID................. 110

NAFEMS Composites.......................................................................................................116OS-V: 0500 Laminated Strip................................................................................... 116OS-V: 0510 Wrapped Thick Cylinder........................................................................ 118OS-V: 0520 Sandwich Shell.................................................................................... 121OS-V: 0530 Composite Shell Bending...................................................................... 123

Response Spectrum Analysis........................................................................................... 128OS-V: 0600 Simply Support Beam...........................................................................128

Elements.......................................................................................................................130OS-V: 0700 Twisted Cantilever Beam.......................................................................130OS-V: 0710 Curved Cantilever Beam....................................................................... 132OS-V: 0720 Straight Cantilever Beam...................................................................... 134OS-V: 0730 Scordelis-Lo Roof................................................................................. 137OS-V: 0750 Radial Stretching of a Cylinder.............................................................. 139

Materials....................................................................................................................... 142OS-V: 0800 Hyperelastic Material Verification............................................................142OS-V: 0810 Hyperelastic Large Displacement Nonlinear Analysis with aPressurized Rubber Disk......................................................................................... 149

Index.................................................................................................................................152

2

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OptiStruct Verification ProblemsTechnical Support p.viii

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Verification Problems 1

Verification Problems

This manual presents solved verification models including NAFEMS problems.

This chapter covers the following:

• Accessing the Model Files (p. 10)

• NAFEMS Linear Elastic (p. 11)

• NAFEMS Thermo-elastic (p. 34)

• NAFEMS Heat Transfer (p. 36)

• NAFEMS Nonlinear Quasi-static Analysis (p. 40)

• NAFEMS Frequency Response Analysis (p. 61)

• NAFEMS Normal Modes Analysis (p. 85)

• NAFEMS Composites (p. 116)

• Response Spectrum Analysis (p. 128)

• Elements (p. 130)

• Materials (p. 142)

The verification problems use model files that are located in the demos directory of the softwareinstallation. In the verification problems, file paths are referenced as /../.

OptiStruct Verification ProblemsVerification Problems p.10

Accessing the Model FilesRequired model files for the models you build.

1. The OptiStruct verification problem model files are located on /demos/hwsolvers/optistruct/verification.

Note: The files may require unzipping before proceeding with the verificationproblem. When extracting zipped files, preserve any directory structure included in thefile package.



NAFEMS Linear Elastic

OS-V: 0010 Elliptic MembraneTest No. LE1The model is a thin plate of thickness 0.1m subjected to a uniform pressure for linear static analysis.OptiStruct examines the direct stress at the point on inside the ellipse on the x-axis.

Figure 1:

Benchmark ModelSecond order Hexahedral, Penta, Tetra, Quad and Tria elements are used to create the coarse and finemesh. A uniform outward pressure of 10 MPa is applied on the outer face. The pressure is converted toforce and is applied to the nodes for Quad8 and Tria6 elements.

The material properties are:

Young's Modulus210 x 103 MPa

Poisson's Ratio0.3

Linear Static Analysis ResultsAll results are normalized with the target value (92.7 MPa).

Direct Stress at Point D(MPa)

Normalized with the TargetValue

Solid Hexahedral

Hex8 coarse 63.15 1.467933492





Hex20 coarse 87.46 1.059913103

Hex8 fine 79.8 1.161654135

Hex20 fine 91.01 1.018569388

Solid Wedges:

Penta6 coarse 48.68 1.904272802

Penta15 coarse 95.21 0.973637223

Penta6 fine 67.3 1.377414562

Penta15 fine 94.28 0.983241409

Solid Tetrahedral:

Tetra4 coarse 53.34 1.737907762

Tetra10 coarse 95.96 0.966027511

Tetra4 fine 66.56 1.392728365

Tetra10 fine 95.28 0.972921914

Quad Shells:

Quad4 coarse 61.83 1.499272198

Quad8 coarse 86.67 1.069574247

Quad4 fine 79.7 1.163111669

Quad8 fine 91.48 1.013336248

Triangular Shells:

Tria3 coarse 54.06 1.714761376

Tria6 coarse 97.07 0.954980942

Tria3 fine 74.21 1.249157795





Tria6 fine 96.64 0.959230132

Model FilesThe model files used in this example include:

/demos/hwsolvers/optistruct/verification

/LE1Hex8C.fem

/LE1Hex20C.fem

/LE1Hex8F.fem

/LE1Hex20F.fem

/LE1Pen6C.fem

/LE1Pen15C.fem

/LE1Pen6F.fem

/LE1Pen15F.fem

/LE1Tet4C.fem

/LE1Tet10C.fem

/LE1Tet4F.fem

/LE1Tet10F.fem

/LE1Quad4C.fem

/LE1Quad8C.fem

/LE1Quad4F.fem

/LE1Quad8F.fem

/LE1Tria3C.fem

/LE1Tria6C.fem

/LE1Tria3F.fem

/LE1Tria6F.fem

ReferenceNAFEMS Ltd, The Standard NAFEMS BENCHMARKS TNSB Rev. 3, NAFEMS Ltd, Scottish EnterpriseTechnology Park, Whitworth Building, East Kilbride, Glasgow, United Kingdom, 1990.



OS-V: 0020 Cylinder Shell PatchTest No. LE2OptiStruct examines the outer surface tangential stress at point E for linear static analysis.

Figure 2:

Benchmark ModelQuad8 elements are used to create a mesh on the cylindrical patch with 4 elements for the loadcase 1 and Quad4 elements are used to create a mesh with 16 elements for the load case 2. Allthe translations and rotations are constrained at edge AB, z-translation and normal rotations areconstrained at the edges AD and BC. For Load case 1 a uniform normal edge moment of 1.0 kNm/m isapplied on the edge DC and for the Load case 2, uniform outward normal pressure of 0.6 MPa is appliedon the mid surface ABCD and a tangential outward normal pressure of 60.0 MPa is applied on edge DC.



Poisson's Ratio0.3


Surface Tangential Stress at Point E (MPa)


Load Case 1: Quad8 58.03 1.033947958



Surface Tangential Stress at Point E (MPa)


Load Case 2: Quad4 58.92 1.018329939



le2quad8lc1.fem

le2quad4lc2.fem




OS-V: 0030 Radial Point Load on a HemisphereTest No. LE3The model is a hemispherical shell subjected to concentrated radial loads at its free edges. It examinesthe performance of the three-dimensional shell to model local bending behavior under conditions wherethe deformations are primarily due to bending.

Figure 3:

Benchmark Model4-node, first order CQUAD4 elements are benchmarked in LE3. The hemisphere is 10m in radius and0.04m in radial thickness. Two pairs of identical loads, 4000N, are applied at the free edge of thehemisphere, and are at right angles to each other. One pair of the loads is directed inwards (towardthe center) of the hemisphere, while the second pair is directed outward from the center, producingdeformation of compression in one direction and elongation in another. Since both the geometry andloads are symmetrical, only a quarter of the hemisphere is modeled. Symmetric boundary constraintsare applied on edges AE and CE. The z-translation at point E is fixed, and all displacements on edge ACare free. The test also requires the mesh of the hemisphere to have equally spaced nodes on edges AC,CE, EA, BG, DG, and FG. The target is x-translation at point A, with a target value of 0.185m.

The material properties for the hemisphere are:

E68.25 GPa

0.3

Linear Static Analysis ResultsAll results are normalized with the target values of x translation at point A.



ElementType

Mesh Configuration nacx nce x nea

CQUAD4 4 x 4 x 4 8 x 8 x 816 x

16 x 1632 x

32 x 3264 x

64 x 64

0.9865 1.0200 1.0076 1.0032 1.0016


/demos/hwsolvers/optistruct/verification/LE3.fem




OS-V: 0040 Z-Section CantileverTest No. LE5OptiStruct examines the axial (x-x) stress (compression) at mid-surface, point A for linear staticanalysis.

Figure 4:

Benchmark ModelQuad4 and Quad8 elements are used to create a uniform mesh of 8 elements along the length with oneelement across width of flange. All the displacements at one end are maintained zero, at the other endtwo uniformly distributed force of 0.6MN each are applied.



Poisson's Ratio0.3

Linear Static Analysis ResultsAll results are normalized with the target value (-108 MPa Compression).

Quadrilateral Shells Axial Stress (x-x) at Mid-surface Point A (MPa)


Quad4 -112.2 0.962566845



Quadrilateral Shells Axial Stress (x-x) at Mid-surface Point A (MPa)


Quad8 -110.9 0.973850316



/LE5Quad4.fem

/LE5Quad8.fem




OS-V: 0050 Skew Plate Normal PressureTest No. LE6OptiStruct examines the maximum principal stress on the lower surface at the plate center point E forlinear static analysis.

Figure 5:

Benchmark ModelQuad4 and Quad8 elements are used to create a uniform mesh on the skew plate with 4 elementsas coarse mesh and 16 elements as fine mesh. The plate is simply supported at all the four edges. ANormal pressure of -0.7 KPa is applied on the face of the plate in the vertical z-direction.



Poisson's Ratio0.3


Quadrilateral Shell Maximum principal stress onthe lower surface at the platecenter point E (MPa)


Quad8 coarse 0.8294 0.96696407

Quad8 fine 0.7869 1.019189224





/le6quad8c.fem

/le6quad8f.fem




OS-V: 0060 Thick Plate PressureTest No. LE10The model is a thick plate subjected to uniform normal pressure of 1MPa on the upper surface of theplate. OptiStruct examines the direct stress at the point D for linear static analysis.

Figure 6:

Benchmark ModelSecond order Hexahedral, Penta and Tetra elements are used to create the coarse and fine mesh. Auniform pressure of 1 MPa is applied on the upper surface of the plate.



Poisson's Ratio0.3




Solid Hexahedral:

Hex20 coarse 5.32 1.011278195





Hex20 fine 5.58 0.964157706

Solid Wedges:

Penta15 coarse 4.91 1.095723014

Penta15 fine 5.94 0.905723906

Solid Tetrahedral:

Tetra10 coarse 5.741 0.937118969

Tetra10 fine 5.029 1.069795188



/LE10Hex20C.fem

/LE10Hex20F.fem

/LE10Pyr15C.fem

/LE10Pyr15F.fem

/LE10Tet10C.fem

/LE10Tet10F.fem




OS-V: 0070 Solid Cylinder/Taper/Sphere - TemperatureTest No. LE11The model is a thick solid cylinder subjected to linear temperature gradient in the radial and axialdirection. OptiStruct examines the direct stress at the point A inside the cylinder on the y axis forlinear static analysis.

Figure 7:

Benchmark ModelSecond order Hexahedral, Penta and Tetra elements are used to create the coarse and fine mesh. ALinear temperature gradient of T°C = (x2 + y2)1/2 + z is applied in the radial and axial direction fromthe center of the cylinder. Only one quarter of the cylinder is considered.


MAT1 Isotropic


Poisson's Ratio0.3

Coefficient of Thermal Expansion2.3 x 10-4/°C

Linear Static Analysis ResultsAll results are normalized with the target value (-105 MPa).



Direct Stress at Point A(MPa)


Solid Hexahedral:

Hex20 coarse -93.21 1.126488574

Hex20 fine -99.12 1.059322034

Solid Wedges:

Penta15 coarse -100.3 1.046859422

Penta15 fine -103.7 1.012536162

Solid Tetrahedral:

Tetra10 coarse -91.97 1.141676634

Tetra10 fine -98.68 1.064045399



/LE11Hex20C.fem

/LE11Hex20F.fem

/LE11Pyr15C.fem

/LE11Pyr15F.fem

/LE11Tet10C.fem

/LE11Tet10F.fem




OS-V: 0080 Buckling of Shells and Composites with OffsetA test of influence of offset on buckling solution for shells, including composite with offset Z0 andelement offset ZOFFS.

Figure 8: FE-Model of the Beam with Boundary Conditions and Loadcases

Benchmark ModelHere, you solve several problems to calculate the critical load on different conditions. The model is asimply supported beam of height 1mm, breadth 2mm and length 100mm with one end constrained inall DOFs and an axial load applied on the other end.

The material properties for the beam are:

MAT1

Young's Modulus1 x 106 N/mm2

Poisson Ratio0.0

Density2 kg/mm3

Thermal Expansion Coefficient1 x 10-4 ºC-1

Reference Temperature for Thermal Loading300ºC

The different case description of the problem are:

1. Buckling without offset.2. Buckling with moment equivalent to offset.3. Buckling with offset created by a frame.4. Buckling with offset applied through ZOFFS.5. Buckling of composite with non-symmetrical layup.6. Buckling of composite with offset.

The theoretical critical buckling load is calculated using the Euler Buckling equation:

(1)

Where,



Maximum or critical force

Modulus of Elasticity

Area moment of Inertia (second moment of area)

Unsupported length of the beam

Column effective length factor (for one end fixed and the other end free, =2)

Results

Figure 9: First Four Buckling Eigenvalues for Non-offset (z0 = -0.5)

Quantity Theoretical No-offset Normalized

cr(1) 4.1123 4.1208 0.997937

cr(2) 16.449 16.513 0.996124

cr(3) 37.011 37.701 0.981698

cr(4) 102.81 108.19 0.950273

Figure 10: First Four Buckling Eigenvalues for Non-offset + Moment(the effect of offset is simulated by adding a moment at the end of the beam)



Quantity TheoreticalNo-offset +Moment

Normalized

cr(1) 4.1123 4.1208 0.997937

cr(2) 16.449 16.513 0.996124

cr(3) 37.011 37.701 0.981698

cr(4) 102.81 108.19 0.950273

Figure 11: First Four Buckling Eigenvalues for C-Frame(the effect of offset is simulated by creating a C-shaped frame)

Quantity Theoretical C-Frame Normalized

cr(1) 4.1123 4.1208 0.997937

cr(2) 16.449 16.513 0.996124

cr (3) 37.011 37.700 0.981724

cr(4) 102.81 108.19 0.950273

Figure 12: First Four Buckling Eigenvalues for z-offset (Zoffs = -0.5)



Quantity Theoretical ZOFFS Normalized

cr(1) 4.1123 4.1208 0.997937

cr(2) 16.449 16.513 0.996124

cr(3) 37.011 37.700 0.981724

cr(4) 102.81 108.19 0.950273

Figure 13: First Four Buckling Eigenvalues for Non-symmetric Layup(since the top layer is very weak, the load is applied to the “strong” layer with an offset of 0.5)

Quantity TheoreticalNon-symmetricLayup

Normalized

cr(1) 4.1123 4.1203 0.998058

cr(2) 16.449 16.510 0.996305

cr(3) 37.011 37.663 0.982689

cr(4) 102.81 107.89 0.952915

Figure 14: First Four Buckling Eigenvalues for Composites with Offset (z0 = -1)



Quantity TheoreticalOffsetComposite

Normalized

cr(1) 4.1123 4.1203 0.998058

cr(2) 16.449 16.510 0.996305

cr(3) 37.011 37.663 0.982689

cr(4) 102.81 107.89 0.952915


/demos/hwsolvers/optistruct/verification/s100_buckl.zip

s100comp_buckl.fem

s100compmom_buckl.fem

s100comp_frame_buckl.fem

s100comp_buckl_zoffs.fem

s100comp2ply_buckl.fem

s100compoffs_buckl.fem



OS-V: 0085 Plane Strain: Analysis of Pressure VesselThis problem examines the expansion of a pressure vessel due to an internal pressure. OptiStructexamines the principal stresses in the pressure vessel, due to the applied loading and boundaryconditions. Two-dimensional plane strain element will be used for this analysis.

Figure 15:

Benchmark ModelQuad4 Plane Strain elements are used to model the quarter symmetric slice of the pressure vessel ofradius 0.1m and thickness 0.020m. Internal pressure of 10,000 Pa which is converted to force andapplied on the nodes of the inner surface of the pressure vessel. A Linear Static analysis is performedon this model.


Young's Modulus207 x 109 Pa

Poisson's Ratio0.27

Linear Static Analysis Results

ModelHoop Stress

(Pa)

Radial Stress

(Pa)

Theoretical 55455 -10000

OptiStruct 54710 -9205.6



ModelHoop Stress

(Pa)

Radial Stress

(Pa)

Normalized 1.013 1.086

Figure 16: First Principle Stress (Hoop Stress)



Figure 17: Third Principle Stress (Radial Stress)


/demos/hwsolvers/optistruct/verification/Pressure_Vessel_LS.fem

ReferenceMacDonald, Bryan J., "Practical Stress Analysis with Finite Elements" (2nd Ed), page 327-329



NAFEMS Thermo-elastic

OS-V: 0100 Membrane with Hot-SpotTest No. T1OptiStruct examines the direct stress in y direction at a point D outside the hot-spot for linear thermoselastic analysis.

Figure 18:

Benchmark ModelQuarter model is considered and Quad4 elements with the specific mesh specifications are used formodel building. The hotspot area is maintained at a temperature 100°C.


Hot-spot Area


Poisson's Ratio0.3

Coefficient of thermal expansion1 x 10-5/°C

Remaining Area

Young's modulus100 x 103 MPa



Poisson's ratio0.3

Coefficient of thermal expansion0.0

Linear Static Analysis ResultsAll results are normalized with the target value (50 MPa).

Quadrilateral ShellsDirect Stress in y Direction at aPoint D, Outside Hot-spot (MPa)

Normalized withthe Target Value

Quad4 45.51 1.098659635


/demos/hwsolvers/optistruct/verification/T1Quad4.fem




NAFEMS Heat Transfer

OS-V: 0110 One Dimensional Transient Heat TransferTest No. T3OptiStruct examines the material temperature at point C, 0.08m from point A and the total simulationtime is 32 seconds for transient heat transfer analysis.

Figure 19:

Benchmark ModelThe 2-noded beam elements, Quad4 elements and Quad8 elements are used to build the modelwith 5 elements each for the coarse mesh and 10 elements each for the fine mesh. At time t=0, alltemperature = zero and at time t>0, at one end temperature is zero and at the other end temperatureis 100 sin(πt/40) °C. There is no heat flux perpendicular to the length of the beam.


Conductivity35.0 W/m°C

Specific Heat440.5 J/kg°C

Density7200 kg/m3

Linear Static Analysis ResultsAll results are normalized with the target value (36.60°C).

Material temperatureat point C, x=0.08m,

time t=32sec (°C)


Beam Elements:

CBEAM coarse 33.6 1.089285714

CBEAM fine 34.58 1.058415269



Material temperatureat point C, x=0.08m,

time t=32sec (°C)


Quadrilateral Element:

Quad4 coarse 33.6 1.089285714

Quad8 coarse 35.1 1.042735043

Quad4 fine 34.58 1.058415269

Quad8 fine 35.19 1.040068201



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OS-V: 0120 Two-Dimensional Heat Transfer withConvectionTest No. T4The model is having zero internal heat generation and OptiStruct examines the temperature at point Efor steady state heat transfer analysis.

Figure 20:

Benchmark ModelA 10x6 mesh configuration is created with QUAD4, QUAD8, TRIA3 and TRIA6 elements. One edge ofthe plate is having a prescribed temperature of 100°C, one end insulated and convection to the ambienttemperature at the other two edges.


Conductivity52.0 W/m°C

Surface Convective Heat Transfer Coefficient750.0 W/m2 °C

Linear Static Analysis ResultsAll results are normalized with the target value (18.30°C).

Shell ElementMaterial Temperature at Point C,x=0.08m, time t=32sec (°C)

Normalized with the Target Value

Quad4 17.77 1.029825549

Quad8 17.1 1.070175439



Shell ElementMaterial Temperature at Point C,x=0.08m, time t=32sec (°C)

Normalized with the Target Value

Tria3 17.29 1.058415269

Tria6 16.83 1.087344029



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NAFEMS Nonlinear Quasi-static Analysis

OS-V: 0200 Simply-Supported Thin Square PlateTest No. 13OptiStruct is used to investigate the repeated eigenvalues.

Figure 21:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.


Young’s Modulus200 x 109 N/m2

Poisson’s Ratio0.3



Density8000 kg/m3

Modal Analysis ResultsThe frequency of each targeted mode is normalized with the closed form solution.

f*Closed form solution

Mode 1 Mode 2 and 3 Mode 4

f* 2.377 Hz f* 5.942 Hz f* 9.507 Hz

HOE 1.021926053 HOE 1.066977913 HOE 1.076061121

LOE 1.013646055 LOE 1.015552897 LOE 1.053290494

Mode 5 and 6 Mode 7 and 8

f* 11.884 Hz f* 15.449 Hz

HOE 1.176750173 HOE 1.175007606



Mode 5 and 6 Mode 7 and 8

LOE 1.002869198 LOE 1.065889334



Test13HOE.fem

Test13LOE.fem




OS-V: 0210 Simply-Supported Thick Square PlateTest 21OptiStruct is used to investigate the repeated eigenvalues and the effect of ‘secondary’ restrains.

Figure 22:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 1.0m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.


Young’s Modulus200 × 109 N/m2


Density8000 kg/m3

Modal Analysis ResultsThe frequency of each targeted mode is normalized with the closed form solution.




Mode 1 Modes 2 and 3 Mode 4

f* 45.897 Hz f* 109.44 Hz f* 167.89 Hz

HOE 1.013827837 HOE 1.044863043 HOE 1.046793653

LOE 1.005851414 LOE 1.002363027 LOE 1.034589005

Modes 5 and 6 Modes 7 and 8

f* 204.51 Hz f* 256.50 Hz

HOE 1.125821617 HOE 1.094829757

LOE 0.993823531 LOE 1.051061511





/Test21HOE.fem

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OS-V: 0220 3D Punch (Rounded Edges)Contacts Benchmark 2For Quasi-static analysis using Linear elastic material, geometric non-linearity and nonlinear boundaryconditions.

OptiStruct FE results examine the plot of contact pressure, tangential stress against radial distance fromthe center of contact and relative tangential slip against distance from the center of contact. OptiStructalso examines the 3D contact, stick/slip behavior along the contact plane, compares the linear andquadratic elements and the plasticity.

Figure 23:

Benchmark ModelHexa8 and Hexa20 elements are used to create one quarter model with punch diameter 100mm, punchheight 100mm, foundation diameter 200mm, foundation height 200mm and fillet radius at the edgeof the punch contact is 10mm. A uniform pressure of 100N/mm2 is applied at the top surface of thepunch. The bottom surface of the foundation is fixed. Two different contact properties are used, onewith coefficient of friction 0.0 and the second with coefficient of friction 0.1. The straight edge of thefoundation is considered as the master surface and the nodes on the bottom edge of the punch areselected as the slave surface.




Epunch210 kN/mm2

Vpunch0.3

Efoundation70 kN/mm2

Nfoundation0.3

Nonlinear Quasi-static Analysis Results

Figure 24: Axial displacement as a function of the radial coordinate (friction coefficient 0.0 and 0.1) obtained withlinear elastic elements



Figure 25: Radial displacement as function of the radial coordinate (friction coefficient 0.0 and 0.1) obtained withlinear elastic elements

Figure 26: Axial displacement along top surface of foundation (with friction)



Figure 27: Radial displacement along top surface of foundation (with friction)

Figure 28: Effect of different friction coefficient and method of fiction handling on the radial displacement of thefoundation edge (Linear Elements)



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OS-V: 0230 3D Loaded PinContacts Benchmark 4For Quasi-static analysis using Linear elastic material, geometric non-linearity and nonlinear boundaryconditions. OptiStruct FE results examine the plot of contact pressure, tangential stress and relativetangential slip against angle .

Figure 29:

Benchmark ModelHexa8 and Hexa20 elements are used to create one quarter model. The length of the sheet from leftside to the center is 200mm, the inner radius of the sheet is 50mm, the outer radius of the sheet is100mm, height of the sheet is 200mm, length of the pin is 20mm and the thickness of the sheet is10mm. The outer surface of the pin and the inner surface of the sheet are in contact. Two equal pointforces, resulting in a total force on the pin of 100kN is acting on both sides of the pin. The left side ofthe sheet is fixed. A frictional coefficient of 0.1 is acting between the contacts. The nodes along the pinboundary are selected as slave nodes, while the nodes along the strip are specified to be the masternodes.

The material properties for the loaded pin are:

Epin210 kN/mm2

Vpin0.3

Esheet70 kN/mm2



Vsheet0.3

Nonlinear Quasi-Static Analysis Results

Figure 30: Displacement as a function of the angles obtained with first order elements for the nodes of the sheetcontact surface

Figure 31: Displacement as a function of the angles obtained with second order elements for the nodes of the sheetcontact surface



Figure 32: Displacement in x-direction for nodes along the pin as a function of the angle



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OS-V: 0240 3D Steel Roller on RubberContacts Benchmark 3For Quasi-static analysis using Linear elastic material, geometric non-linearity and nonlinear boundaryconditions.

OptiStruct FE results examine the horizontal displacement of the point A after 360 degrees’ motion.OptiStruct also examines the 3D deformable-deformable contact, Rolling contact and Incompressiblematerial feature.

Figure 33:

Benchmark ModelHexa8 elements are used to create one half of the model. The Steel is of 20mm width and 30mmradius, the rubber mat is 22mm wide, 20mm in high and 360mm long. The steel roller starts rollingfrom a point 60mm from the left-hand side of the rubber mat. The center of the roller is fixed inhorizontal and vertical direction, for a time period of 0-1 second the bottom surface of the rubber isdisplaced 3mm in the negative y direction, the sheet x-displacement is fixed and there is no rollerrotation. For the time period of 1-2 second the bottom surface of the rubber sheet is held at 3mm y-displacement and rotation of 360 degrees is prescribed to the steel roller where the sheet is free tomove in horizontal direction. There is no force applied on the system and the coefficient of frictionbetween the two surface is 0.3. The nodes on the outer surface of the roll are selected as master nodes,while the nodes on the top surface of the mat are specified as the slave nodes.


Esteel210 kN/mm2

Nsteel0.3

C10, rubber10 N/mm2 (Neo Hookean material description)



D1rubber0.0001


Horizontal Displacement (mm) NAFEMS OptiStruct Normalized

3D first order elements 182.9 182.06 1.00461386

Figure 34: Vertical Forces on the Roller versus Time



Figure 35: 3D Analysis – Undeformed and Contour Plots of Contact Pressure on Deformed Structure


/demos/hwsolvers/optistruct/verification/contb5H8.fem




OS-V: 0250 3D Sheet Metal FormingContacts Benchmark 3For Quasi-static analysis using Elastic plastic material, geometric non-linearity and nonlinear boundaryconditions.

OptiStruct FE results examine the forming angle and angle after the punch is released. OptiStruct alsoexamines the contact features of the rigid and deformable bodies and sliding contact around the circularsurfaces.

Figure 36:

Benchmark ModelHexa8 elements are used to create the half model of the sheet and Quad4 elements are used to modelthe punch and the die. The punch radius is 23.5mm, the die radius is 25mm, the die shoulder radiusis 4mm, width of the tool is 50mm, length of sheet is 120mm, sheet thickness is 1mm and the widthof the sheet is 30mm. The punch stroke is 28.5mm. The bottom surface is fixed. Two different contactproperties are used, one with coefficient of friction 0.0 and the second with coefficient of friction0.1342. For the contacts between the punch and the sheet, punch is considered as master surface andthe sheet as slave and for the contacts between the die and the sheet die is considered as master andsheet as slave.




E70.5 kN/mm2

0.342

0 (Initial yield stress)194 N/mm2

Hollomon hardening=K x n

K = 550.4 N/mm2

n = 0.223

Nonlinear Quasi Static Analysis ResultsCharacteristic angles during process.

Frictional Coefficient=0 NAFEMSOptiStruct

ResultsNormalized

Forming angle 21.88 20.50 1.067317

Angle after release 48.38 45.53 1.062596

Frictional Coefficient=0.1348 NAFEMSOptiStruct

ResultsNormalized

Forming angle 21.84 22.437 0.973392

Angle after release 54.45 43.22 1.259833



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OS-V: 0260 Shell Bending under a Tip LoadA beam is analyzed for bending due to tip load. OptiStruct investigates the vertical steady-statedisplacement at the tip of the beam.

Figure 37:

Benchmark ModelTwo Beams are analyzed, Beam1 without follower load and Beam 2 with follower load. Shell elementsare used to model the beams which is 400mm long consists of 40 elements and a cross section of20mm. All the nodes are constrained for the 3,4 and 5 degrees of freedom and the ends of the beamsare constrained in all degrees of freedom. Both the beams are loaded at the edge by a point force of125N on each node in the negative y direction. The load on the Beam1 is not having a follower forcewhereas the load on the Beam2 is a follower force. Nonlinear Quasi-static analysis is performed withLarge displacement.


Young's Modulus1000 MPa

Poisson's Ratio0.0

Density10000 kg/m3


Non-Follower Loady-Displacement

(mm) Follower Loady-Displacement

(mm)

Bisshopp and Drucker 240 Bisshopp and Drucker 291

CBEAM 242 CBEAM 277



Non-Follower Loady-Displacement

(mm) Follower Loady-Displacement

(mm)

Normalized 0.99173554 Normalized 1.05054152


/demos/hwsolvers/optistruct/verification/Tiploadfllwer.fem

ReferenceBisshopp, K. E., and D. C. Drucker, “Large Deflections of Cantilever Beams,” Quarterly of AppliedMathematics, vol. 3 272, 1945



NAFEMS Frequency Response Analysis

OS-V: 0300 Deep Simply-Supported Beam HarmonicForced Vibration ResponseTest 5HOptiStruct is used to investigate the Peak Displacement in y-direction and extreme fiber bending stressat undamped Natural Frequency (at the mid-span node).

Figure 38:

Benchmark ModelTimoshenko beam and Engineer’s beam elements are used to model the simply-supported beam whichconsists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x directionare fixed at the end A. In addition, the displacements in y and z direction are constrained at end B. Asteady state harmonic forced vibration F=F0 sin ωt is induced in the y-direction. (F0=106 N/m uniformlydistributed, ω=2πf, f=0 to 4.16 Hz). For modal analysis solution, a damping ratio of 0.02 is applied in all16 modes and for direct solution, Rayleigh damping factor α1=5.36 and α2=7.46×10

-5 are given.




Density8000 kg/m3

Frequency Response SummaryThe frequency of each targeted mode is normalized with the closed form solution.




Peak Displacement(mm)

Peak Stress (N/mm2)

Frequency (Hz)

Reference Solution 13.45 241.9 42.65

PBEAML

Direct Solution 13.42 236.1 43.02

Normalized 1.002235469 1.024565862 0.991399349

Modal Solution 13.56 238.61 43.16

Normalized 0.991887906 1.01378819 0.988183503

PBEAM


Normalized 1.096169519 1.015490534 0.941916961

Modal Solution 12.3 238.89 45.34

Normalized 1.093495935 1.012599941 0.94067049



Test5HPBEAMLD.fem

Test5HPBEAMLM.fem

Test5HPBEAMD.fem

Test5HPBEAMM.fem




OS-V: 0310 Deep Simply-Supported Beam Periodic ForcedVibration ResponseTest 5POptiStruct is used to investigate the Peak Displacement in y-direction and extreme fiber bending stressat the mid-span node.

Figure 39:

Benchmark ModelTimoshenko beam and Engineer’s beam elements are used to model the simply-supported beam whichconsists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x directionare fixed at the end A. In addition, the displacements in y and z direction are constrained at end B. Asteady state periodic forced vibration F=F0 (sin ωt-sin 3ωt) is induced in the y-direction. (F0=10

6 N/muniformly distributed, ω=2ωf, f=20 Hz). For modal analysis solution, a damping ratio of 0.02 is appliedin all 16 modes and for direct solution, Rayleigh damping factor α1=5.36 and α2=7.46×10

-5 are given.




Density8000 kg/m3




Peak Stress (N/mm2)

Reference Solution 0.951 17.10

HOE:




Peak Stress (N/mm2)

Direct Solution 0.982 17.367

Normalized 0.968431772 0.984626015

Modal Solution 0.982 17.369

Normalized 0.968431772 0.984512637

LOE:

Direct Solution 1 19.6

Normalized 0.951 0.87244898

Modal Solution 1 0.951

Normalized 19.6 0.87244898



Test5PPBEAMLD.fem

Test5PPBEAMLM.fem

Test5PPBEAMD.fem

Test5PPBEAMM.fem




OS-V: 0320 Deep Simply-Supported Beam Random ForcedVibration ResponseTest 5ROptiStruct is used to investigate the Peak Displacement in y-direction and extreme fiber bending stressat undamped Natural Frequency (at the mid-span node).

Figure 40:

Benchmark ModelTimoshenko beam and Engineer’s beam elements are used to model the simply-supported beam whichconsists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x directionare fixed at the end A. In addition, the displacements in y and z direction are constrained at end B.A steady state random forcing with uniform power spectral density (of force) PSD= (106 N/m)2/Hzis induced in the y-direction. For modal analysis solution, a damping ratio of 0.02 is applied in all 16modes and for direct solution, Rayleigh damping factor α1=5.36 and α2=7.46×10

-5 are given.




Density8000 kg/m3




Peak Stress (N/mm2)

Frequency (Hz)

Reference Solution 180.90 58516 42.65

HOE:




Peak Stress (N/mm2)

Frequency (Hz)

Direct Solution 184.03 56999.67 43.14

Normalized 0.982991903 1.026602435 0.988641632

Modal Solution 183.93 56973.24 43.16

Normalized 0.983526342 1.027078678 0.988183503

LOE:

Direct Solution 150.9 56917.35 45.34

Normalized 1.198807157 1.028087218 0.94067049

Modal Solution 151.4 57105.24 45.34

Normalized 1.194848085 1.024704563 0.94067049



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OS-V: 0330 Deep Simply-Supported Beam TransientForced Vibration ResponseTest 5TOptiStruct is used to investigate the Peak Displacement in y-direction, the time at the peakdisplacement, extreme fiber bending stress at undamped Natural Frequency and the Staticdisplacement at the mid-span node.

Benchmark ModelTimoshenko beam and Engineer’s beam elements are used to model the simply-supported beam whichconsists of 10 elements. The displacements in x, y, and z direction, as well as the rotation in x directionare fixed at the end A. In addition, the displacements in y and z direction are constrained at end B.A suddenly applied step load F0=10

6 N/m is induced in the y-direction. For modal analysis solution, adamping ratio of 0.02 is applied in all 16 modes at a time step of 0.0001 secs and for direct solution,Rayleigh damping factor α1=5.36 and α2=7.46×10

-5 at a time step of 0.0001 secs are given.




Density8000 kg/m3



PeakDisplacement(mm)

Time at PeakDisplacement(sec)

Peak Stress (N/mm2)

StaticDisplacement(mm)

Reference Solution 1.043 0.0117 18.76 0.538

HOE:





Peak Stress (N/mm2)


Direct Solution 1.036 0.0116 18.02 0.533

Normalized 1.006756757 1.00862069 1.041065483 1.009380863

Modal Solution 1.03 0.0115 17.99 0.533

Normalized 1.012621359 1.017391304 1.042801556 1.009380863

LOE:

Direct Solution 0.94 0.0109 18.02 0.48

Normalized 1.109574468 1.073394495 1.041065483 1.120833333

Modal Solution 0.939 0.01109 17.92 0.48

Normalized 1.110756124 1.055004509 1.046875 1.120833333



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OS-V: 0340 Simply-Supported Thin Square Plate HarmonicForced Vibration ResponseTest 13HOptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat undamped Natural Frequency (at the center of the plate).

Figure 41:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A steady state harmonic forced vibration F=F0 sin ωt is induced in the z-direction. (F0=100 N/m

2 overwhole plate, ω=2ωf, f=0 to 4.16 Hz). For modal analysis solution, a damping ratio of 0.02 is applied inall 16 modes and for direct solution, Rayleigh damping factor α1=0.299 and α2=1.339×10

-3 are given.






Density8000 kg/m3




Peak Stress (N/mm2) Frequency (HZ)


HOE:


Normalized 0.961188471 0.799307958 1.023245803


Normalized 0.959442332 0.797821467 1.022805508

LOE:


Normalized 1.004422822 0.973735409 1.011919966


Normalized 0.999339934 0.969335055 1.013646055



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OS-V: 0350 Simply-Supported Thin Square Plate PeriodicForced Vibration ResponseTest 13POptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat the center of the plate.

Figure 42:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10. Asteady state harmonic forced vibration F=F0 (sin ωt-sin 3ωt) is induced in the z-direction. (F0=100 N/m2 over whole plate, ω=2πf, f=1.2 Hz). For modal analysis solution, a damping ratio of 0.02 is appliedin all 16 modes and for direct solution, Rayleigh damping factor α1=0.299 and α2=1.339×10

-3 aregiven.






Density8000 kg/m3



Peak Displacement (mm) Peak Stress (N/mm2)


HOE:


Normalized 0.977800546 0.834574028


Normalized 0.977466712 0.831821929

LOE:


Normalized 1.013451327 1.031697342


Normalized 1.013092711 1.029066803



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OS-V: 0360 Simply-Supported Thin Square Plate HarmonicForced Vibration ResponseTest 13ROptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat undamped Natural Frequency (at the center of the plate).

Figure 43:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A steady state random forcing with uniform power spectral density (of force) PSD= (100 N/m2)2/Hzis induced in the z-direction. For modal analysis solution, a damping ratio of 0.02 is applied in all 16modes and for direct solution, Rayleigh damping factor α1=0.299 and α2=1.339×10-3 are given.




Density8000 kg/m3





Peak DisplacementPSD (mm2/Hz)

Peak Stress PSD ((N/mm2)2/Hz)

Frequency (Hz)


HOE

Direct Solutions 2232.98 1411.14 2.322

Normalized 0.923967075 0.726674887 1.023686477

Modal Solution 2241.33 1416.89 2.324

Normalized 0.920524867 0.723725907 1.022805508

LOE

Direct Solutions 2045.23 951.00 2.349

Normalized 1.008786298 1.078275499 1.011919966

Modal Solution 2065.73 960.22 2.345

Normalized 0.998775251 1.067921935 1.013646055



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OS-V: 0365 Simply-Supported Thin Square Plate TransientForced Vibration ResponseTest 13TOptiStruct is used to investigate the Peak Displacement in z-direction, the time at the peakdisplacement, extreme fiber bending stress at undamped Natural Frequency and the Staticdisplacement at the center of the plate.

Figure 44:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A suddenly applied step load F0=100 N/m

2 is induced in the z-direction. For modal analysis solution,a damping ratio of 0.02 is applied in all 16 modes at a time step of 0.002 secs and for direct solution,Rayleigh damping factor α1=0.299 and α2=1.339×10







Density8000 kg/m3





Peak Stress (N/mm2)



HOE:

Direct Solution 3.637 0.212 2.784 1.832

Normalized 0.968655485 0.990566038 0.892241379 0.991812227

Modal Solution 3.643 0.210 2.820 1.832

Normalized 0.967060115 1 0.880851064 0.991812227

LOE:

Direct Solution 3.465 0.210 2.244 1.780

Normalized 1.016738817 1 1.106951872 1.020786517

Modal Solution 3.454 0.214 2.203 1.779

Normalized 1.019976838 0.981308411 1.127553336 1.021360315



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OS-V: 0370 Simply-Supported Thick Square PlateHarmonic Forced Vibration ResponseTest 21HOptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat undamped Natural Frequency (at the center of the plate).

Figure 45:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 1.0m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A steady state harmonic forced vibration F=F0 sin ωt is induced in the z-direction. (F0=10

6 N/m2 overwhole plate, ω=2πf, f=0 to 78.17 Hz). For modal analysis solution, a damping ratio of 0.02 is applied inall 16 modes and for direct solution, Rayleigh damping factor α1=5.772 and α2=6.926×10

-5 are given.

The material properties are;



Density8000 kg/m3






Peak Stress (N/mm2) Frequency (Hz)


HOE:


Normalized 0.931298197 0.848601736 1.01526211

Modal Solution 62.67 944.57 45.23

Normalized 0.930748364 0.847793176 1.014813177

LOE:

Direct Solution 60 774.73 45.62

Normalized 0.972166667 1.033650433 1.006137659

Modal Solution 60.05 774.54 45.59

Normalized 0.971357202 1.033903995 1.006799737



/Test21HHOED.fem

/Test21HHOEM.fem

/Test21HLOED.fem

/Test21HLOEM.fem




OS-V: 0375 Simply-Supported Thick Square Plate PeriodicForced Vibration ResponseTest 21POptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat the center of the plate.

Figure 46:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 0.05m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10. Asteady state harmonic forced vibration F=F0 (sin ωt-sin 3ωt) is induced in the z-direction. (F0=10

6 N/m2

over whole plate, ω=2πf, f=20 Hz). For modal analysis solution, a damping ratio of 0.02 is applied in all16 modes and for direct solution, Rayleigh damping factor α1=5.772 and α2=6.929×10

-5 are given.






Density8000 kg/m3



Peak Displacement (mm) Peak Stress (N/mm2)


HOE:


Normalized 0.960070121 0.853772395


Normalized 0.960070121 0.8534386

LOE:


Normalized 0.982655502 1.033445327


Normalized 0.98226385 1.031633509



/Test21PHOED.fem

/Test21PHOEM.fem

/Test21PLOED.fem

/Test21PLOEM.fem




OS-V: 0380 Simply-Supported Thick Square Plate RandomForced Vibration ResponseTest 21ROptiStruct is used to investigate the Peak Displacement in z-direction and extreme fiber bending stressat undamped Natural Frequency (at the center of the plate).

Figure 47:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 1.0m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A steady state random forcing with uniform power spectral density (of force) PSD= (106 N/m2)2/Hzis induced in the z-direction. For modal analysis solution, a damping ratio of 0.02 is applied in all 16modes and for direct solution, Rayleigh damping factor α1=5.772 and α2=6.929×10

-5 are given.






Density8000 kg/m3



Peak DisplacementPSD (mm2/Hz)

Peak Stress PSD ((N/mm2)2

Frequency (Hz)


HOE:

Direct Solution 3929.62 892303.43 45.24

Normalized 0.865684214 0.718589639 1.014588859

Modal Solution 3928.88 892421.36 45.27

Normalized 0.865847264 0.718494681 1.013916501

LOE:

Direct Solution 3607.25 600979.1 45.62

Normalized 0.943048028 1.066925622 1.006137659

Modal Solution 3606.23 600094.2 45.63

Normalized 0.943314764 1.068498912 1.00591716



/Test21RHOED.fem

/Test21RHOEM.fem

/Test21RLOED.fem

/Test21RLOEM.fem




OS-V: 0385 Simply-Supported Thick Square PlateTransient Forced Vibration ResponseTest 21TOptiStruct is used to investigate the Peak Displacement in z-direction, the time at the peakdisplacement, extreme fiber bending stress at undamped Natural Frequency and the Staticdisplacement at the center of the plate.

Figure 48:

Benchmark ModelThe 2nd order and 1st order quad elements are used to model the square plate of thickness 1.0m. Thez-rotation and x, y translations are fixed for all the nodes, z-translation is fixed along all four edges, x-rotation is fixed along the edge x=0 and x=10 and y-rotation is fixed along the edge y=0 and y=10.A suddenly applied step load F0=10

6 N/m2 is induced in the z-direction. For modal analysis solution, adamping ratio of 0.02 is applied in all 16 modes at a time step of 0.0001 secs and for direct solution,Rayleigh damping factor α1=5.772 and α2=6.929×10







Density8000 kg/m3





Peak Stress (N/mm2)



HOE:

Direct Solution 4.838 0.011 72.67 2.42

Normalized 0.935097148 0.981818182 0.854685565 0.964049587

Modal Solution 4.870 0.011 75.16 2.42

Normalized 0.928952772 0.981818182 0.82637041 0.964049587

LOE:

Direct Solution 4.604 0.0108 57.98 2.34

Normalized 0.982623805 1 1.071231459 0.997008547

Modal Solution 4.611 0.0107 58.44 2.341

Normalized 0.981132075 1.009345794 1.062799452 0.996582657



/Test21THOED.fem

/Test21THOEM.fem

/Test21TLOED.fem

/Test21TLOEM.fem




NAFEMS Normal Modes Analysis

OS-V: 0400 Pin-ended Double CrossTest No. FV2A pin-ended

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