femap structural - verification guide

Version 8.2

Verification Guide

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Table of ContentsProprietary and Restricted Rights Notice

Overview

Linear Statics Verification Using Theoretical SolutionsNodal Loads on a Cantilever Beam ....................................................................................4Axial Distributed Load on a Linear Beam ..........................................................................6Distributed Loads on a Cantilever Beam ............................................................................9Moment Load on a Cantilever Beam ................................................................................12Thermal Strain, Displacement, and Stress on Heated Beam ............................................15Uniformly Distributed Load on Linear Beam ..................................................................18Membrane Loads on a Plate .............................................................................................21Thin Wall Cylinder in Pure Tension .................................................................................24Thin Shell Beam Wall in Pure Bending ...........................................................................27Strain Energy of a Truss ...................................................................................................30

Linear Statics Verification Using Standard NAFEMS BenchmarksElliptic Membrane ............................................................................................................34Cylindrical Shell Patch Test .............................................................................................39Laminate Strip ..................................................................................................................42Hemisphere-Point Loads ..................................................................................................44Z–Section Cantilever ........................................................................................................47Skew Plate Normal Pressure .............................................................................................49Thick Plate Pressure .........................................................................................................53Solid Cylinder/Taper/Sphere–Temperature ......................................................................58

Normal Modes/Eigenvalue Verification Using Theoretical SolutionsUndamped Free Vibration - Single Degree of Freedom ...................................................65Two Degrees of Freedom Undamped Free Vibration - Principle Modes .........................68Three Degrees of Freedom Torsional System ..................................................................71Two Degrees of Freedom Vehicle Suspension System ....................................................73Cantilever Beam Undamped Free Vibrations ...................................................................76Natural Frequency of a Cantilevered Mass ......................................................................78

Normal Modes/Eigenvalue Verification Using Standard NAFEMS Bench-marksBar Element Test Cases ....................................................................................................82

Pin-ended Cross - In-plane Vibration ........................................................................83Pin-ended Double Cross - In-plane Vibration ...........................................................86Free Square Frame - In-plane Vibration ....................................................................89

Cantilever with Off-Center Point Masses ................................................................. 92Deep Simply-Supported Beam .................................................................................. 95Circular Ring - In-plane and Out-of-plane Vibration ................................................ 98Cantilevered Beam .................................................................................................. 101

Plate Element Test Cases ................................................................................................ 104Thin Square Cantilevered Plate -Symmetric Modes ............................................... 105Thin Square Cantilevered Plate - Anti-symmetric Modes ...................................... 108Free Thin Square Plate ............................................................................................ 111Simply-Supported Thin Square Plate ...................................................................... 114Simply-Supported Thin Annular Plate .................................................................... 117Clamped Thin Rhombic Plate ................................................................................. 121Cantilevered Thin Square Plate with Distorted Mesh ............................................. 124Simply-Supported Thick Square Plate, Test A ....................................................... 129Simply-Supported Thick Square Plate, Test B ........................................................ 133Clamped Thick Rhombic Plate ............................................................................... 136Simply-Supported Thick Annular Plate .................................................................. 140Cantilevered Square Membrane .............................................................................. 144Cantilevered Tapered Membrane ............................................................................ 148Free Annular Membrane ......................................................................................... 152Cantilevered Thin Square Plate ............................................................................... 156Cantilevered Thin Square Plate #2 .......................................................................... 161

Axisymmetric Solid and Solid Element Test Cases ....................................................... 164Free Cylinder - Axisymmetric Vibration ................................................................ 165Thick Hollow Sphere - Uniform Radial Vibration .................................................. 168Simply-Supported Annular Plate -Axisymmetric Vibration ................................... 171Deep Simply-Supported Solid Beam ...................................................................... 174Simply-Supported Solid Square Plate ..................................................................... 178Simply-Supported Solid Annular Plate ................................................................... 182Cantilevered Solid Beam ......................................................................................... 186

Verification Test Cases from the Societe Francaise des MechaniciensMechanical Structures - Linear Statics Analysis with Bar or Rod Elements ................. 191

Short Beam on Two Articulated Supports .............................................................. 192Clamped Beams Linked by a Rigid Element .......................................................... 194Transverse Bending of a Curved Pipe ..................................................................... 196Plane Bending Load on a Thin Arc ......................................................................... 199Nodal Load on an Articulated Rod Truss ................................................................ 201Articulated Plane Truss ........................................................................................... 203Beam on an Elastic Foundation ............................................................................... 206

Mechanical Structures - Linear Statics Analysis with Plate Elements ........................... 209Plane Shear and Bending Load on a Plate ............................................................... 210Infinite Plate with a Circular Hole .......................................................................... 212Uniformly Distributed Load on a Circular Plate ..................................................... 215Torque Loading on a Square Tube .......................................................................... 218Cylindrical Shell with Internal Pressure .................................................................. 221

Uniform Axial Load on a Thin Wall Cylinder ........................................................225Hydrostatic Pressure on a Thin Wall Cylinder ........................................................229Gravity Loading on a Thin Wall Cylinder ..............................................................232Pinched Cylindrical Shell ........................................................................................236Spherical Shell with a Hole .....................................................................................239Uniformly Distributed Load on a Simply-Supported Rectangular Plate .................242Uniformly Distributed Load on a Simply-Supported Rhomboid Plate ...................247Shear Loading on a Plate .........................................................................................251

Mechanical Structures - Linear Statics Analysis with Solid Elements ...........................254Solid Cylinder in Pure Tension ...............................................................................255Internal Pressure on a Thick-Walled Spherical Container ......................................261Internal Pressure on a Thick-Walled Infinite Cylinder ...........................................268Prismatic Rod in Pure Bending ...............................................................................274Thick Plate Clamped at Edges .................................................................................279

Mechanical Structures - Normal Modes/Eigenvalue Analysis .......................................284Lumped Mass-Spring System ..................................................................................285Short Beam on Simple Supports ..............................................................................288Axial Loading on a Rod ..........................................................................................291Cantilever Beam with a Variable Rectangular Section ...........................................294Thin Circular Ring ...................................................................................................297Thin Circular Ring Clamped at Two Points ............................................................300Vibration Modes of a Thin Pipe Elbow ...................................................................303Cantilever Beam with Eccentric Lumped Mass ......................................................307Thin Square Plate (Clamped or Free) ......................................................................311Simply-Supported Rectangular Plate ......................................................................314Thin Ring Plate Clamped on a Hub .........................................................................317Vane of a Compressor - Clamped-free Thin Shell ..................................................320Bending of a Symmetric Truss ................................................................................323Hovgaard’s Problem - Pipes with Flexible Elbows .................................................326Rectangular Plates ...................................................................................................328

Stationary Thermal Tests - Steady State Heat Transfer Analysis ...................................330Hollow Cylinder - Fixed Temperatures ...................................................................331Hollow Cylinder - Convection ................................................................................334Cylindrical Rod - Flux Density ...............................................................................337Hollow Cylinder with Two Materials - Convection ................................................340Wall - Convection ....................................................................................................344Wall - Fixed Temperatures ......................................................................................347L-Plate .....................................................................................................................350Hollow Sphere - Fixed Temperatures, Convection .................................................353Hollow Sphere with Two Materials -Convection ....................................................356

Thermo-mechanical Test - Linear Statics Analysis ........................................................360Thermal Gradient on a Thin Pipe ............................................................................361

Index ...............................................................................................................................365

OverviewThis guide contains verification test cases for the FEMAP Structural solver. These test cases verify the function of the different FEMAP Structural analysis types using theoretical and benchmark solutions from well–known engineering test cases. Each test case contains test case data and information, such as element type and material properties, results, and refer-ences.

The guide contains test cases for:

• Linear Statics verification using theoretical solutions

• Linear Statics verification using standard NAFEMS benchmarks

• Normal Modes/Eigenvalue verification using theoretical solutions

• Normal Modes/Eigenvalue verification using standard NAFEMS benchmarks

• Verification Test Cases from the Societe Francaise des Mechaniciens

Linear Statics Verification Using Theoretical Solutions

The purpose of these linear statics test cases is to verify the function of the FEMAP Structural Statics Analysis software using theoretical solutions. The test cases are relatively simple in form and most of them have closed–form theoretical solutions.

The theoretical solutions shown in these examples are from well–known engineering texts. For each test case, a specific reference is cited. All theoretical reference texts are listed at the end of this topic.

The finite element method is very flexible in the types of physical problems represented. The verification tests provided are not exhaustive in exploring all possible problems, but represent common types of applications.

This overview provides information on the following:

• understanding the test case format

• understanding comparisons with theoretical solutions

• references

Understanding the Test Case FormatEach test case is structured with the following information:

• test case data and information

- physical and material properties

- finite element modeling (modeling procedure or hints)

- units

- solution type

- element type

- boundary conditions (loads, constraints)

• results

• references (text from which a closed–form or theoretical solution was taken)

Note: . The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ.

In addition to these example problems, test cases from NAFEMS (National Agency for Finite Element Methods and Standards, National Engineering Laboratory, Glasgow, U.K.) have been executed. Results for these test cases can be found in the next section, Linear Stat-ics Analysis Verification Using NAFEMS Standard Benchmarks.

Understanding Comparisons with Theoretical Solutions

While differences in finite element and theoretical results are, in most cases, negligible, some tests would require an infinite number of elements to achieve the exact solution. Ele-ments are chosen to achieve reasonable engineering accuracy with reasonable computing times.

Results reported here are results which you can compare to the referenced theoretical solu-tion. Other results available from the analyses are not reported here. Results for both theoret-ical and finite element solutions are carried out with the same significant digits of accuracy.

The closed–form theoretical solution may have restrictions, such as rigid connections, that do not exist in the real world. These limiting restrictions are not necessary for the finite ele-ment model, but are used for comparison purposes. Verification to real world problems is more difficult but should be done when possible.

The actual results from the FEMAP Structural software may vary insignificantly from the results presented in this document. This variation is due to different methods of performing real numerical arithmetic on different systems. In addition, it is due to changes in element formulations which SDRC has made to improve results under certain circumstances.

ReferencesThe following references have been used in the Linear Statics Analysis verification prob-lems presented:

1. Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.)

2. Harris, C. O., Introduction to Stress Analysis, (1959.)

3. Roark, R. and Young, W., Formulas for Stress and Strain, 5th Edition, (New York: McGraw–Hill Book Company, 1975.)

4. Shigley, J. and Mitchel L., Mechanical Engineering Design, 4th Edition, (New York: McGraw–Hill Book Company, 1983.)

5. Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, (New YorK: Van Norstrand Reinhold Company, 1955.)

Nodal Loads on a Cantilever BeamThe complete model and results for this test case are in file mstvl001.neu.

Determine the deflection of a beam at the free end. Determine the stress at the end of the beam.

Test Case Data and Information

Element Types bar

Units Inch

Model GeometryLength=480 in

Cross Sectional Properties• Area = 30 x 30 in

• Iy =Iz = 67500 in4

Material Properties• E = 30 E+06 psi

Finite Element Modeling • 5 nodes

• 4 successive bar elements along the X axis

Boundary Conditions

ConstraintsConstrain the left end (node 1) of the beam in all six degrees.

LoadsSet nodal force to 50,000 lb. in the negative Y direction.

Solution Type Statics

Results

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p.

716.

Beam End A1 Z Shear Force Stress

(Node 1)T2 Translation (Node 5)

Bench Value 5333.3 0.91022

FEMAP Structural 5333.3 0.913

Difference 0% 0.30%

Axial Distributed Load on a Linear Beam

The complete model and results for this test case are in file mstvl002.neu.

Determine the stress, elongation, and constraint force due to an axial loading along a linear beam.


Element Typebar

Units Inch

Model GeometryLength = 300 in

Cross Sectional Properties• Area = 9 in2

• square cross section (3 in x 3 in)

• I = 6.75 in4

Material PropertiesE = 30E+6 psi


• 30 bar elements along the X axis, each 10 inches long.

Boundary Conditions

ConstraintsConstrain one end of the beam (node 1) in all translations and rotations.

LoadsSet the axial distributed load (force per unit length) to 1000lb/in for the 10–inch long ele-ment (element 30) furthest from the constrained end.


Results

Reference• Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p.

76.

Beam End A1 Axial Stress

(Node 1)

T1 Translation(Node 2)

T1 Constraint Force

(Node 1)

Bench value 1111.1 0.0111111 -10,000

FEMAP Structural 1111.1 0.0109258 -10,000

Difference 0 1.6% 0

Distributed Loads on a Cantilever Beam


Determine the deflection of a beam at the free end. Determine the stress at the midpoint of the beam and the reaction force at the restrained end.


Element Typebar

Units Inch

Model Geometry• Length = 480 in



• Iy = Iz = 67500 in4

Material PropertiesE = 30 E+06 psi


• 8 successive bar elements along the X axis

Boundary Conditions

ConstraintsConstrain the left end of the beam (node 1) in all translations and rotations.

LoadsDefine a distributed load on the elements of 250 lb/in in the negative Y direction.


Results

Beam End A1Z Bend Stress

(node 1)

Total Translation(node 5)

Total Constraint Force(lb)

Bench Value 6,400.0 0.8190 120,000

FEMAP Structural 6,400.0 0.8225* 120,000

Difference 0.0% 0.43% 0

* Includes shear deformation which is neglected in theoretical value.


716.

Moment Load on a Cantilever BeamThe complete model and results for this test case are in file mstvl004.neu.

Determine the deflection of a beam at the free end. Determine the bending stress of the beam and the reaction force at the restrained end.


Element Typebar

Units Inch




• Iy = Iz = 67500 in4

Material PropertiesE = 30 E+06 psi


• 8 successive bar elements along the X axis.

Boundary Conditions

ConstraintsConstrain the left end of the beam (node 1) in all translations and rotations.

LoadsSet the Z–moment of the end node (node 5) to 2.5e+6 in–lb.


Results

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill Inc., 1992.) p.

716.

Beam End A1 Z Bend Stress (psi)

(node 1)

Total Translation (in)(node 5)

Total Constraint Moment (lb.)

(node 1)

Bench Value 555.6 0.1422 2.5E+06

FEMAP Structural 555.6 0.1422 2.5E+06

Difference 0 0 0

Thermal Strain, Displacement, and Stress on Heated Beam


A beam originally 1 meter long and at -50° C is heated to 25° C. Determine the displacement and thermal strain on a cantilever beam. In case 1, fix the beam at the free end. In case 2, fix the beam at both ends. In both cases, determine the displacement, constraint forces, and stresses along the beam.


Element Typebar

Units SI - meter

Model GeometryLength = 1 m

Cross Sectional PropertiesArea = 0.01 m2

Material Properties• E = 2.068E+11 PA

• Coeff. of thermal expansion = 1.2E-05 m/(m-C)

• v = 0.3


• 10 bar elements on the X axis.

Boundary Conditions

Constraints• Case 1: Constrain the node on one end (node 1) of the beam in all translations and rota-

tions.

• Case 2: Constrain the nodes on both ends (nodes 1 and 11) of the beam in all translations and rotations.

Loads Set the temperature on all nodes to 25°C. Set the reference temperature to -50°C.


Results

Case: One Fixed End

Case: Both Ends Fixed


65.

Total Translation (Node 11)(m)

Beam End A1 Axial Strain

Bench Value 9E-04 9E-04

FEMAP Structural 9E-04 9E-04

Difference 0 0

Total Translation (m)Total Constraint

Force(N)(node 1)

Beam End A1 Axial Stress(Pa)

Bench Value 0 1.86+06 –1.86E+08

FEMAP Structural 0 1.86+06 –1.86E+08

Difference 0 0 0

Uniformly Distributed Load on Lin-ear Beam


A beam 40 feet long is restrained and loaded with a distributed load of –833 lb. Determine the beam end torque stress and the deflection at the middle of the beam.


Element Typebar

Units Inch


Cross Sectional Properties• Rectangular cross section (1.17 in x 43.24 in)

• Iz = 7892 in4

Material Properties• E = 30E6 psi


• 4 successive bar elements that are each 10 feet long

Boundary Conditions

ConstraintsConstrain nodes 2 and 4 in five degrees of freedom. Do not constrain rotation about Z.

Loads Define a distributed load (force per unit length) of -833 lb. (global negative Y direction) on the elements 1 and 4.


Results

Reference• Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, (New

York: Van Norstrand Reinhold Company, 1955.) p. 98.

Total Translation (in)(node 3)

Beam End A1 Z Bend Stress (psi)

(node 3)

Bench Value 0.182 16,439

FEMAP Structural 0.182 16,439

Difference 0 0

Membrane Loads on a PlateThe complete model and results for this test case are in file mstvl009.neu.

A circle is scribed on an unstressed aluminum plate. Forces acting in the plane of the plate cause normal stresses. Determine the change in the length of diameter AB and of diameter CD.


Element Types plate

Units Inch


• Diameter = 9 in

• Thickness = 3/4 in

Material Properties• E = 10 E+06 psi

• Poisson’s ratio = 1/3

• F(x)/l = 9,000 lb./in

• F(z)/l = 15,000 lb./in

Finite Element Modeling Create 1/4 of the model and apply symmetry boundary conditions. Then multiply the answer by 2 for correct results. Remember to account for the ratio of the circle diameter to plate length.

Boundary Conditions

ConstraintsConstrain nodes along adjacent sides of the plate to allow only translation along the corre-sponding axis.

• Node 1: Fully constrain in all translations and rotation.

• Nodes 2-6: Constrain in the Y and Z translations and the X and Z rotations.

• Nodes 12, 13, 19, 25, 31: Constrain in the X and Y translations and the X and Z rotations.

Loads Set the elemental edge load to 9,000 lb./in in the X direction and 15,000 lb/in in the Z direc-tion.


Results

Post Processing• (T1 translation at node 7 - T1 translation at node 10) x2 = (.004-.0016) x2 = .0048

• (T3 translation at node 7 - T3 translation at node 24) x2 = (.012-.0048) x2 = .0144


85.

T1 Translation (in) T3 Translation (in)

Bench Value 4.8E-03 14.4E-03

FEMAP Structural 4.8E-03 14.4E-03

Difference 0 0

Thin Wall Cylinder in Pure TensionThe complete model and results for this test care are in file mstvl014.neu.

Determine the stress and deflection of a thin wall cylinder with a uniform axial load.


Element Type linear quadrilateral plate

Units Inch

Model Geometry• R = 0.5 in

• Thickness = 0.01 in

• y = 1.0 in

Material Properties• E = 10000 psi

• v = 0.3

Finite Element Modeling• 25 nodes

• Create 1/4 model of the cylinder with 16 linear quadrilateral plate elements and symmetry boundary conditions.

Boundary Conditions

Constraints• Constrain node 1 in the X and Z translation and the Z rotation.

• Constrain nodes 2-4 in the Z translation.

• Constrain node 5 in the Y and Z translation and Z rotation.

• Constrain nodes 6, 11, 16, and 21 in the X translation and Z rotation.

• Constrain nodes 10, 15, 20, and 25 in the Y translation and Z rotation.

Loads• Nodal forces of p/(pi)D = 3.1831 where p = 10 psi; Apply the following nodal forces:

• Nodes 21, 25: .9757 pounds

• Nodes 22, 23, 24: 1.9509 pounds


Results

Top Y Normal Stress(psi)

T3 Translation (in) T1 Translation (in)

Bench Value 1000.0 0.1 -0.015

FEMAP Structural 1000.0 0.1 -0.015

Difference 0 0 0

Reference • Roark, R. and Young, W., Formulas for Stress and Strain, 6th Edition, (New York:

McGraw–Hill Book Company, 1989.) p. 518, Case 1a.

Thin Shell Beam Wall in Pure Bend-ing


Determine the maximum stress, maximum deflection, and strain energy of a thin shell beam wall with a uniform bending load.


Element Type linear quadrilateral plate

Units Inch


• Width = 5 in

• Thickness = 0.1 in

Material Properties• E = 30E6 psi

• v = 0.03


• 6 linear quadrilateral plate elements

Boundary Conditions

ConstraintsConstrain the nodes at one end (nodes 7 and 14) in all translations and rotations.

Out–of–plane LoadsApply nodal forces (nodes 1 and 8) of p/w = 1.2 lbs/in. where p = 6.0 lb


Results

Reference • Shigley, J. and Mitchel L., Mechanical Engineering Design, 4th Edition, (New York:

McGraw–Hill, Inc., 1983.) pp. 134, 804.

T3 Translation (in)Node 1

Plate Bottom Major Stress(psi)

Node 7

Total Strain Energy (lb in)

Bench Value 4.320 21600 12.96

FEMAP Structural 4.242 20983 12.73

Difference 2.17% 1.39% 2.16%

Strain Energy of a TrussThe complete model and results for this test case are in file mstvl016.neu.

Determine the strain energy of a truss. The cross–sectional area of the diagonal members is twice the cross–sectional area of the horizontal and vertical members.


Element Type rod

Units Inch


Cross Sectional PropertiesCross sectional area (A) = 0.01 in2

Material PropertiesE = 30E6 psi


• 5 rod elements

Boundary Conditions

Constraints• Constrain node 1 in the X, Y, and Z translations and the X and Y rotations.

• Constrain node 3 in the Y and Z translations and the X and Y rotations.

Loads• Apply nodal force in Y direction on node 2; p = 300 lb


Results


588.

Total Strain Energy (lb in)

Bench Value 5.846

FEMAP Structural 5.846

Difference 0

Linear Statics Verification Using Standard NAFEMS Benchmarks

The purpose of these linear statics test cases is to verify the function of the FEMAP Structural Statics Analysis software using standard benchmarks published by NAFEMS (National Agency for Finite Element Methods and Standards, National Engineering Laboratory, Glas-gow, U.K.).

These standard benchmark tests were created by NAFEMS to stretch the limits of the finite elements in commercial software. All results obtained using the FEMAP Structural Statics Analysis software compare favorably with other commercial finite element analysis software. Results of these test cases using other commercial finite element analysis software programs are available from NAFEMS.

A detailed discussion of the linear statics NAFEMS benchmarks can be found in the NAFEMS publication Background to Benchmarks, cited below. The results for all of these test cases illustrate the need for adequate mesh refinement for obtaining accurate stresses, especially when using linear elements. The linear triangular and linear tetrahedral elements are particularly poor performers for stress analysis and are not generally recommended.





- units

- finite element modeling information

- boundary conditions (loads and constraints)

- solution type

• results

• reference

ReferencesThe following references have been used in these test cases:

Note: The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ.

• NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.)

• Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glas-gow: NAFEMS, 1993).

Elliptic MembraneThe complete model and results for this test case are in the following files:

• le101.neu (quadrilateral plane strain)

• le102.neu (triangular plane strain)

• le103.neu (quadrilateral plate)

This test is a linear elastic analysis of an elliptic membrane (shown below) using coarse and fine meshes of plane strain elements and plate elements. The plane strain elements use a plane stress element formulation. It provides the input data and results for NAFEMS Standard Benchmark Test LE1.

Ellipses:


Physical and Material Properties• Thickness = 0.1 m

• Isotropic material

• E = 210 x 103 MPa

A

B

C DX

Y

Ellipse AC: x2---

2y

2+ 1= Ellipse BD:

x3.25----------

2 y2.75----------

2+ 1=

• v = 0.3

UnitsSI

Finite Element Modeling• plane strain (with plane stress element formulation) - linear and parabolic quadrilaterals -

coarse and fine mesh

• plane strain (with plane stress element formulation) - linear and parabolic triangles - coarse and fine mesh

• plate - linear and parabolic quadrilaterals - coarse and fine mesh

The fine mesh is created by approximately halving the coarse mesh.

Boundary Conditions

Constraints• Constrain the nodes along edge AB in the X translation.

• Constrain the nodes along edge CD in the Y translation.

Linear Triangle Parabolic Triangle

Fine Mesh

Coarse Mesh

Linear Quadrilateral Parabolic Quadrilateral

Fine Mesh

Coarse Mesh

A

B

C D

A

B

C D

A

B

C D

A

B

C D

A

B

C D

A

B

C D

A

B

C D

A

B

C D

Loads• Uniform outward pressure on the elements on outer edge BD = 10MPa

• Inner curved edge AC is unloaded

Solution TypeStatics

ResultsOutput - Plate Mid Y Normal Stress at point D

Node # Element Type & Mesh

NAFEMSBenchValue(MPa)

FEMAP Structural

Result(MPa)

Plane Strain Elements with a Plane Strain Formulation (le101):

Node 4 linear quad - coarse mesh 92.7 62.8Node 204 linear quad - fine mesh 92.7 80.3Node 104 parabolic quad - coarse mesh 92.7 88.3Node 304 parabolic quad - fine mesh 92.7 90.7

Plane Strain Elements with a Plane Strain Formulation (le102):

Node 4 linear triangle - coarse mesh 92.7 54.2Node 204 linear triangle - fine mesh 92.7 72.0Node 104 parabolic triangle - coarse mesh 92.7 93.0Node 304 parabolic triangle – fine mesh 92.7 94.0

References• NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks,

(Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE1.


Plate Elements (le 103):Node 4 linear quad - coarse mesh 92.7 66.4Node 204 linear quad - fine mesh 92.7 82.3Node 104 parabolic quad - coarse mesh 92.7 88.6Node 304 parabolic quad - fine mesh 92.7 91.7

Cylindrical Shell Patch TestThe complete model and results for this test case are in the following files:

• le201a.neu (linear plate, case 1)

• le201b.neu (parabolic plate, case 1)

• le202a.neu (linear plate, case 2)

• le202b.neu (parabolic plate, case 2)

This test is a linear elastic analysis of a cylindrical shell (shown below) using plate elements and two different loadings. It provides the input data and results for NAFEMS Standard Benchmark Test LE2.




• E = 210 x 103 MPa

• v = 0.3

UnitsSI

Finite Element Modeling• le201a and le202a: 9 nodes, 4 linear quadrilateral plates

• le201b and le202b: 21 nodes, 4 parabolic quadrilateral plates

Linear Quadrilaterals Parabolic Quadrilaterals

A B

E

D C

A B

E

D C

Boundary Conditions

ConstraintsFully constrain the nodes on edge AB in all translations and rotations.

Constrain the nodes on edge AD and edge BC in the Z translation and X and Y rotations.

Case 1 Loading:• Nodal moments along DC = 1.0 kNm/m:

Node 3 = -125

Node 4 = -250

Node 9 = -125

Case 2 Loading:• Nodal forces:

Nodes 3, and 9 = 75,000N

Node 4 = 150,000N

• Apply an elemental pressure on elements 1-4 = 600,000Pa


ResultsOutput - Plate Top Major Stress at point E (node 2)

*Since the shapes of the plates are an approximation to a cylindrical surface, an edge load will not be in the correct direction. To get this result, the edge load must be input as nodal loads in the tangential direction.




Plate Element & Loading

NAFEMS Bench Value (MPa)

FEMAP Structural Result (MPa)

linear plate - case 1 (le201a) 60.0 57.9linear plate - case 2 (le202a) 60.0 66.0 *parabolic plate - case 1 (le201b) 60.0 54.8parabolic plate - case 2 (le202b) 60.0 55.7 *

Laminate StripThe complete model and results for this test case are in the following file:

• r0031.neu

This test is a linear statics analysis of plate using plate elements with a laminate material. It provides the input data and results for NAFEMS Report R0031.


Geometry

Material PropertiesLaminate material:

0° fiber direction

10 15 15 10

XY

XZ

1

E

10N/mm

A B

0°90°0°

90°

0°90°0°

0.10.10.1

0.4

0.10.10.1

C

E

DF

E 1.0E5 MPa= ν12 0.4= E2 5.0E3 MPa= ν12

E1--------

ν21

E2--------=

G12 3.0E3 MPa= ν23 0.3= G33 2.0E3 MPa=

UnitsSI

Finite Element Modeling8 x 40 4-noded shells (quarter model)

Boundary Conditions

ConstraintsThe one quarter model is:

• simply supported at A (Z=0)

• reflective symmetry about X=25 and Y=5

LoadsLine load of 10N/mm at C (X=25, Z=1).


Results

*Value extrapolated from FEMAP Structural results at F. (FEMAP Structural calculates stress at the center of the ply (F)).

**Recovered from post-processing.

Reference• NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks,

(Glasgow: NAFEMS, Rev. 3, 1990.) Test No. R0031.

Results


FEMAP Structural Result (MPa)

Z deflection at E -1.06 -1.06Bending stress at E 683.9 *668Bending stress at F - 601Interlaminar shear stress at D -4.1 **-4.1Shear stress at F - -2.2

Hemisphere-Point LoadsThe complete model and results for this test care are in the following files:

• le301.neu (linear quadrilateral plate, coarse mesh)

• le302.neu (linear quadrilateral plate, fine mesh)

• le303.neu (parabolic quadrilateral plate, coarse mesh)

• le304.neu (parabolic quadrilateral plate, fine mesh)

This test is a linear elastic analysis of hemisphere point loads (shown below) using coarse and fine meshes of plate elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE3.




• E = 68.25 x 103 MPa

• v = 0.3

UnitsSI

Finite Element Modelingplate - linear & parabolic quadrilaterals - coarse & fine mesh

equally spaced nodes on AC, CE, EA

Point G at X = Y = Z = 10

3

12---

------

Node 7

Boundary Conditions

Constraints• Fully constrain point E in all translations and rotations.

• Constrain the nodes along edge AE (symmetry about X–Z plane) in the Y translation, and X and Z rotations.

• Constrain the nodes along edge CE (symmetry about Y–Z plane) in the X translation, and Y and Z rotations.

Loads• Concentrated radial load outward at A = 2KN

Coarse Mesh Fine Mesh

A C

E

AC

E

F

G

D

B

F

G

D

B

• Concentrated radial load inward at C = 2KN


Results




Test Case Number

Plate Element & MeshNAFEMS

Bench Value(m)

FEMAP Structural Result at node 1

(point A) T1 Translation

(m)

le301 linear quadrilateral plate - coarse mesh 0.185 0.113le302 linear quadrilateral plate - fine mesh 0.185 0.185le303 parabolic quadrilateral plate - coarse mesh 0.185 0.0861le304 parabolic quadrilateral plate - fine mesh 0.185 0.171

Z–Section CantileverThe complete model and results for this test case are in the following files:

• le501.neu (linear quadrilateral plate)

• le502.neu (parabolic quadrilateral plate)

This test is a linear elastic analysis of a Z–section cantilever (shown below) using plate ele-ments. It provides the input data and results for NAFEMS Standard Benchmark Test LE5.




• E = 210 x 103 MPa

• v = 0.3

UnitsSI

Finite Element Modeling• Test 1: 36 nodes, 24 linear quadrilateral plate elements

• Test 2: 95 nodes, 24 parabolic quadrilateral plate elements

Boundary Conditions

Constraints• Fully constrain the nodes on edges B1, B2, B3 in all translations and rotations.

Loads• Torque of 1.2MN applied at end C by two nodal forces (at nodes 9 and 27) of 0.6MN


ResultsOutput - Plate Top Von Mises Stress (σxx), point A, node 30 (compression)




Plate Element & LoadingNAFEMS Bench

Value (MPa)

FEMAP Structural Result

(MPa)

linear quad - point A/node 30 -108 -117.3parabolic quad - point A/node 30 -108 -109.2

B1

B2

B3C

Skew Plate Normal PressureThe complete model and results for this test case are in the following files:

• le601.neu (linear and parabolic quadrilateral)

• le602.neu (linear and parabolic triangle)

This test is a linear elastic analysis of a plate (shown below) using plate elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE6.


Physical and Material Properties• Thickness = 0.01m


• E = 210 x 103 MPa

• v = 0.3

UnitsSI

A B

CD

E

150o

30o

10m

Finite Element Modeling• plate - linear and parabolic quadrilaterals - coarse and fine mesh

• plate - linear and parabolic triangles - coarse and fine mesh

Boundary Conditions

Constraints (le601)• Constrain nodes 1, 10, 35, and 44 in the X, Y, and Z translations.

• Constrain nodes 4, 13, 38, 47 in the X and Z translations.

• Constrain all other nodes in the Z translation.

Constraints (le602)• Fully constrain nodes 1, 10, 35, 44 in all directions and rotations.

• Constrain all other nodes in the Z translation.

Loads• Elemental pressure = -0.7KPa in the Z–direction


ResultsOutput - Plate Bottom Major Stress on the bottom surface at the plate center.




Test Case Name

Node # Plate Element & Mesh


FEMAP Structural

Result (MPa)

le601 Node 9 linear quad - coarse mesh 0.802 0.365le601 Node 18 linear quad - fine mesh 0.802 0.714le601 Node 43 parabolic quad - coarse mesh 0.802 1.055le601 Node 52 parabolic quad - fine mesh 0.802 0.791le602 Node 9 linear triangle - coarse mesh 0.802 0.390le602 Node 18 linear triangle - fine mesh 0.802 0.709le602 Node 43 parabolic triangle - coarse mesh 0.802 0.847le602 Node 52 parabolic triangle - fine mesh 0.802 0.822

Thick Plate PressureThe complete model and results for this test case are in the following files:

• le1001.neu (linear and parabolic brick)

• le1002.neu (linear and parabolic wedge)

• le1003.neu (linear and parabolic tetrahedron)

This article provides the input data and results for NAFEMS Standard Benchmark Test LE10. This test is a linear elastic analysis of a thick (shown below) using coarse and fine meshes of solid elements.

Ellipses:


Physical and Material Properties• Isotropic material

• E=210x103 MPa

• v = 0.3

A

B

CD

A

B

CD

A’

B’

C’D’

Ellipse AD: x2---

2y

2+ 1= Ellipse BC:

x3.25----------

2 y2.75----------

2+ 1=

UnitsSI

Finite Element Modeling• Solid brick

• Solid wedge

• Solid tetrahedron

Solid BrickLinear and parabolic, coarse and fine mesh.

Solid WedgeLinear and parabolic, coarse and fine mesh.

Solid TetrahdronLinear and parabolic, fine mesh.

Boundary Conditions

Constraints• Constrain the nodes on faces DCD’C’ and ABA’B’ in the X and Y translations.

• Constrain the nodes on face BCB’C’ in the X and Y translation.

• Constrain the nodes along the mid–plane in the Z translation.

Loads• Uniform normal elemental pressure on the elements on the upper surface of the plate =

1MPa

• Inner curved edge AD unloaded


ResultsOutput - Solid Y normal stress at point D3σyy



• Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glas-gow: NAFEMS, 1993)

Test Case Name

Node#

Element Type & Mesh



(MPa)

le1001 N4 linear brick - coarse mesh -5.38 -6.31le1001 N204 linear brick - fine mesh -5.38 -6.01le1001 N104 parabolic brick - coarse mesh -5.38 -5.73le1001 N304 parabolic brick - fine mesh -5.38 -5.84le1002 N4 linear wedge - coarse mesh -5.38 -3.52le1002 N204 linear wedge - fine mesh -5.38 -4.97le1002 N104 parab wedge - coarse mesh -5.38 -5.53le1002 N304 parab wedge - fine mesh -5.38 -6.10le1003 N40 linear tetra - fine mesh -5.38 -2.41le1003 N171 parabolic tetra - fine mesh -5.38 -5.29

Solid Cylinder/Taper/Sphere–Tem-perature

The complete model and results for this test case are in the following files:

• le1101a.neu (linear brick, coarse mesh)

• le1101b.neu (linear brick, fine mesh)

• le1102a.neu (parabolic brick, coarse mesh)

• le1102b.neu (parabolic brick, fine mesh)

• le1103a.neu (linear wedge, coarse mesh)

• le1103b.neu (linear wedge, fine mesh)

• le1104a.neu (parabolic wedge, coarse mesh)

• le1104b.neu (parabolic wedge, fine mesh)

• le1105a.neu (linear tetrahedron, coarse mesh)

• le1105b.neu (linear tetrahedron, fine mesh)

• le1106a.neu (parabolic tetrahedron, coarse mesh)

• le1106b.neu (parabolic tetrahedron, fine mesh)

This test is a linear elastic analysis of a solid cylinder with a temperature gradient (shown below) using coarse and fine meshes of solid elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE11.


Physical and Material Properties• Isotropic material

• E = 210 x 103 MPa

• v = 0.3

• a = 2.3 x 10-4/oC

UnitsSI

Finite Element Modeling• Solid brick - linear (8–noded) and parabolic (20–noded) - coarse and fine mesh

• Solid tetrahedron - linear (4–noded) and parabolic (10–noded) - coarse and fine mesh

• Solid wedge - linear (6–nodes) and parabolic (15–noded) - coarse and fine mesh

Solid BrickCoarse and fine mesh:

Coarse and fine mesh:

Boundary Conditions

Constraints• Constrain the nodes on the XZ plane and on the opposite face in the Y translation.

• Constrain the nodes on the YZ plane in the Z translation.

• Constrain the nodes on the XY plane in the X translation.

Loads• Nodal temperatures: linear temperature gradient in the radial and axial direction

T°C X2

Y2

+( )

12---

Z+=


ResultsOutput - Solid Y Normal Stress at point A.

Note that the Y direction in the models corresponds to the Z direction in NAFEMS.

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks,



CaseNode # at Point A

Element Type & Mesh


FEMAP Structural

Result (MPa)

le1101a 30 linear brick - coarse mesh -105 -95.7le1101b 71 linear brick - fine mesh -105 -99.5le1102a 67 parabolic brick - coarse mesh -105 -93.9le1102b 159 parabolic brick - fine mesh -105 -105.9le1103a 33 linear wedge - coarse mesh -105 -9.49le1103b 74 linear wedge - fine mesh -105 -46.9le1104a 71 parabolic wedge - coarse mesh -105 -88.5le1104b 187 parabolic wedge - fine mesh -105 -96.8le1105a 8 linear tetra - coarse mesh -105 -31.4le1105b 8 linear tetra - fine mesh -105 -65.2le1106a 8 parabolic tetra - coarse mesh -105 -89.6le1106b 8 parabolic tetra - fine mesh -105 -97.2

Normal Modes/Eigenvalue Verifica-tion Using Theoretical Solutions

The purpose of these normal mode dynamics test cases is to verify the function of the FEMAP Structural Normal Modes/Eigenvalue Analysis software using theoretical solutions. The test cases are relatively simple in form and most of them have closed–form theoretical solutions.

The theoretical solutions shown in these examples are from well known engineering texts. For each test case, a specific reference is cited. All theoretical reference texts are listed at the end of this topic.

The finite element method is very flexible in the types of physical problems represented. The verification tests provided are not exhaustive in exploring all possible problems, but represent common types of applications.

This overview provides information on the following:

• understanding the test case format

• understanding comparisons with theoretical solutions

• references





- units

- solution type

- element type


• results

• references (text from which a closed–form or theoretical solution was taken)


Understanding Comparisons with Theoretical SolutionsWhile differences in finite element and theoretical results are, in most cases, negligible, some tests would require an infinite number of elements to achieve the exact solution. Ele-ments are chosen to achieve reasonable engineering accuracy with reasonable computing times.

Results reported here are results which you can compare to the referenced theoretical solu-tion. Other results available from the analyses are not reported here. Results for both theoret-ical and finite element solutions are carried out with the same significant digits of accuracy.

The closed–form theoretical solution may have restrictions, such as rigid connections, that do not exist in the real world. These limiting restrictions are not necessary for the finite ele-ment model, but are used for comparison purposes. Verification to real world problems is more difficult but should be done when possible.

The actual results from the FEMAP Structural software may vary insignificantly from the results presented in this document. This variation is due to different methods of performing real numerical arithmetic on different systems. In addition, it is due to changes in element formulations which SDRC has made to improve results under certain circumstances.

ReferencesThe following references have been used in the Normal Mode Dynamics Analysis verifica-tion problems presented:

• Blevins, R., Formulas For Natural Frequency and Mode Shape, 1st Edition, (New York: Van Norstrand Reinhold Company, 1979.)

• Timoshenko and Young, Vibration Problems in Engineering, (New York: Van Norstrand Reinhold Company, 1955.)

• Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, Theory and Applications, (Boston: Allyn and Bacon, Inc., 1978.)

• Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.)

Undamped Free Vibration - Single Degree of Freedom

The complete model and results for this test case are in file mstvn002.neu.

Determine the natural frequency of the system.


Element Types • rigid

• mass

• DOF springs

Units SI - meter

Model Geometry• Length = 0.5 m

• a = 0.3 m

Physical Properties• mass = 20 Kg

• k = 8 KN/m

Finite Element Modeling • Create 5 rigid elements along the X axis. Each rigid should be 0.1m long.

• Create a mass element on the end node.

• Create 3 DOF spring elements 0.2m from the mass element.

Boundary Conditions

ConstraintsConstrain node 6 in all directions except the Z rotation.

Constrain all other nodes in the X and Y translations and in the Z rotation.

Solution Type Normal Modes/Eigenvalue – Guyan method

Results

Frequency (Hz)

Bench Value 1.90985 FEMAP Structural 1.90986 Difference 0.0%

Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, Theory and Applications,

(Boston: Allyn and Bacon, Inc., 1978.) p. 75.

Two Degrees of Freedom Undamped Free Vibration - Princi-ple Modes


Determine the natural frequencies of a dynamic system with two degrees of freedom.


Element Types • DOF springs

• mass

Units SI- meter

Physical Properties• mass = 1 kg

• k = 1 N/m

Finite Element Modeling • Create four nodes on the Y axis.

• Create DOF three springs with stiffness of 1 N/m and with a stiffness reference coordinate system being uniaxial.

• Create mass elements with a mass of 1 kg.

Boundary Conditions

Constraints • Constraint Set 1: Constrain nodes 1 and 4 in all DOF. On the other nodes, constrain all

DOF except the Y translation.

• Constraint Set 2: On the inner nodes, constrain the Y translation. Use this set as the Mas-ter (ASET) DOF set.

Solution TypeNormal Modes/Eigenvalue – Guyan method

Results

Reference• Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn

and Bacon, Inc., 1978.) pp. 145-149.

Frequency of Mode 1

(Hz)

Frequency of Mode 2

(Hz)

Bench Value 0.159155 0.2756644FEMAP Structural 0.159155 0.2756644 Difference 0.00% 0.00%

Three Degrees of Freedom Tor-sional System


Determine the natural frequencies of a dynamic system with three degrees of freedom.


Element Types • DOF springs

• mass

Units SI - meter

Physical Properties• J = J1 = J2 = J3 = 0.1 (mass)

• k = k1 = k2 = k3 = 1 N*m (stiffness)

Finite Element Modeling • Create four nodes on the X axis.

• Create three DOF springs with stiffness of 1 N*m and with a stiffness reference coordinate system being uniaxial.

• Create three mass elements with a mass coordinate system = 1 and with mass inertia sys-tem of: 0.1, 0.0, 0.0, 0.0, 0.0, 0.0.

Boundary Conditions

Constraints • Constraint Set 1: On one end node (node 1), constrain all DOF. On the other nodes, con-

strain all DOF except RX.

• Constraint Set 2: On the other nodes (nodes 2-4), constrain the DOF in RX. Use this set as the Master (ASET) DOF set.


Results

Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn

and Bacon, Inc., 1978.) pp. 153–155

Frequency of Mode 1

(Hz)

Frequency of Mode 2

(Hz)

Frequency of Mode 3

(Hz)

Bench Value 0.223986 0.627595 0.906901 FEMAP Structural 0.223986 0.627595 0.906901 Difference 0.00% 0.00% 0.00%

Two Degrees of Freedom Vehicle Suspension System


Determine the natural frequencies of dynamic system with two degrees of freedom. Degrees of freedom are one translational and one rotational.


Element Types 5 nodes, 4 elements:

• 2 DOF springs

• 1 mass element

• 1 rigid element

Units SI - meter

Model Geometry• Length1 = 1.6 m

• Length2 = 2.0 m

• r = 1.4 m (radius of gyration; J=m*r*r)

Physical Properties• mass = 1800 kg

• K1 = 42000 N/m

• K2 = 48000 N/m

Finite Element Modeling • Create five nodes in the X–Y plane with coordinates:

N1 = (0, 0)

N2 = (L2, 0)

N3 = (-L1, 0)

N4 = (L2, -1)

N5 = (-L1, -1)

• Create a DOF spring with stiffness of k1 between nodes 3 and 5.

• Create a DOF spring with stiffness of k2 between nodes 2 and 4.

• Create a mass element with a mass coordinate system = 1 and with mass inertia system of: 0.0, 0.0, 3528, 0.0, 0.0, 0.0.

• Create a three–noded rigid element using node 1 as the master node and nodes 2 and 3 as the slave nodes.

Boundary Conditions

Constraints • Constraint Set 1:

Constrain nodes 1-3 in the X and Z translation and X and Y rotations.

Constrain nodes 4-5 in the X, Y, and Z translations.

• Constraint Set 2 (Master (ASET) DOF Set):

Constrain nodes 1-3 in the Y translation and Z rotation.


Results


and Bacon, Inc., 1978.) pp. 150-153.

Frequency of Mode 1

(Hz)

Frequency of Mode 2

(Hz)

Bench Value 1.086347 1.495612FEMAP Structural 1.086347 1.495612 Difference 0.00% 0.00%

Cantilever Beam Undamped Free Vibrations


Determine the natural frequencies of a cantilever beam.


Element Type bar

Units Inch


• Height = 2 in

Physical and Material Properties• w = 1 lb/in

• J = .10

• Poisson’s ratio = .3

Calculated Data • A = h2 = 4 in2

• I = h4/12 = 1.33333

• G = E/2 x 1/1+nu = 11538461.54

• m = w/g = 2.59067375E-3

• Ip = Ixx + Iyy = 2.66666

Finite Element Modeling • Create 11 nodes on X axis.

• Create 10 bars between the nodes.

Boundary Conditions

Constraints • Fully constrain one end node (node 1) in all directions and rotations.

Solution Type Normal Modes/Eigenvalue – SVI method

Results

Reference • Blevins, R., Formulas For Natural Frequency and Mode Shape, 1st Edition, (New York:

Van Norstrand Reinhold Company, 1979) pp. 108,193.

Mode Bench Values (Hz)FEMAP Structural

(Hz)Difference

1 & 2 6.9533571 6.951037 -0.033%3 & 4 43.575945 43.54267 -0.076%5 64.684410 64.66795 -0.254%6 & 7 122.01391 121.8567 -0.128%8 193.85388 195.6024 0.901%9 & 10 238.75784 238.6964 -0.026%

Natural Frequency of a Cantilevered Mass


Determine the natural frequencies of a dynamic system consisting of a massless bar element and a mass element at the end.


Element Types • bar

• mass

Units Inch


Physical and Material Properties• Mass = 0.5 lbm

• E = 30E6 psi

• Density = 1.0E-06

• I = 1.5 in 4

Finite Element Modeling • Create 2 nodes on the X axis with coordinates (0,0,0) and (30,0,0).

• Create a bar between nodes with shear area ratio=0.

• Create a mass on one node with mass of 0.5 lbm.

Boundary Conditions

Constraints • −Constraint Set 1: On the wall end (at node 1), constrain all DOF. On the mass end, con-

strain the DOF in Z, RX, and RY.

• Constraint Set 2: On the mass end node, constrain the DOF in Z, Y, and RZ. Use this set as the Master (ASET) DOF set.


Results

Natural Frequency (Hz)

Bench Value 15.9155FEMAP Structural 15.9154Difference 0.00%


and Bacon, Inc., 1978.) p. 72

Normal Modes/Eigenvalue Verifica-tion Using Standard NAFEMS Benchmarks

The purpose of these normal mode dynamics test cases is to verify the function of the FEMAP Structural Normal Modes/Eigenvalue solver using standard benchmarks published by NAFEMS (National Agency for Finite Element Methods and Standards, National Engineer-ing Laboratory, Glasgow, U.K.).

These standard benchmark tests were created by NAFEMS to stretch the limits of the finite elements in commercial software. All results obtained using the FEMAP Structural software compare favorably with other commercial finite element analysis software. Results of these test cases using other commercial finite element analysis software programs are available from NAFEMS.



- units

- material properties



- solution type

• results

• reference

ReferenceThe following reference has been used in these test cases:

• NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.)


Bar Element Test CasesThe normal mode dynamics test cases using the standard NAFEMS benchmarks include these bar element test cases:

• "Pin-ended Cross - In-plane Vibration"

• "Pin-ended Double Cross - In-plane Vibration"

• "Free Square Frame - In-plane Vibration"

• "Cantilever with Off-Center Point Masses"

• "Deep Simply-Supported Beam"

• "Circular Ring - In-plane and Out-of-plane Vibration"

• "Cantilevered Beam"

Pin-ended Cross - In-plane Vibra-tion

The complete model and results for this test case are in file nf001ac.neu.

This test is a normal modes/eigenvalue analysis of a pin–ended cross (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Bench-marks for Natural Frequency Analysis, Test 1.

Attributes of this test are:

• coupling between flexural and extensional behavior

• repeated and close eigenvalues


UnitsSI

Cross Sectional PropertiesKey–in section:

• Area = .015625 m2

Shear ratio:

A

B

C

D

5.0 m

.125 m

.125 m

• Y = 0

• Z = 0

Material Properties


• 16 bar elements; four elements per arm

Boundary Conditions

Constraints• Constrain points A, B, C, D (nodes 2, 3, 4, 5) in all directions except for the Z rotation.

• Constrain node point Z (node 1) in the Z translation and X rotation.

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

η 0.29 Poissons ratio( )=

G 8.01x1010

=

• Constrain all other nodes (6-17) in the Z translation and X and Y rotations.

Solution TypeNormal Modes/Eigenvalue – SVI method

Results

Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and

Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 1.

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural (Hz)

1 11.336 linear 11.336 11.336 2, 3 17.709 linear 17.687 17.687 4 17.709 linear 17.715 17.715 5 45.345 linear 45.477 45.477 6, 7 57.390 linear 57.364 57.364 8 57.390 linear 57.683 57.683

Note: Reference value (Ref. Value) refers to the accepted solution to the problem.

Pin-ended Double Cross - In-plane Vibration

The complete model and results for this test case are in file nf002ac.neu.

This test is a normal modes/eigenvalue analysis of a pin–ended double cross (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 2.





UnitsSI

Cross Sectional PropertiesKey–in section:

• Area = .015625 m2

Shear ratio:

A

B

C

D

E

F

G

H

5.0 m

5.0m

.125 m

.125 m

• Y = 0

• Z = 0

Material Properties



Boundary Conditions

Constraints• Constrain points A, B, C, D, E, F, G, H (nodes 2-9) in all directions except for the Z rota-

tion.

• Constrain all other nodes 1, (10-33) in the Z translation and X and Y rotations.

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

The following figure shows the boundary conditions.


Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural Result (Hz)

1 11.336 linear 11.336 11.336 2, 3 17.709 linear 17.687 17.687 4,5, 6,7,8

17.709 linear 17.715 17.715

9 45.345 linear 45.477 45.477 10, 11 57.390 linear 57.364 57.364 12,13, 14,15, 16

57.390 linear 57.683 57.683


Free Square Frame - In-plane Vibra-tion

The complete model and results for this test are in file nf003ac.neu.

This test is a normal modes/eigenvalue analysis of a free square frame (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 3.



• rigid body modes (3 modes)



UnitsSI

Cross Sectional PropertiesShear ratio:

• Y = 1.0

• Z = 1.0

10.0 m

10.0m

.125 m

.125 m

Material Properties



Boundary Conditions

Constraints• Constraint Set 1: Constrain all nodes in the Z translation and X and Y rotations.

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

• Constraint Set 2 (Kinematic DOF): Constrain nodes 1 and 3 in the X and Y translation and the Z rotation.


Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural (Hz)

4 3.261 linear 3.262 3.259 5 5.668 linear 5.665 5.662 6, 7 11.136 linear 11.145 11.127 8 12.849 linear 12.833 12.793 9 24.570 linear 24.664 24.611 10, 11 28.695 linear 28.813 28.700


Cantilever with Off-Center Point Masses

The complete model and results for this test is in file nf004a.neu.

This test is a normal modes/eigenvalue analysis of a cantilever with off–center point masses (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 4.


• coupling between torsional and flexural behavior

• inertial axis non–coincident with flexibility axis

• discrete mass, rigid links

• close eigenvalues


UnitsSI


• Y = 1.128

• Z = 1.128

Material Properties


• 9 elements

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

five bar elements along cantilever

two mass elements

two rigid elements

Boundary Conditions

Constraints• Fully constrain point A (node 1) in all directions.


Results


Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 4

Mode # Ref. Value (Hz)NAFEMS Target

Value (Hz)


(Hz)

1 1.723 1.723 1.722 2 1.727 1.727 1.726 3 7.413 7.413 7.410 4 9.972 9.972 9.947 5 18.155 18.160 18.051 6 26.957 26.972 26.712


Deep Simply-Supported BeamThe complete model and results for this test are in file nf005ac.neu.

This test is a normal mode dynamic analysis of a deep simply–supported beam. This docu-ment provides the input data and results for NAFEMS Selected Benchmarks for Natural Fre-quency Analysis, Test 5.


• shear deformation and rotary inertial (Timoshenko beam)

• possibility of missing extensional modes when using iteration solution methods

• repeated eigenvalues


UnitsSI

Cross Sectional PropertiesShear Ratio

• Y = 1.176923

• Z = 1.176923

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 5 bar elements

Boundary Conditions

Constraints• Constrain the X, Y, Z translation an X rotation at point A (node 1)

• Constrain the Y and Z translation at point B (node 10)

The boundary conditions are shown in the following diagram.

Solution TypeNormal Modes/Eigenvalues – SVI method

Results

Mode # Ref. Value (Hz)NAFEMS

Target Value (Hz)

FEMAP Structural Result (Hz)

1, 2 42.649 42.568 42.710 3 77.542 77.841 77.841 4 125.00 125.51 125.52 5, 6 148.31 145.46 150.76 7 233.10 241.24 241.24 8, 9 284.55 267.01 301.08

Circular Ring - In-plane and Out-of-plane Vibration

The complete model and results for this test are in file nf006ac.neu.

This test is a normal modes/eigenvalue analysis of a circular ring using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 6.


• rigid body modes (six modes)



UnitsSI


• Y = 1.128205

• Z = 1.128205

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 20 bar elements

Boundary Conditions

Constraints• Constraint Set 1 (Kinematic DOF): Constrain nodes 10 and 11 in all directions and rota-

tions.


Results



Mode # Ref. Value (Hz)NAFEMS

Target Value (Hz)


(Hz)

7, 8(out of plane)

51.849 52.290 52.211

9, 10(in plane)

53.382 53.971 53.775

11, 12(out of plane)

148.77 149.70 148.92

13, 14(in plane)

150.99 152.44 151.25

15(out of plane)

286.98 288.25 285.33

16(in plane)

289.51 288.25 285.33


Cantilevered BeamThe complete model and results for this test case are in the following files:

• nf071a.neu (Test 1)

• nf071b.neu (Test 2)

• nf071c.neu (Test 3)

This test is a normal modes/eigenvalue analysis of a cantilevered beam. This document pro-vides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 71.


• ill–conditioned stiffness matrix


UnitsSI

Material Properties

Finite Element ModelingThree tests - all use 8 bar elements and 9 nodes

• Test 1: a=b

• Test 2: a = 10b

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

• Test3: a = 100b

Boundary ConditionsConstraints

• Fully constrain point A (node 1) in all directions and rotations.

• Constrain all other nodes in the Z translation and X and Y rotations.


Bar elements always use a consistent mass formulation.

Results



Mode #Ref. Value

(Hz)Mesh

FEMAP Structural

Result(Hz)

1 1.010 a = b a = 10b a = 100b

1.0095 1.0095 1.0095

2 6.327 a = b a = 10b a = 100b

6.3223 6.3260 6.3289

3 17.716 a = b a = 10b a = 100b

17.693 17.791 17.819

4 34.717 a = b a = 10b a = 100b

34.675 34.854 35.061

5 57.390 a = b a = 10b a = 100b

57.422 60.595 64.751

6 85.730 a = b a = 10b a = 100b

86.135101.673104.654


Plate Element Test CasesThe normal mode dynamics test cases using the standard NAFEMS benchmarks include these plate element test cases:

• "Thin Square Cantilevered Plate -Symmetric Modes"

• "Thin Square Cantilevered Plate - Anti-symmetric Modes"

• "Free Thin Square Plate"

• "Simply-Supported Thin Square Plate"

• "Simply-Supported Thin Annular Plate"

• "Clamped Thin Rhombic Plate"

• "Cantilevered Thin Square Plate with Distorted Mesh"

• "Simply-Supported Thick Square Plate, Test A"

• "Clamped Thick Rhombic Plate"

• "Simply-Supported Thick Square Plate, Test B"

• "Simply-Supported Thick Annular Plate"

• "Cantilevered Square Membrane"

• "Cantilevered Tapered Membrane"

• "Free Annular Membrane"

• "Cantilevered Thin Square Plate"

• "Cantilevered Thin Square Plate #2"

Thin Square Cantilevered Plate -Symmetric Modes


• nf011alc.neu (linear quadrilateral plate, consistent mass)

• nf011all.neu (linear quadrilateral plate, lumped mass)

• nf011apc.neu (parabolic quadrilateral plate, consistent mass)

• nf011apl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a thin, square, cantilevered plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 11a.


• symmetric modes, symmetric boundary conditions along the cutting plane


UnitsSI

Material Properties

Finite Element ModelingTest 1 and Test 2 (nf011alc and nf011all)

• 45 nodes

• 32 linear quadrilateral plate elements - thickness = 0.05m

Test 2 and Test 3 (nf011apc and nf011apl)

• 37 nodes

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 8 parabolic quadrilateral plate elements - thickness = 0.05m

Mesh only half the plate (10m x 5m).

Boundary Conditions• Constraints (all tests)

• Fully constrain nodes 1-5 in all translations and rotations.

• Constrain nodes 6, 11, 16, 21, 26, 31, 36, 41 in the X and Y translations and X and Z rotations.

• Constrain all other nodes in the X and Y translations and Z rotation.


Results were obtained two different ways:

• using lumped mass

• using consistent mass

Linear Quadrilateral Plates Parabolic Quadrilateral Plates

Results


Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 11a.

Mode #Ref.

Value (Hz)Mesh


(lumped mass) (Hz)

FEMAP Structural Result (consistent

mass) (Hz)

1 0.421 linearparabolic

0.4150.414

0.4180.418


2.5072.444

2.6232.569


3.1173.081

3.3153.281


5.9846.018

6.5736.551


7.2416.954

7.9797.525


10.38710.493

12.11211.950


Thin Square Cantilevered Plate - Anti-symmetric Modes

The complete model and results for this test case are the in following files:

• nf011blc.neu (linear quadrilateral plate, consistent mass)

• nf011bll.neu (linear quadrilateral plate, lumped mass)

• nf011bpc.neu (parabolic quadrilateral plate, consistent mass)

• nf011bpl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a thin, square, cantilevered plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 11b.


• anti–symmetric modes


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf011blc.neu and nf011bll.neu)

• 45 nodes, 32 linear quadrilateral plate elements - thickness = 0.05m

Tests 3 and 4 (nf011bpc.neu and nf011bpl.neu)

• 37 nodes, 8 parabolic quadrilateral plate elements - thickness = 0.05m

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Mesh only half the plate (10m x 5m).

Boundary ConditionsConstraints (all tests)

• Fully constrain nodes 1-5 in all directions.

• Constrain nodes 6, 11, 16, 21, 26, 31, 36, 41 in the X, Y, Z translations and Z rotation.






Results


Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 11b.

Mode # Ref.

Value (Hz)

Mesh NAFEMS

Target Value (Hz)


(lumped mass) (Hz)

FEMAP Structural Result (consistent mass)

(Hz)


1.0191.018

0.9930.999

1.0121.024


3.8393.710

3.5533.541

3.7503.728


8.3137.768

7.1306.847

8.1627.846


9.4248.483

8.0827.894

9.0798.693

5 not available

linearparabolic

11.72811.185

9.8059.954

11.52611.451

6 not available

linearparabolic

17.81815.755

13.08713.724

17.19216.918


Free Thin Square PlateThe complete model and results for this test case are in the following files:

• nf012lc.neu (linear quadrilateral plate, consistent mass)

• nf012ll.neu (linear quadrilateral plate, lumped mass)

• nf012pc.neu (parabolic quadrilateral plate, consistent mass)

• nf012pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a free thin square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Bench-marks for Natural Frequency Analysis, Test 12.


• rigid body modes (three modes)


• use of kinematic DOF for the rigid body mode calculation with the SVI eigensolver


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf012lc.neu and nf012ll.neu)


Tests 3 and 4 (nf012pc.neu and nf012pl.neu)


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Boundary ConditionsConstraints

• Constraint Set 1: Constrain all the nodes in the X and Y translations and Z rotation.

• Constraint Set 2 (Kinematic DOF): Constrain nodes 1 and 3 in all directions and rota-tions.





Results

Reference NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 12.

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target

Value (Hz)

FEMAP Structural Result (lumped

mass) (Hz)


mass) (Hz)

4 1.622 linear parabolic

1.6321.532

1.5701.567

1.6151.619


2.4022.356

2.2462.183

2.3942.364


3.0062.861

2.8152.750

2.9902.930

7, 8 4.233 linear parabolic

4.2514.122

3.9123.879

4.2184.186


7.8597.363

6.9026.586

7.7517.494

10 notavailable

linear parabolic

8.0277.392

6.9036.586

7.8847.494


Simply-Supported Thin Square Plate






This test is a normal modes/eigenvalue analysis of a simply–supported thin square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 13.


• well established



UnitsSI

Material Properties




E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=


Boundary Conditions

Constraints• Constrain all nodes in the X and Y translations and Z rotation.

• Constrain the nodes along edges X = 0 and X = 10m in the Z translation and X rotation.

• Constrain the nodes along edges Y = 0 and Y = 10m in the Z translation and Y rotation.

• Fully constrain the DOF on the four corner nodes (9, 13, 41, 68).





Results



Mode # Ref. Value

(Hz) Mesh

FEMAP Structural Result (lumped mass)

(Hz)


mass) (Hz)

1 2.377 4–noded 8–noded

2.3382.375

2.3992.383

2, 3 5.942 4–noded 8–noded

5.8205.932

6.2066.034

4 9.507 4–noded 8–noded

8.9099.392

9.8739.831

5, 6 11.884 4–noded 8–noded

11.77011.879

13.37512.590

7, 8 15.449 4–noded 8–noded

14.21515.033

16.78616.734


Simply-Supported Thin Annular Plate






This test is a normal modes/eigenvalue analysis of a simply–supported thin annular plate meshed with shell elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 14.


• curved boundary (skewed coordinate system)



UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf014lc and nf014ll):


Tests 3 and 4 (nf014pc and nf014pl)

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=


Boundary Conditions• Constraint Set 1 (All Tests):

Constrain all nodes in in the X and Y translation and Z rotation.

Additionally constrain all nodes around the model’s circumference in the Z transla-tion and X rotation.

• Constraint Set 2 (Kinematic DOF):

Tests 1 and 2: Constrain nodes 258 and 290 in the X and Y translations.


Solution TypeNormal Mode Dynamics - SVI method




Results

Mode #Ref. Value

(Hz)Mesh


mass) (Hz)


mass) (Hz)


1.8591.840

1.8771.873


5.1755.111

5.2495.151


9.6869.672

9.9839.713


14.18813.946

15.41214.924


15.32615.547

16.17615.708


17.59417.380

19.08818.521


Clamped Thin Rhombic PlateThe complete model and results for this test case are in the following files:





This test is a normal modes/eigenvalue analysis of a clamped thin rhombic plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 15.


• distorted elements


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf015lc.neu and nf015ll.neu):


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Tests 3 and 4 (nf015pc.neu and nf015pl.neu):


Boundary Conditions

Constraints• Completely constrain the nodes along all four edges of the part in all directions and rota-

tions.

• Constrain all other nodes in the X and Y translation and Z rotation.

Solution TypeNormal Modes/Eigenvalue - SVI method




1

2

3

4

5

6

Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)


mass) (Hz)


mass) (Hz)

7.938 linear parabolic

8.1427.873

7.8187.902

7.9557.929


13.89112.480

12.83112.851

13.38813.008


20.03617.312

17.80717.952

19.07218.472


20.16518.738

18.55418.964

19.23919.168


27.70427.950

23.66523.879

26.18525.226


32.04625.883

27.69827.910

29.81628.810


Cantilevered Thin Square Plate with Distorted Mesh


• nf016a1.neu (16 parabolic quadrilateral plate, lumped mass)

• nf016a2.neu (16 parabolic quadrilateral plate, consistent mass)

• nf016b1.neu (16 parabolic quadrilateral plate, lumped mass)

• nf016b2.neu (16 parabolic quadrilateral plate, consistent mass)

• nf016c1.neu (4 parabolic quadrilateral plate, lumped mass)

• nf016c2.neu (4 parabolic quadrilateral plate, consistent mass)

• nf016d1.neu (4 parabolic quadrilateral plate, lumped mass)

• nf016d2.neu (4 parabolic quadrilateral plate, consistent mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate meshed with distorted plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 16.


• distorted meshes


UnitsSI

Material Properties

Finite Element ModelingAll tests - parabolic quadrilateral plate elements - thickness = 0.05m

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Four tests:

• Test 1 (nf016a1, nf016a2) - 65 nodes, 16 elements

• Test 2 (nf016b1, nf016b2) - 65 nodes, 16 elements with specified nodes at the following XY coordinates:

X Coordinate Y Coordinate

4.0 4.02.25 2.254.75 2.57.25 2.757.5 4.757.75 7.255.25 7.252.25 7.252.5 4.75

• Test 3 (nf016c1, nf016c2) - 21 nodes, 4 elements

• Test 4 (nf016d1, nf016d2) - 21 nodes, 4 elements with a specified node at X=4.0, Y=4.0.

Boundary Conditions

Constraints (nf016a1 and nf016a2)• Constrain the nodes along the model’s Y axis in the X, Y, and Z translations and in the Y

and Z rotations.

• Constrain all other nodes in the Z rotation only.

Constraints (nf016b1 and nf016b2)• Fully constrain the nodes along the model’s Y axis in all directions.

• Constrain all other nodes in the Z rotation only.





Results



Mode #Ref. Value

(Hz)Test

NAFEMS Target Value

(Hz)

FEMAP Structural

Result (lumped mass) (Hz)


mass) (Hz)

1 0.421 1 2 3 4

0.41740.41740.41440.4145

0.41390.41350.40210.3999

0.41810.41820.41890.4192

2 1.029 1 2 3 4

1.0201.0200.9991.002

0.99850.99670.93470.9202

1.0241.0241.0211.025

3 2.582 1 2 3 4

2.5642.5712.5542.565

2.4442.4452.1322.112

2.5692.5662.7082.698

4 3.306 1 2 3 4

3.3023.3173.4013.424

3.0823.0722.7072.697

3.2813.2803.4493.430

5 3.753 1 2 3 4

3.7693.7803.6973.714

3.5403.5353.1363.077

3.7283.7313.9133.881

6 6.555 1 2 3 4

6.8056.8835.4555.133

6.0185.9945.4585.459

6.5516.5527.1086.858


Simply-Supported Thick Square Plate, Test A


• nf021alc.neu (linear quadrilateral plate, consistent mass)

• nf021all.neu (linear quadrilateral plate, lumped mass)

• nf021apc.neu (parabolic quadrilateral plate, consistent mass)

• nf021apl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thick square plate meshed with shell elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 21a.


• well–established


• effect of secondary restraints


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf021alc.neu and nf021all.neu)

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=


Tests 3 and 4 (nf021apc.neu and nf021apl.neu)


Boundary Conditions

Constraints• Fully constrain the corner nodes in all directions and rotations.

• Constrain the nodes along edges X=0 and X=10m in all directions, except the Y rotation.

• Constrain the nodes along edges Y=0 and Y=10m in all directions, except the X rotation.






Results

Mode #Ref. Value

(Hz)Mesh

NAFEMSTarget Value

(Hz)

FEMAP Structural



(Hz)


46.65945.936

45.5046.165

46.3545.830


115.84110.41

108.70110.32

114.12109.38


177.53170.38

160.63167.30

174.29169.75


233.40212.81

204.75204.59

227.05208.20


Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 21a.


283.60269.96

240.84249.26

276.88268.40


371.11344.77

298.18311.32

364.30319.40


371.11344.77

320.41347.63

385.84319.40


Simply-Supported Thick Square Plate, Test B


• nf021blc.neu (linear quadrilateral plate elements, consistent mass)

• nf021bll.neu (linear quadrilateral plate elements, lumped mass)

• nf021bpc.neu (parabolic quadrilateral plate elements, consistent mass)

• nf021bpl.neu (parabolic quadrilateral plate elements, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thick square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 21b.


• well–established


• effect of secondary restraints


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf021blc.neu and nf021bll.neu)


E 200X109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Tests 3 and 4 (nf021plc.neu and nf021pll.neu)


Boundary Conditions

Constraints• Constrain the nodes along all edges in the X,Y, and Z translations and Z rotation.






Results


Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 21b.

Mode #Ref. Value

(Hz) Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural Result (lumped mass)

(Hz)


(Hz)


44.74544.134

44.1444.815

44.9644.493


112.94107.85

106.96108.52

112.25107.57


170.28164.19

156.96163.57

170.17165.70


230.2320.07

203.40203.12

225.51206.46


274.19260.32

237.31245.71

272.47263.61


355.98342.80

293.95307.16

358.43318.56


355.98342.80

319.64346.85

384.78318.58


Clamped Thick Rhombic PlateThe complete model and results for this test case are in the following files:





This test is a normal modes/eigenvalue analysis of a thick clamped thick rhombic plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 22.


• distorted elements


UnitsSI

Material Properties


E 200X109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=




Boundary Conditions

Constraints• Fully constrain the nodes along all four edges in all directions and rotations.


Solution TypeNormal Mode Dynamics - SVI




Results

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural



(consistent mass) (Hz)


137.80133.86

133.33134.51

135.17132.48


218.48203.34

204.42204.30

213.06200.28


295.42271.38

269.23270.17

288.08266.06


296.83283.68

279.75283.95

289.05273.65


383.56346.41

337.92338.90

377.05338.88

6 notavailable

linear parabolic

426.59386.62

381.87381.90

411.28369.79

Simply-Supported Thick Annular Plate






This test is a normal modes/eigenvalue analysis of a simply–supported thick annular plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 23.





UnitsSI

Material Properties


E 200X109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=




Boundary Conditions

Constraints• Constrain the nodes around the circumference in the X, Y, and Z translations and X and Z

rotations.






Results

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural

Result (lumped mass)

(Hz)


(Hz)


18.8218.59

18.4918.32

18.6118.59


49.8249.02

49.8948.99

50.3549.13


96.0692.90

93.4393.19

95.4492.42




148.34140.86

136.71134.27

145.39139.41

7, 8 notavailable

linear parabolic

153.68146.63

145.21146.87

151.28145.37


174.52167.31

163.74160.43

174.10166.11


Cantilevered Square MembraneThe complete model and results for this test case are in the following files:





This test is a normal modes/eigenvalue analysis of a cantilevered square membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 31.

Attributes of this test are well established.


UnitsSI

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=




Boundary Conditions

Constraints• Fully constrain the nodes along the Y axis in all directions and rotations.

• Constrain all other nodes in the Z translation and X, Y, and Z rotations.





Results

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value (Hz)

FEMAP Structural


(Hz)


(Hz)


52.90552.635

52.4752.16

52.7752.39


126.11125.87

125.59125.18

126.06122.48


143.20141.47

139.54138.28

142.83138.02


228.85224.59

214.61209.01

227.04214.95


247.90243.26

239.84239.16

247.25227.48


260.61256.76

252.06251.31

259.46236.73

Cantilevered Tapered MembraneThe complete model and results for this test case are in the following files:





This test is a normal modes/eigenvalue analysis of a cantilevered tapered membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 32.


• shear behavior

• irregular mesh

• symmetry


UnitsSI

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=




Boundary Conditions

Constraints• Fully constrain the nodes along the Y axis in all directions and rotations.

• Constrain all other nodes in the Z translation and the X, Y, and Z rotations.





Results

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


44.90544.636

44.7344.84

44.8245.14


132.12130.14

129.92129.05

131.28130.50


162.83162.72

162.61162.37

162.80161.37


252.99246.63

244.62241.80

250.56245.00


393.31382.02

375.09369.61

391.79374.78


396.26391.55

389.81388.11

393.11375.77

Free Annular MembraneThe complete model and results for this test case are in the following files:





This test is a normal modes/eigenvalue analysis of a free annular membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 33.




• kinematically incomplete suppressions


UnitsSI

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=




Boundary Conditions

Constraints• Constraint Set 1 (DOF set):



• Constraint Set 2: Constrain all other nodes in the Z translation and X, Y, and Z rotations.





Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


129.51126.48

127.71126.66

128.70126.15


225.46224.27

224.52222.82

225.22218.17


234.92232.95

229.67230.12

234.94225.14


272.13264.81

263.86262.45

270.83257.67


340.34335.70

328.44329.09

339.93311.38


391.98378.60

368.15368.48

389.38361.52


Cantilevered Thin Square PlateThe complete model and results for this test case are in the following files:

• nf073ac.neu (Test 1 - parabolic quadrilateral plate, consistent mass)

• nf073al.neu (Test 2 - parabolic quadrilateral plate, lumped mass)

• nf073bc.neu (Test 3 - parabolic quadrilateral plate, consistent mass)

• nf073bl.neu (Test 4 - parabolic quadrilateral plate, lumped mass)

• nf073cc.neu (Test 5 - parabolic quadrilateral plate, consistent mass)

• nf073cl.neu (Test 6 - parabolic quadrilateral plate, lumped mass)

• nf073dc.neu (Test 7 - parabolic quadrilateral plate, consistent mass)

• nf073dl.neu (Test 8 - parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate. This docu-ment provides the input data and results for NAFEMS Selected Benchmarks for Natural Fre-quency Analysis, Test 73.


• effect of master DOF selection on frequencies


UnitsSI

Material Properties

Finite Element Modeling65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

Boundary Conditions

Constraints• Constraint Set 1: Constrain the nodes along the Y axis in the X, Y, and Z translations and

Y rotation.

• Constraint Set (DOF set) 2: Create a constraint set to define a Master (ASET) DOF set (in Z direction) - four different placements:

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Tests 1 and 2:

Tests 3 and 4:

Tests 5 and 6:

Tests 7 and 8:





Results

Mode #Ref. Value

(Hz)DOF Set

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)

1 0.421 test 1 test 2 test 3 test 4

0.4174 0.4174 0.4175 0.4184

0.4139 0.4139 0.4140 0.4147

0.41820.41820.41830.4191


1.020 1.020 1.021 1.032

0.999 1.000 1.001 1.009

1.0251.0261.0271.036


2.564 2.597 2.677 2.850

2.449 2.476 2.524 2.670

2.5802.6102.6752.844




3.302 3.345 3.365 3.571

3.095 3.126 3.140 3.325

3.3143.3523.3623.555


3.769 3.888 4.035 5.466

3.563 3.663 3.765 4.816

3.7813.8914.0235.414


6.805 7.517 7.495 -----

6.126 6.694 6.675 ------

6.7987.4987.479------


Cantilevered Thin Square Plate #2The complete model and results for this test case are in the following files:

• nf074c.neu (parabolic quadrilateral plate, consistent mass)

• nf074l.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate. This docu-ment provides the input data and results for NAFEMS Selected Benchmarks for Natural Fre-quency Analysis, Test 74.


UnitsSI

Material Properties

Finite Element Modeling65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

Boundary Conditions

ConstraintsConstrain the nodes along the Y axis in the X, Y, and Z translations and the Y rotation.

Solution TypeNormal Modes/Eigenvalues - SVI




Results

Mode # Ref. Value (Hz) FEMAP Structural

Result(lumped mass) (Hz)

FEMAP Structural Result(consistent

mass) (Hz)

1 0.471 0.4139 0.4181 2 1.029 0.999 1.024 3 2.582 2.444 2.569 4 3.306 3.082 3.281 5 3.753 3.540 3.728


Axisymmetric Solid and Solid Ele-ment Test Cases

The normal mode dynamics test cases using the standard NAFEMS benchmarks include these axisymmetric solid and solid element test cases:

• "Free Cylinder - Axisymmetric Vibration"

• "Simply-Supported Annular Plate -Axisymmetric Vibration"

• "Thick Hollow Sphere - Uniform Radial Vibration"

• "Simply-Supported Solid Square Plate"

• "Simply-Supported Solid Annular Plate"

• "Deep Simply-Supported Solid Beam"

• "Cantilevered Solid Beam"

Free Cylinder - Axisymmetric Vibra-tion


• nf041lc.neu (linear axisymmetric solid quadrilateral, consistent mass)

• nf041ll.neu (linear axisymmetric solid quadrilateral, lumped mass)

• nf041pc.neu (parabolic axisymmetric solid quadrilateral, consistent mass)

• nf041pl.neu (parabolic axisymmetric solid quadrilateral, lumped mass)

This test is a normal modes/eigenvalue analysis of a free cylinder meshed with axisymmetric elements. This document provides the input data and results for NAFEMS Selected Bench-marks for Natural Frequency Analysis, Test 41.


• rigid body modes (one mode)

• coupling between axial, radial, and circumferential behavior

• close eigenvalues


UnitsSI

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 68 nodes, 48 linear axisymmetric quadrilateral solid elements

Tests 3 and 4 (nf041pc.neu and nf041pl.neu):

• 43 nodes, 8 parabolic axisymmetric quadrilateral solid elements

Boundary Conditions

Constraints• Tests 1 and 2: Create a constraint set (Kinematic DOF set) to constrain nodes 1 and 68 in

the X and Z translations.

• Tests 3 and 4: Create a constraint set (Kinematic DOF set) to constrain nodes 1 and 51 in the X and Z translations.





Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


244.01243.50

243.18243.24

243.96243.50


379.41377.46

370.86356.49

378.15377.46


395.41394.28

379.31356.88

394.42394.30


401.35397.94

385.92 375.85

398.00397.97


421.87406.41

389.56393.65

406.85406.44


Thick Hollow Sphere - Uniform Radial Vibration






This test is a normal modes/eigenvalue analysis of a thick, hollow sphere using axisymmetric solid elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 42.



• constraint equations


UnitsSI

Material Properties


• 22 nodes, 10 linear axisymmetric solid quadrilateral elements - α = 5°


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 53 nodes, 10 parabolic axisymmetric solid quadrilateral elements

Boundary Conditions

Constraints• Constraint Set 1: Constrain all nodes in the Z translation.

• Constraint Equations: Constrain all nodes at the same R’ are constrained to have same r’ displacement





Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


370.64370.01

369.91369.49

370.08369.83


841.20838.08

831.80832.72

839.49837.77


1473.11453.0

1421.31433.7

1470.51450.85


2192.22131.7

2030.52072.9

2188.62117.3


2975.72852.8

2604.22706.3

2970.92799.5


Simply-Supported Annular Plate -Axisymmetric Vibration






This test is a normal modes/eigenvalue analysis of a simply–supported annular plate meshed with axisymmetric elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 43.




UnitsSI

Material Properties


E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 80 nodes, 60 linear axisymmetric solid quadrilateral elements


• 28 nodes, 5 parabolic axisymmetric solid quadrilateral elements

Boundary Conditions

ConstraintsConstrain point A (node 1) in the Z translation





Results

Reference NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 43.

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


18.71118.582

18.54218.429

18.57018.582


145.46145.56

138.66135.97

140.24140.56


224.22224.18

224.20224.00

224.20224.18


385.59374.05

361.50 353.62

371.48374.05


689.34686.04

643.34633.16

673.79686.05


Deep Simply-Supported Solid BeamThe complete model and results for this test case are in the following files:

• nf051lc.neu (linear solid brick, consistent mass)

• nf051ll.neu (linear solid brick, lumped mass)

• nf051pc.neu (parabolic solid brick, consistent mass)

• nf051pl.neu (parabolic solid brick, lumped mass)

This test is a normal mode dynamic analysis of a deep, solid beam meshed with bricks. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 51.


• skewed coordinate system

• skewed restraints


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf051lc.neu, nf051ll.neu)

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 88 nodes, 30 linear solid brick elements

Tests 3 and 4 (nf051pc.neu, nf051pl.neu)

• 68 nodes, 5 parabolic solid brick elements

Boundary Conditions

Constraints, Tests 1 and 2:• Constrain node 7 in the X, Y, and Z translations.

• Constrain node 8 in the X and Z translations.

• Constrain node 87 in the Y and Z translations.

• Constrain node 88 in the Z translation.

• Constrain all other nodes along the plane Y’ in the Y translation.

Constraints, Tests 3 and 4:• Constrain node 10 in the X, Y, and Z translations

• Constrain nodes 12 and 35 in the X and Z translations.

• Constrain node 30 in the Y and Z translations.

• Constrain node 71 in the Z translation.

• Constrain all other nodes along the plane Y’ in the Y translation.





Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value (Hz)

FEMAP Structural

Result(lumped mass) (Hz)


(consistent mass) (Hz)


42.88138.821

37.96437.788

38.28238.269


93.81788.451

83.40787.027

83.97787.659


170.67159.34

152.84150.53

157.63157.49


286.12259.20

251.76243.10

265.02259.00


318.86307.92

288.20281.27

298.43306.02


Simply-Supported Solid Square Plate






This test is a normal modes/eigenvalue analysis of a simply–supported solid square plate meshed with bricks. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 52.




• kinematically incomplete suppressions


UnitsSI

Material Properties

Finite Element ModelingTests 1 and 2 (nf052lc.neu, nf052ll.neu)

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• 324 nodes, 192 linear solid brick elements

Tests 3 and 4 (nf052pc.neu, nf052pl.neu)

• 155 nodes, 16 parabolic solid brick elements

Boundary Conditions

Constraints• Constraint Set 1: Constrain all the nodes along the four edges on the plane ZS = -0.5m in

the Z translation.

• Constraint Set 2 (Kinematic DOF):

Tests 1 and 2: Constrain nodes 36 and 264 in the X, Y, and Z translations.

Tests 3 and 4: Constrain nodes 27 and 219 in the X and Y translation.





Results

Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


51.65444.762

44.11544.502

45.31844.796

5, 6 109.44 linearparabolic

132.73110.52

106.73107.94

113.96110.54


194.37169.08

156.48161.44

173.30169.11


197.18193.93

193.58193.16

196.77193.92

9, 10 206.19 linearparabolic

210.55206.64

200.14185.60

209.56206.65

Simply-Supported Solid Annular Plate






This test is a normal modes/eigenvalue analysis of a solid annular plate using solid elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 53.



• constraint equations


UnitsSI

Material Properties

Finite Element Modeling• 160 nodes, 60 linear solid bricks:

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

α 5°=

• 68 nodes, 5 solid parabolic bricks

Boundary Conditions

Constraints, Tests 1 and 2:• Constrain nodes 76-80 and 156-160 in the Y and Z translations.

• Constrain all other nodes in the Y translation.

α 10°=

• Constraint equations: Constrain nodes at same R and Z are constrained to have same z displacement.

Constraints, Tests 3 and 4:• Constrain nodes 11, 22, 33, 44, 66, 77, 88, and 99 in the Y and Z translations and X, Y,

and Z rotations.

• Constrain all other nodes in the Y translation and X, Y, and Z rotations.

• Constraint equations: Constrain nodes at same R and Z are constrained to have same z displacement

Solution TypeNormal Modes/Eigenvalues - SVI method




Results



Mode #Ref. Value

(Hz)Mesh

NAFEMS Target Value

(Hz)

FEMAP Structural


(Hz)


(Hz)


19.65918.582

18.61218.409

18.64118.629


146.42140.42

140.13134.21

141.78141.44


224.25224.18

224.34223.62

224.48224.33


386.70374.04

369.74345.98

380.74380.03


689.47686.02

668.73616.01

690.09688.59


Cantilevered Solid BeamThe complete model and results for this test case are in the following files:

• nf072ac.neu (conventional numbering, consistent mass)

• nf072al.neu (conventional numbering, lumped mass)

• nf072bc.neu (unconventional numbering, consistent mass)

• nf072bl.neu (unconventional numbering, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered solid beam. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 72.


• highly populated stiffness matrix


UnitsSI

Material Properties

Finite Element ModelingTwo tests - both use solid parabolic brick elements

E 200x109 N

m2

-------=

ρ 8000kg

m3

-------=

ν 0.3=

• Test 1: conventional node numbering

• Test 2: unconventional node numbering

Boundary Conditions

Constraints• Constrain all nodes on the X=0 plane in the X, Y, and Z translations.

• Constrain all nodes on the Y=1m plane in the Y translation.

Solution TypeNormal Modes/Eigenvalue – SVI Method

Results



Mode # MeshNAFEMS

Target Value (Hz)

FEMAP Structural

(lumped mass)(Hz)

FEMAP Structural

(consistent mass)(Hz)

1 Test 1 Test 2

16.007 16.007

15.800 15.800

16.00716.007

2 Test 1 Test 2

87.226 87.226

82.235 82.235

87.22687.226

3 Test 1 Test 2

125.96 125.96

125.03 125.03

125.96125.96

4 Test 1 Test 2

209.56 209.56

189.33 189.33

209.56209.56

5 Test 1 Test 2

351.11 351.11

299.30 299.32

351.11351.11

6 Test 1 Test 2

375.81 375.81

352.39 352.40

375.82375.81


Verification Test Cases from the Societe Francaise des Mech-aniciens

The purpose of these test cases is to verify the function of the FEMAP Structural software using standard benchmarks published by SFM (Societe Francaise des Mecaniciens, Paris, France) in “Guide de validation des progiciels de calcul de structures.”

Included here are:

• test cases on mechanical structures using linear statics analysis and normal modes/eigen-value analysis

• stationary thermal test cases using heat transfer analysis

• a thermo–mechanical test case using linear statics analysis

Results published in “Guide de validation des progiciels de calcul de structures” are compared with those computed using the FEMAP Structural software.



- units

- material properties



- solution type

• results

• reference

ReferenceThe following reference has been used in these test cases:


• Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.)

Mechanical Structures - Linear Stat-ics Analysis with Bar or Rod Ele-ments

The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these bar and rod element test cases:

• "Short Beam on Two Articulated Supports"

• "Clamped Beams Linked by a Rigid Element"

• "Transverse Bending of a Curved Pipe"

• "Plane Bending Load on a Thin Arc"

• "Nodal Load on an Articulated Rod Truss"

• "Articulated Plane Truss"

• "Beam on an Elastic Foundation"

Short Beam on Two Articulated Supports

The complete model and results for this test case are in file ssll02.neu.

This test is a linear statics analysis of a short, straight beam with plane bending and shear loading. It provides the input data and results for benchmark test SSLL02/89 from “Guide de validation des progiciels de calcul de structures.”

• area = 31E-4m2

• inertia = 2810E-8m4

• Shear area ratio = 2.42


UnitsSI

Material Properties

Finite Element Modeling• 10 bar elements

• 11 nodes

The mesh is shown in the following figure:

E 2E11 Pa=

ν 0.3=

Boundary Conditions

Constraints• Constrain the nodes at both free ends of the beam (nodes 1 and 2) in all directions except

for the Z rotation.

Loads• On nodes 1-10, apply a load = 1E5 N/m in -Y direction

The boundary conditions are shown in the following figure:


Results

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de

structures, (Paris, Afnor Technique,1990.) Test No. SSLL02/89.

Total Translation at point B(Node 7)

Bench Value -1.25926E-3FEMAP Structural Value -1.25926E-3Difference 0.00%

Clamped Beams Linked by a Rigid Element


This test is a linear statics analysis of a straight, cantilever beam with plane bending and a rigid element. It provides the input data and results for benchmark test SSLL05/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties• E = 2E11 Pa

• I = (4/3)E-8m4


• 1 rigid element

• 26 nodes


Boundary Conditions

Constraints• Nodes 1 and 4: Fully constrained in all directions.

Loads• Node 3: Set nodal force = 1000 N in -Y direction



Results



Node#

DisplacementReaction Force

BenchValue

FEMAP Structural

Difference

Node 6 Displacement Y(T2 Translation)

-0.125 -0.125 0.00%

Node 3 Displacement Y(T2 Translation)

-0.125 -0.125 0.00%

Node 1 Force Y (N)(T2 Constraint Force)

500 500 0.00%

Node 1 Moment Rz (Nm)(R3 Constraint Moment)

500 500 0.00%

Node 4 Force Y (N)(T2 Constraint Force)

500 500 0.00%

Node 4 Rz moment (Nm) (R3 Con-straint Moment)

500 500 0.00%

Transverse Bending of a Curved Pipe

The complete model and results for this test case are the following files:

• ssll07a.neu (linear beam)

• ssll07b.neu (curved beam)

This test is a linear statics analysis (three–dimensional problem) of a curved pipe with trans-verse bending and bending–torque loading. It provides the input data and results for bench-mark test SSLL07/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingTest 1 (ssll07a)

• 90 bar elements

• 91 nodes

Test 2 (ssll07b)

• 90 curved beam elements

• 91 nodes

E 2E11 Pa=

ν 0.3=

The mesh for Test 1 is shown in the following figure:

Boundary Conditions

Constraints• Fully constrain node 91 in all translations and rotations.

Loads• Create a nodal force at node 1 = 100 N in Z direction



Results

Mf = bending moment

Mt = torsional moment

*See “Post Processing” below

Post Processing

Bar Element (ssll07a)List beam forces on element 167, second end

• Mf=Bar End BX2 Moment

• Mt=Bar End BX1 Moment

Curved Beam Element (ssll07b)List beam forces on element 166, second end

• Mf=Bar End BX2 Moment

• Mt=Bar End BX1 Moment



Node # PointDisplacement

MomentBenchValue

TestNumber

FEMAP Structural

Difference

Node 1 Displacement Z(T3 Translation)

0.13462 1 0.13465 0.02%

Node 1 Displacement Z(T3 Translation)

2 0.13464 0.01%

θ=15° Mt (Nm)* 74.1180 1 76.6709 3.44% Mt (Nm)* 2 75.8109 1.02% Mf (Nm) -96.5925 1 -96.3680 0.23% Mf (Nm) 2 -95.2869 1.35%

Plane Bending Load on a Thin ArcThe complete model and results for this test case are in file ssll08.neu.

This test is a linear statics analysis (plane problem) of a thin arc with plane bending. It pro-vides the input data and results for benchmark test SSLL08/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 10 bar elements


Boundary Conditions

Constraints• Node 2: Constrain the X, Y, and Z translations.

• Node 1: Constrain the Y and Z translation only.

E 2E11 Pa=

ν 0.3=

• Nodes 3-11: Constrain in the Z translation only.

Loads• Force=100N in -Y direction



Results



Node # DisplacementBenchValue

FEMAP Structural

Difference

Node 2 Rz (rad)(R3 Rotation)

-3.0774E-2 -3.1097E-2 1.05%

Node 1 Rz (rad)(R3 Rotation)

3.0774E-2 3.1097E-2 1.05%

Node 7 Y (m)(T2 Translation)

-1.9206E-2 -1.9342E-2 0.71%

Node 1 X (m)(T1 Translation)

5.3913E-2 5.3735E-2 0.33%

Nodal Load on an Articulated Rod Truss


This test is a linear statics analysis of a plane truss with an articulated rod. It provides the input data and results for benchmark test SSLL11/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties• E = 1.962E11 Pa


• 4 rod elements


Boundary Conditions

Constraints• Nodes 3 and 17: Constrained in the X, Y, and Z translations only.

• Nodes 2 and 18: Constrained in the Z translation only.

Loads• Node 2: Set Nodal force = 9.81E3 N in -Y direction



Results



Node # DisplacementBenchValue

FEMAP Structural

Difference


0.26517E-3 0.26517E-3 0.00%


0.08839E-3 0.08839E-3 0.00%


3.47902E-3 3.47903E-3 ~0.00%


-5.60084E-3 -5.6004E-3 ~0.00%

Articulated Plane TrussThe complete model and results for this test case are in the following files:

• ssll14a.neu (4 bar elements)

• ssll14b.neu (10 bar elements)

This test is a linear statics analysis of a straight cantilever beam with plane bending and ten-sion–compression. It provides the input data and results for benchmark test SSLL14/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI


Finite Element ModelingTest 1 (ssll14a)

• 4 bar elements

• 5 nodes

Test 2 (ssll14b)

• 10 linear beam elements

• 11 nodes

The mesh for Test 1 is shown in the following figure:

Boundary Conditions

Test 1 (ssll14a)• Constraints

Nodes 1 and 4: Constrain in the X, Y, and Z translations.

Nodes 2, 3, 8: Constrain in the Z translation only.

• Loads

Set forces and moments to the following numeric values:

p = -3,000N/m (on element 4); F1 = -20,000N (on node 8); F2 = -10,000N (on node 2); M = -100,000Nm (on node 2)

Test 2 (ssll14b)• Constraints

Nodes 1 and 4: Constrain in the X, Y, and Z translations.

Nodes 2, 3, 5-13: Constrain in the Z translation only.

• Loads (ssll14b)

Set forces and moments to the following numeric values:

p = -3,000N/m (on elements 5-7); F1 = -20,000N (on node 8); F2 = -10,000N (on node 2); M = -100,000Nm (on node 2)



Results



Node #Displacement Reaction

ForceBenchValue

TestNumber

FEMAP Structural

Difference

1 V vertical (Y)reaction (N) (T2 Constraint Force)

31500.0 12

33233.133233.1

5.50%5.50%

1 horizontal (x) reaction (N) (T1 Constraint Force)

20239.4 12

20609.220609.3

1.82%1.83%

8 Y (m) (T2 Translation) -0.03072 12

-0.03106-0.03161

1.10%2.90%

Note: The software takes shear effect into account.

Beam on an Elastic FoundationThe complete model and results for this test case are in file ssll16.neu.

This test is a linear statics analysis (plane problem) of a straight beam with plane bending and an elastic support. It provides the input data and results for benchmark test SSLL16/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI


• K = 8.4E5 N/m2

• Each spring stiffness is set to: K*L/ (number of DOF spring elements).


• 49 DOF spring elements

• 51 nodes


Boundary Conditions

Constraints• Nodes 1 and 51: Constrain in the X, Y, and Z translations.

• Nodes 2-49: Constrain in the Z translation and X and Y rotations only.

Loads• Set forces, moments, and distributed loads on element to the following numeric values:

F = -10000 N (node 26) ; p = -5000 N/m (elements 1-50) ; M = 15000 Nm (node 51); M= -15000 Nm (node 1).

The distributed loads are shown below:

The forces and moments are shown below:


Results

*On element 26, second end



NodeDisplacement

Force, MomentBenchValue

FEMAP Structural Difference

51 rotation(rad) Rz(R3 rotation)

-0.003045 -0.003041 0.36%

reaction force (N) Y(T2 Constraint Force)

11674 11646 0.78%

26 disp. Y (m) (T2 Translation) -0.423326E-2 -0.42270E-2 0.41%26 M moment (Nm)* (Bar End

BX3 Moment)33840 33286 1.63%

Mechanical Structures - Linear Stat-ics Analysis with Plate Elements

The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these plate element test cases:

• "Plane Shear and Bending Load on a Plate"

• "Infinite Plate with a Circular Hole"

• "Uniformly Distributed Load on a Circular Plate"

• "Torque Loading on a Square Tube"

• "Cylindrical Shell with Internal Pressure"

• "Uniform Axial Load on a Thin Wall Cylinder"

• "Hydrostatic Pressure on a Thin Wall Cylinder"

• "Gravity Loading on a Thin Wall Cylinder"

• "Pinched Cylindrical Shell"

• "Spherical Shell with a Hole"

• "Uniformly Distributed Load on a Simply-Supported Rectangular Plate"

• "Shear Loading on a Plate"

• "Uniformly Distributed Load on a Simply-Supported Rhomboid Plate"

Plane Shear and Bending Load on a Plate

The complete model and results for this test case are in file sslp01.neu.

This test is a linear statics analysis (plane problem) of a plate with plane bending. It provides the input data and results for benchmark test SSLP01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 100 linear quadrilateral plate elements

• 126 nodes


Boundary Conditions

Constraints• Nodes 121-126: Fully constrain in all translations and rotations.

E 3E10 Pa=

ν 0.25=

Loads• Set a shear force with parabolic distribution on width and constant distribution on thick-

ness

• Resultant force: p = 40 N.



Results

The displacements are shown in the following figure:


structures, (Paris, Afnor Technique,1990.) Test No. SSLP01/89.

Node #Point

CoordinatesCenterline

DisplacementBenchValue

FEMAP Structural

Difference

3 (L,y) Y (mm)(T2 Translation)

0.3413 0.3408 0.15%

Infinite Plate with a Circular HoleThe complete model and results for this test case are in file sslp02.neu.

This test is a linear statics analysis (plane problem) of a plate with tension–compression and a membrane effect. It provides the input data and results for benchmark test SSLP02/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingMapped meshing (with biasing)


• 121 nodes


E 3E10 Pa=

ν 0.25=

Boundary Conditions

Constraints• Nodes 1-11: Constrain in Y translation and X and Z rotations only.

• Nodes 12-110: Constrain in Z translation only.

• Nodes 111-121: Constrain in X translation and Y and Z rotations only.

Loads• Tension force P = 2.5 N/mm**2 (in plane force of 2500 N/m)



Results


structures, (Paris, Afnor Technique,1990.) Test No. SSLP02/89.

PointCoordinates

Node#

StressBenchValue

FEMAP Structural

Difference

(a,0) 1

Plate Top Y Normal Stress

7.5 7.52 0.26%

56 (N/mm**2)Plate Top Y Normal Stress

2.5 2.61 4.40%

111 Plate Top Y Normal Stress -2.5 -2.38 4.80%

σθ

aπ4---,

aπ2---,

Uniformly Distributed Load on a Cir-cular Plate


• ssls03a.neu (linear quadrilateral)

• ssls03b.neu (linear triangle)

This test is a linear statics analysis (three–dimensional problem) of a circular plate fixed at the edge with transverse bending and a uniform load. It provides the input data and results for benchmark test SSLS03/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingTest 1 (ssl03a) - Free meshing:


• 50 nodes

E 2.111×10 Pa=

ν 0.3=

Test 2 (ssl03a) - Free meshing:

• 53 linear triangular plate elements

• 38 nodes

Only 1/4 of the plate is meshed.

Boundary Conditions

Constraints• Constrain node 1 in all directions except for the Z translation.

• Fully constrain nodes 2-3 and nodes 15-21 in all directions.

• Constrain nodes 4-8 in the X translation and Y and Z rotations.

• Constrain nodes 9-13 in the Y translation and X and Z rotations.

Loads• Uniform elemental pressure p = -1000 Pa.

Note: Symmetric conditions are applied to the sides.

Test 1 boundary conditions:


Results


structures, (Paris, Afnor Technique,1990.) Test No. SSLS03/89.

Node # PointT3 Translation(Displacement

Z)

BenchValue

TestNumber

FEMAP Structural

Difference

Node 1 Center O w (m) -0.0065 1 -0.0065 0.00%Node 1 Center O -0.0065 2 -0.0065 0.00%

Torque Loading on a Square TubeThe complete model and results for this test case are in file ssls05.neu.

This test is a linear statics analysis (three–dimensional problem) of a thin–walled tube loaded in torsion by pure shear at the free end. It provides the input data and results for benchmark test SSLS05/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingMapped meshing


• 176 nodes


E 2.111×10 Pa=

ν 0.3=

Boundary Conditions

Constraints• Completely constrain nodes 1-5, 57-60, 112-115, and 167-169 in all translations and

rotations.

Loads• Torque equal to 10 Nm on the free end.



Results

Note: This translates into an equivalent nodal force of ±12.5N.

Node #Displacement

and StressBenchValue

FEMAP Structural

Difference

193 T2 Translation (m) -0.617E-7 -0.617E-7 0.00%193 R1 Rotation (rad) 0.123E-4 0.123E-4 0.00%193 Plate Bottom Minor Stress

(Pa)-0.11E6 -0.11E6 0.00%

208 T2 Translation (m) -0.987E-7 -0.988E-7 0.10%208 R1 Rotation (rad) 0.197E-4 0.197E-4 0.00%208 Plate Bottom Minor Stress

(Pa)-0.11E6 -0.11E6 0.00%

Cylindrical Shell with Internal Pres-sure

The complete model and results for this test are in the following files:

• ssls06a.neu (linear quadrilateral, test 1)

• ssls06b.neu (linear quadrilateral, test 2)

This test is a linear statics analysis of the thin cylinder loaded by internal pressure. It provides the input data and results for benchmark test SSLS06/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling

Test 1 (ssls06a) - Mapped meshing• 100 linear quadrilateral plate elements

• 121 nodes

E 2.111×10 Pa=

ν 0.3=

Test 2 (ssls06b) - Mapped meshing• 400 linear quadrilateral plate elements

• 441 nodes

Boundary Conditions

Constraints for Test 1 (ssls06a)• Constrain node 1 in all directions except for the Y translation.

• Constrain nodes 2-10 in the Z translation and X and Y rotations.

• Constrain node 11 in all directions except for the X translation.

• Constrain nodes 12, 23, 34, 45, 56, 67, 78, 89, 100, and 111 in the X translation and Y and Z rotations only.

• Constrain nodes 22, 33, 44, 55, 66, 77, 88, 99, 110, 121 in the Y translation and X and Z rotations.

Constraints for Test 2 (ssls06b)• Constrain node 1 in all directions except for the Y translation.


• Constrain node 21 in all directions except for the X translation.

• Constrain nodes 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 232, 253, 274, 295, 316, 337, 358, 379, 400, and 421 in the X translation and Y and Z rotations only.

• Constrain nodes 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441 in the Y translation and X and Z rotations only.

Loads for Test 1 and Test 2• Internal pressure on the elements = 10000 Pa.



Results

Node #Displacement


TestNumber

FEMAP Structural

Difference

11


0.0 1 1.32

21


2 -0.139

111

Plate Top X Normal Stress

5.00E5 1 4.98E5 0.40%

421 σ22(Pa)


2 4.99E5 0.20%

σ11 Pa )( )

σ11 Pa )( )

σ22 Pa )( )

σ22 Pa )( )

All results are averages.



121

T1 Translation

2.38E-6 1 2.37E-6 0.42%

441

T1 Translation

2 2.38E-6 0.00%

121

T3 Translation

-1.43E-6 1 -1.42E-6 0.70%

441

T3 Translation

2 -1.43E-6 0.00%

∆R m( )

∆R m( )

∆L m( )

∆L m( )

Uniform Axial Load on a Thin Wall Cylinder

The complete model and results for this test are in the following files:

• ssls07a.neu (parabolic quadrilateral plate, test 1)

• ssls07b.neu (parabolic triangle plate, test 2)

This test is a linear static analysis of a thin cylinder loaded axially. It provides the input data and results for benchmark test SSLS07/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 • Meshed by revolving a meshed beam

• 200 parabolic quadrilateral plate elements

E 2.111×10 Pa=

ν 0.3=

• 661 nodes

Test 2 • Meshed by free meshing on 1/8 of a cylinder

• 400 parabolic triangular plate elements

Boundary Conditions

Constraints• Constrain the nodes along one long edge in the Y translation and X and Z rotations.

• Constrain the nodes along the other long edge in the X translation and the Y and Z rota-tions.

• Constrain the nodes along the top short edge in the Z translation only.

• Constrain node 1 in the Y and Z translations and the X and Z rotations.

• Constrain node 21 in the X and Z translations and Y and Z rotations.

Loads• Uniform axial elemental pressures, q = 10000 N/m



Results

Node #Displacement


TestNumber

FEMAP Structural

Difference

641


5.00E5 1 5.00E5 0.00%

641


5.00E5 2 5.00E5 0.00%

σ11 Pa( )

σ11 Pa( )

All results are averages.



641


0.0 1 0.0

641


0.0 2 0.0

641

T1 Translation

-7.14E-7 1 -7.14E-7 0.0%

641

T1 Translation

-7.14E-7 2 -7.14E-7 0.0%

641

T3 Translation

9.52E-6 1 9.52E-6 0.0%

641

T3 Translation

9.52E-6 2 9.52E-6 0.0%

σ22 Pa( )

σ22 Pa( )

∆R m( )

∆R m( )

∆L m( )

∆L m( )

Hydrostatic Pressure on a Thin Wall Cylinder

The complete model and results for this test case are in file ssls08.neu.

This test is a linear statics analysis of a thin cylinder loaded by hydrostatic pressure. It pro-vides the input data and results for benchmark test SSLS08/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 200 parabolic quadrilateral plate elements

• 661 nodes

Cylinder is meshed by revolving a meshed beam.


E 2.111×10 Pa=

ν 0.3=

Boundary Conditions

Constraints• Constrain the nodes on side A (from node 21 to node 661) in the X translation and Y and

Z rotations.

• Constrain the nodes on side B (from node 1 to node 641) in the Y translation,and X and Z rotation.

Loads• Internal elemental pressures, p = p0*Z/L with p0=20000 Pa



Results

ψ represents the rotation of a generator



Node PointDisplacement


FEMAP Structural

Difference

Node 321 Any


0.0 -0.0054E5

Node 321 x=L/2


5.0E5 4.98E5 0.40%

Node 321 x=L/2

T1 Translation

2.38E-6 2.38E-6 0.00%

Node 1 x=L

T3 Translation

-2.86E-6 1.486E-6 0.00%

Node 321

R2 Rotation

1.19E-6 1.19E-6 0.00%

σ11 Pa( )

σ22 Pa( )

∆R m( )

∆L m( )

ψ rad( )

Gravity Loading on a Thin Wall Cyl-inder

The complete model and results for this test case are in file ssls09.neu.

This test is a linear statics analysis of a thin cylinder loaded by its own weight. It provides the input data and results for benchmark test SSLS09/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 65 linear quadrilateral plate elements (mapped meshing)

• 84 nodes

E 2.111×10 Pa=

ν 0.3=

γ 7.8511×10 Pa=

mass 8002kg

m3

-------=


Boundary Conditions

Constraints• Nodes 1, 5-16: Constrain in the Y translation and X and Z rotations.

• Node 2: Constrain in all directions except for the X translation and Y rotation.

• Nodes 3, 21-32: Constrain the X translation and Y and Z rotations.

• Node 4: Constrain in the X and Z translations and the Y and Z rotations.

• Nodes 33-36: Constrain in the Z translation only.

Loads• Body load: Translational acceleration in the Z direction



Results

Node # PointDisplacement



Node 2 x=0


3.14E5 3.02E5 3.82%

Node 1 Any


0.0 -1578 to 1578

σ11 Pa( )

σ22 Pa( )



Node 2 x=0

T1 Translation

-4.49E-7 -4.39E-7 2.00%

Node 1 x=L

T3 Translation

2.99E-6 2.99E-6 0.00%

Node 10 x-L

R2 Rotation

-1.12E-7 -1.12E-7 0.00

∆R m( )

z∆ m( )

ψ rad( )

Pinched Cylindrical ShellThe complete model and results for this test case are in the following files:

• ssls20a.neu (linear triangle plate)

• ssls20b.neu (linear quadrilateral plate)

This test is a linear statics analysis of a cylindrical shell with nodal forces, F, pinching as shown. It provides the input data and results for benchmark test SSLS20/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingTest 1 (ssls20a) - Free meshing

• 296 linear triangle plate elements

• 173 nodes

E 10.5x106Pa=

ν 0.315=

Test 2 (ssls20b) - Mapped meshing


• 165 nodes

Boundary Conditions

Constraints• Free conditions. To set free boundary conditions, use symmetry about XY, XZ and YZ

planes.

• Node 1: Fully constrain except for the X translation.

• Node 2, 5-13: Constrain in the Y translation and the X and Z rotations.

• Node 3: Fully constrain except for the Y translation.

• Node 4, 27-35: Constrain in the X translation and the Y and Z rotations.

• Nodes 14-26: Constrain the Z translation and the X and Y rotations.

Loads• Nodal forces Fy = -25 N at point D



Results



Point DisplacementBenchValue

TestNumber

FEMAP Structural

Difference

D Displacement Y (Node 3) (T2 Translation)

-113.9E-3 1 -114.4E-3 0.44%

D Displacement Y (Node 3) (T2 Translation)

-113.9E-3 2 -113.3E-3 0.53%

Spherical Shell with a HoleThe complete model and results for this test case are in the following files:

• ssls21a.neu (Test 1, linear quadrilateral plate)

• ssls21b.neu (Test 2, linear triangular plate)

• ssls21c.neu (Test 3, parabolic quadrilateral plate)

This test is a linear statics analysis of a spherical shell with a hole with nodal forces. It pro-vides the input data and results for benchmark test SSLS21/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (ssls21a)• 100 linear quadrilateral plate elements

• 121 nodes

E 6.285x107Pa=

ν 0.3=

Test 2 (ssls21b) • 200 linear triangular plate elements

• 121 nodes

Test 3 (ssls21c)• 100 parabolic quadrilateral plate elements

• 341 nodes

All tests are executed with mapped meshing.

Boundary Conditions

Constraints • Constrain nodes 1-11 in the X translation and Y and Z rotations.


• Free condition

s

Loads• Nodal forces F = 2 Newtons

Due to the symmetric boundary conditions, only half of the load is applied.



Results



Note: To set free boundary conditions, use symmetry about XY and YZ planes.

PointT1 Translation

u (m) BenchValue

TestNumber

FEMAP Structural

Difference

A(R,0,0) node 111 94.0E-3 1 103.3E-3 9.91%node 111 94.0E-3 2 103.7E-3 10.32%node 421 94.0E-3 3 98.6E-3 4.89%

Uniformly Distributed Load on a Simply-Supported Rectangular Plate


• ssls24a.neu (Test 1, coarse mesh)

• ssls24b.neu (Test 2, fine mesh)

• ssls24c.neu (Test 3, very fine mesh)

This test is a linear statics analysis of a plate with pressure loading and simple supports. It pro-vides the input data and results for benchmark test SSLS24/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (ssls24a): length/thickness=1• 100 linear quadrilateral plate elements

E 1.0x107Pa=

ν 0.3=

• 121 nodes

Test 2 (ssls24b): length/thickness=2• 200 linear quadrilateral plate elements

• 231 nodes

Test 3 (ssls24c): length/thickness=5• 500 linear quadrilateral plate elements

• 561 nodes

Boundary Conditions


• Constrain the nodes on all edges in the Z translation only.

Loads• Set pressure = 1 N/m**2 in the -Z direction



Results

Center NodeLength/Thickne

ssParameter

BenchValue

TestFEMAP

StructuralDifference

61z direction(T3 Translation)

1.0 0.00444 1 0.00453 2.03%


2.0 0.01110 2 0.01110 0.0%


5.0 0.1417 3 0.01402 1.06%

α

α

α

Where:

q= distributed load

b = dimension

t = thickness

E = elastic modules

β values of reference from the “Guide de Validation” are incorrect. The correct values are extracted from “Formulas for Stress and Strain (Roark/Young)”.

Note that the plate top surface corresponds to the side of the plate with negative global z coordinates.



61x component top surface (Plate Top X Normal Stress)

1.0 2874 1 2905 1.00%

116x component top surface (Plate Top Y Normal Stress)

2.0 6102 2 6065 0.61%

281x component top surface (Plate Top Y Normal Stress)

5.0 7476 3 7332 1.93%

β

β

α

Max σ σbβqb

2

t2

------------= =

Max yαqb

4–

Et3

----------------=

Uniformly Distributed Load on a Simply-Supported Rhomboid Plate


• ssls25a.neu (Test 1)

• ssls25b.neu (Test 2)

This test is a linear statics analysis (three–dimensional problem) of a plate with pressure and transverse bending. It provides the input data and results for a test similar to benchmark test SSLS25/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• Length/thickness=2

• linear quadrilateral plate elements

Test 1 (ssls25a)

E 36.0x106Pa=

ν 0.3=

θ 30°=

Test 2 (ssls25b)

Boundary Conditions


• Constrain the nodes along the edges of the mesh in the Z translation.

θ 45°=

Loads• Elemental pressure = 1 N/m**2 in the -Z direction


Results

Test Case ParametersBench Center location

ValueFEMAP Structural Difference

Test 1ssls25a

Z displacement

-3.277x10E-3m

Z displacement (T3 Trans-lation) at node 116-3.137x10E-3m

4.27%

ssls25a Y stress

-5.70x10E3N/m2

Y stress (Plate Top Y Nor-mal Stress) at node 116-5.761x10E3N/m2

1.07%

Test 2ssls25b

Z displacement

-3.0x10E-3m

Z displacement (T3 Trans-lation) at node 116-2.894x10E-3m

3.53%

ssls25b Y stress

-5.39x10E3N/m2

Y stress (Plate Top Y Nor-mal Stress) at node 116-5.349x10E3N/m2

0.76%

α 0.118=

θ 30°=

β 0.570=

α 0.108=

θ 45°=

β 0.539=

Where:

q= distributed load

b = dimension

t = thickness

E = elastic modules

Values of reference from the “Guide de validation” are incorrect. The correct values are extracted from “Formulas for Stress and Strain (Roark/Young),” table 26, case number 14a.



Max σ =βqb2

Max yαqb

4

Et3

-------------=

Shear Loading on a PlateThe complete model and results for this test case are in the following files:

• ssls27a.neu (Test 1)

• ssls27b.neu (Test 2)

• ssls27c.neu (Test 3)

This test is a linear statics analysis of a thin plate with torque and shear loading. It provides the input data and results for benchmark test SSLS27/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (ssls27a) - Mindlin (element formulation)• 6 linear quadrilateral plate elements

• 14 nodes

Test 2 (ssls27b) - Kirchhoff (element formulation)• 6 linear quadrilateral plate elements

E 1.0x107Pa=

ν 0.25=

• 14 nodes

Test 3 (ssls27c) - Mindlin (element formulation)• 48 linear quadrilateral plate elements

• 75 nodes

All tests are executed with mapped meshing.

Boundary Conditions

Constraints• Fully constrain the nodes on side AD in all translations and rotations.

Loads• Create a nodal force Fz = -1N at point B.

• Create a nodal force -Fz = 1N at point C.



D

A

C

D

Results at Location C



DisplacementNode

(Total T3 Translation)

BenchValue

TestNumber

FEMAP Structural

Difference

14 3.537E-2 1 5.335E-2 50.83%14 3.537E-2 2 3.382E-2 4.38%75 3.537E-2 3 3.750E-2 6.02%

Mechanical Structures - Linear Stat-ics Analysis with Solid Elements

The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these solid element test cases:

• "Solid Cylinder in Pure Tension"

• "Internal Pressure on a Thick-Walled Spherical Container"

• "Internal Pressure on a Thick-Walled Infinite Cylinder"

• "Prismatic Rod in Pure Bending"

• "Thick Plate Clamped at Edges"

Solid Cylinder in Pure TensionThe complete model and results for this test case are in the following files:

• sslv01a.neu (parabolic tetrahedron, free meshing)

• sslv01b.neu (linar brick, mapped meshing)

• sslv01c.neu (linear quadrilateral axisymmetric solid, mapped meshing)

• sslv01d.neu (linear triangular axisymmetric solid, free meshing)

This test is a linear statics analysis of a solid cylinder with tension–compression. It provides the input data and results for benchmark test SSLV01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (sslv01a) - Free meshing• 155 parabolic tetrahedron elements

• 342 nodes

E 2.0x1011

Pa=

ν 0.30=

Test 2 (sslv01b) - Mapped meshing• 192 linear brick elements

• 259 nodes

Test 3 (sslv01c) - Mapped meshing• 48 linear quadrilateral axisymmetric solid elements

• 65 nodes

Test 4 (sslv01d) - Free meshing• 28 linear triangular axisymmetric solid elements

• 24 nodes

Boundary Conditions

Constraints• Uniaxial deformation of the cylinder section

Constraints (sslv01a)• Nodes 1, 17-19: Constrain in the Y and Z translations.

• Nodes 2, 14-16: Constrain in the X and Z translations.

• Node 3: Constrain in the X, Y, and Z translations.

• Nodes 4, 59-63: Constrain in the X and Y translations.

• Nodes 5, 20-22, 33-45, 200-226: Constrain in the Y translation.

• Nodes 6, 23-25, 46-58, 173-199: Constrain in the X translation.

• Nodes 7-13, 64-72: Constrain in the Z translation.

Constraints (sslv01b)• Constrain node 1, 10,19, and 28 in the Y and Z translation.

• Constrain nodes 2-8, 11-17, 20-26, and 29-35 in the Z translation.

• Constrain nodes 9, 18, 27, and 36 in the X and Z translation.

• Constrain node 37 in the X, Y, and Z translations.

• Constrain nodes 54, 63, 72, 81, 99, 108, 117, 126, 144, 153, 162, 171, 189, 198, 207, 216, 234, 243, 252, 261, 279, 288, 297, and 306 in the X translation.

• Constrain nodse 82, 127, 172, 217, and 307 in the X and Y translation.

• Constrain nodes 46, 55, 64, 73, 91, 100, 109, 118, 136, 145, 154, 163, 181, 190, 199, 208, 226, 235, 244, 253, 271, 280, 289, and 298 in the Y translation.

Constraints (sslv01c)• Constrain nodes 13, 26, 39, and 52 in the Z translation.

• Constrain node 65 in the X and Z translations.

Constraints (sslv01d)• Constrain node 1 in the X and Z translation

• Constrain nodes 2, 5, 6, and 7 in the Z translation.

Loads (all tests)• Set uniformly distributed force -F/A on the free end in the Z direction

• Elemental pressure, F/A = 100 MPa

Loads, Tests 1 and 2

Loads, Tests 3 and 4:


Results

Node # DisplacementsBench Value

Test #FEMAP


6 T3 Translation 1.5E-3 1 1.5E-3 0.00%279 T3 Translation 1.5E-3 2 1.5E-3 0.00%1 T3 Translation 1.5E-3 3 1.5E-3 0.00%4 T3 Translation 1.5E-3 4 1.5E-3 0.00%4 T3 Translation 1.5E-3 1 1.5E-3 0.00%307 T3 Translation 1.5E-3 2 1.5E-3 0.00%53 T3 Translation 1.5E-3 3 1.5E-3 0.00%


structures, (Paris, Afnor Technique,1990.) Test No. SSLV01/89/89.

3 T3 Translation 1.5E-3 4 1.5E-3 0.00%37 T3 Translation 1E-3 1 1E-3 0.00%189 T3 Translation 1E-3 2 1E-3 0.00%5 T3 Translation 1E-3 3 1E-3 0.00%25 T3 Translation 1E-3 4 1E-3 0.00%41 T3 Translation 0.5E-3 1 0.5E-3 0.00%99 T3 Translation 0.5E-3 2 0.5E-3 0.00%9 T3 Translation 0.5E-3 3 0.5E-3 0.00%29 T3 Translation 0.5E-3 4 0.5E-3 0.00%6 T2 Translation -0.15E-3 1 -0.15E-3 0.00%279 T2 Translation -0.15E-3 2 -0.15E-3 0.00%1 T1 Translation -0.15E-3 3 -0.15E-3 0.00%4 T1 Translation -0.15E-3 4 -0.15E-3 0.00%37 T1 Translation -0.15E-3 1 -0.15E-3 0.00%189 T1 Translation -0.15E-3 2 -0.15E-3 0.00%5 T2 Translation -0.15E-3 3 -0.15E-3 0.00%25 T1 Translation -0.15E-3 4 -0.15E-3 0.00%41 T1 Translation -0.15E-3 1 -0.15E-3 0.00%99 T2 Translation -0.15E-3 2 -0.15E-3 0.00%9 T1 Translation -0.15E-3 3 -0.15E-3 0.00%29 T1 Translation -0.15E-3 4 -0.15E-3 0.00%

Internal Pressure on a Thick-Walled Spherical Container


• sslv03a.neu (Test 1, linear solids)

• sslv03b.neu (Test 2, parabolic solids)

• sslv03c.neu (Test 3, linear axisymmetric solids)

• sslv03d.neu (Test 4, parabolic axisymmetric solids)

This test is a linear statics analysis of a thick sphere with internal pressure. It provides the input data and results for benchmark test SSLV03/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (sslv03a) - Mapped meshing• 1600 linear brick elements

E 2.0x105Pa=

ν 0.30=

• 1898 nodes

Test 2 (sslv03b) - Mapped meshing• 250 parabolic brick elements

• 1256 nodes

Test 3 (sslv03c) - Mapped meshing• 400 linear quadrilateral axisymmetric solid elements

• 451 nodes

Test 4 (sslv03d) - Mapped meshing• 400 parabolic quadrilateral axisymmetric solid elements

• 1301 nodes

Boundary Conditions

Constraints• The equivalent of the center of the sphere being fixed is modeled via symmetric bound-

ary conditions.

Constraints - Tests 1 and 2:

Constraints - Tests 3 and 4:

Loads• Uniform radial elemental pressure = 100 MPa


Pressure -Tests 1 and 2:

Pressure - Tests 3 and 4:


Results

Results for Point R = 1m

Point Node #Displacement

StressBenchValue

TestNumber

FEMAP Structural

Difference

r=1 m 1

Solid Z Normal Stress

-100 1 -90.07 9.93%

1 Solid Z Normal -100 2 -104.33 4.33%41 Axisym C1 Radial

Stress-100 3 -95.50 4.50%

41 Axisym C1 Radial Stress

-100 4 -94.81 5.19%

1

Solid Y Normal Stress

71.43 1 72.04 0.85%

1


71.43 2 73.70 3.18%

41

Axisym C1 Azi-muth Stress

71.43 3 69.20 3.12%

41

Axisym C1 Azi-muth Stress

71.43 4 69.50 2.70%

1 u (m)T3 Translation

0.4E-3 1 0.40E-3 0.00%

σΠ MPa( )

σθ MPa( )

σθ MPa( )

σθ MPa( )

σθ MPa( )

Results for Point R = 2m


0.4E-3 2 0.40E-3 0.00%


0.4E-3 3 0.41E-3 2.50%


0.4E-3 4 0.40E-3 0.00%

Point Node #Displacement

StressBenchValue

TestNumber

FEMAP Structural

Difference

r=2 m 1826


0 1 -.041 N/A

2221


0 2 -.649 N/A


0 3 -.233 N/A


0 4 -.430 N/A

1826


21.43 1 21.18 1.16%

2221


21.43 2 21.76 1.53%


21.43 3 21.39 0.19%


21.43 4 21.58 0.70%

σΠ MPa( )

σΠ MPa( )

σθ MPa( )

σθ MPa( )

All results are averaged.


structures, (Paris, Afnor Technique,1990.) Test No. SSLV03/89.


1.5E-4 1 1.50E-4 0.00%


1.5E-4 2 1.50E-4 0.00%


1.5E-4 3 1.53E-4 2.00%


1.5E-4 4 1.50E-4 0.00%

Internal Pressure on a Thick-Walled Infinite Cylinder


• sslv04a.neu (solid, linear brick)

• sslv04b.neu (solid, parabolic brick)

• sslv04c.neu (solid, axisymmetric quadrilateral)

• sslv04d.neu (solid, axisymmetric parabolic)

This test is a linear statics analysis of a thick cylinder with internal pressure. It provides the input data and results for benchmark test SSLV04/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingAll tests are executed with mapped meshing.

Test 1 (sslv04a) - Mapped meshing• 400 solid (linear brick) elements

• 902 nodes

Test 2 (sslv04b) - Mapped meshing• 240 solid (parabolic brick) elements

• 1873 nodes

E 2.0x105Pa=

ν 0.30=

FE Model - Tests 1 and 2:

Test 3 (sslv04c) - Mapped meshing• 600 axisymmetric (linear quadrilateral solid) elements

• 656 nodes

Test 4 (sslv04d) - Mapped meshing• 600 axisymmetric (parabolic quadrilateral solid) elements

• 1911 nodes


Boundary Conditions

Constraints (sslv04a)• Nodes 1-41, 452-492: Constrain in the X translation.

• Nodes 411-451, 862-902: Constrain in the Z translation.

Constraints (sslv04b)• Nodes 1-61, 1038-1098, 2075-2135: Constrain in the X translation.

• Nodes 977-1037, 2014-2074, 3051-3111: Constrain in the Z translation.

Constraints (sslv04c)• Nodes 1-41: Constrain in the Z translation.

Constraints (sslv04d)• Nodes 1-81: Constrain in the Z translation.

Loads (all tests)• Unlimited cylinder

• Internal elemental pressure p = 60 MPa

Boundary Conditions - Tests 1 and 2:



ResultsAll results are averaged.

Results for R=0.1m

Test Case

PointDisplacement

StressBenchValue

Node #FEMAP


sslv04a r=0.1 m

Solid X Normal Stress

-60 411 -57.07 4.88%

sslv04b Solid X Normal Stress

-60 977 -60.97 1.62%

sslv04c Axisymm C1 Radial Stress

-60 616 -58.03 3.28%

sslv04d Axisymm C1 Radial Stress

-60 1831 -59.98 0.03%

sslv04a


100 411 99.69 0.31%

sslv04b


100 977 100.98 0.98%

sslv04c Axisymm C1 Azi-muth Stress

100 616 100.77 0.77%

sslv04d Axisymm C1 Azi-muth Stress

100 1831 99.98 0.02%

sslv04a

Solid Max Shear Stress

80 411 79.35 0.81%

σr MPa( )

σθ MPa( )

σθ MPa( )

τmax MPa( )

Results for R=0.2m

sslv04b Solid Max Shear Stress

80 977 80.97 1.21%

sslv04a u (m)T1 Translation

59E-6 411 59E-6 0.00%

sslv04b T1 Translation 59E-6 977 59E-6 0.00%sslv04c T1 Translation 59E-6 616 59E-6 0.00%sslv04d T1 Translation 59E-6 1831 59E-6 0.00%

Test Case

PointDisplacement

StressBenchValue

Node#

FEMAP Structural

Difference

sslv04a r=0.2m

Solid X Normal Stress

0 451 -.006 NA

sslv04b Solid X Normal Stress

0 1037 -.250 NA

sslv04c Axisymm C1 Radial Stress

0 656 -.253 NA

sslv04d Axisymm C1 Radial Stress

0 1911 .002 NA

sslv04a


40 451 39.70 0.75%

sslv04b Solid Z Normal Stress

40 1037 40.25 0.62%

sslv04c Axisymm C1 Axi-muth Stress

40 656 40.61 1.53%

sslv04d Axisymm C1 Axi-muth Stress

40 1911 39.90 0.25%

sslv04a

Solid Max Shear Stress

20 451 20.10 0.50%

σr MPa( )

σθ MPa( )

τmax MPa( )



sslv04b Solid Max Shear Stress

20 1037 20.25 1.25%

sslv04a u (m)T1 Translation

40E-6 451 40E-6 0.00%

sslv04b T1 Translation 40E-6 1037 40E-6 0.00%sslv04c T1 Translation 40E-6 656 39.9E-6 0.25%sslv04d T1 Translation 40E-6 1911 40E-6 0.00%

Prismatic Rod in Pure BendingThe complete model and results for this test case are in the following files:

• sslv08a.neu (Test 1, solid elements, linear tetrahedrons)

• sslv08b.neu (Test 2, solid elements, parabolic tetrahedrons)

• sslv08c.neu (Test 3, solid elements, linear bricks)

• sslv08d.neu (Test 4 solid elements, parabolic bricks)

This test is a linear statics analysis of a solid rod with bending. It provides the input data and results for benchmark test SSLV08/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 (sslv08a) - Free meshing• 198 solid (linear tetrahedron) elements

• 76 nodes

Test 2 (sslv08b) - Free meshing• 198 solid (parabolic tetrahedron) elements

• 409 nodes

E 2.0x105Pa=

ν 0.30=


Test 3 (sslv08c) - Mapped meshing• 48 solid (linear brick) elements

• 117 nodes

Test 4 (sslv08d) - Mapped meshing• 48 solid (parabolic brick) elements

• 381 nodes


Boundary Conditions

Constraints (sslv08a)• Nodes 29, 33: Constrain in the X and Z translations.

• Nodes 30-32, 34, 39, 40: Constrain in the Z translation.


Constraints (sslv08b)• Nodes 127, 131: Constrain in the X and Z translations.

• Nodes 128-130, 132-146, 188-195: Constrain in the Z translation only.


Constraints (sslv08c)• Nodes 1-4, 6-9: Constrain in the Z translation.


Constraints (sslv08d)• Nodes 1-8, 10, 12, 14-21: Constrain in the Z translation.

• Nodes 9, 13: Constrain in the X translation.

• Nodes 11: Constrain in the X, Y, and Z translations.

Loads (all tests)• Set moment Mx equal to (4/3)E+7 N.m




Results

Test#

Node#

Displacement/Stress

BenchValue

FEMAP Structural

Difference

1 5 Solid Z Normal Stress (Pa)

-10E6 -4.268E6 57.00%


-10E6 10.03E6 0.30%


-10E6 10.07E6 0.70%


-10E6 10.01E6 0.10%

1 26 T2 Translation 4E-4 2.964E-4 26.00%2 90 T2 Translation 4E-4 4E-4 0.00%3 77 T2 Translation 4E-4 4E-4 0.00%4 251 T2 Translation 4E-4 4.044E-4 1.10%1 19 T3 Translation 2E-4 1.460E-4 27.00%2 40 T3 Translation 2E-4 2E-4 0.00%3 76 T3 Translation 2E-4 2E-4 0.00%4 249 T3 Translation 2E-4 2.010E-4 0.50%1 5 T1 Translation 0.15E-4 7.449E-6 50.34%2 5 T1 Translation 0.15E-4 0.1514E-4 0.93%3 75 T1 Translation 0.15E-4 0.1480E-4 1.33%4 245 T1 Translation 0.15E-4 0.1511E-4 0.73%



1 8 T1 Translation -0.15E-4 -6.2620E-6 58.20%2 8 T1 Translation -0.15E-4 -0.1509E-4 0.60%3 73 T1 Translation -0.15E-4 -0.1480E-4 1.33%4 241 T1 Translation -0.15E-4 -0.1511E-4 0.73%

Thick Plate Clamped at EdgesThe complete model and results for this test case are in the following files:

• sslv09a10.neu (Test 1, parabolic brick, length/thickness =10)





• sslv09b10.neu (Test 2, linear plate, length/thickness =10)





This test is a linear statics analysis of a square thick plate with pressure and transverse bend-ing. It provides the input data and results for benchmark test SSLV09/89 from “Guide de vali-dation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Test 1 - Mapped meshing• 25 parabolic brick elements

• 228 nodes

E 2.1x1011

Pa=

ν 0.30=

• length/thickness =10, 20, 50, 75, 100

Test 2 - Mapped meshing• 25 linear quadrilateral plate elements

• 36 nodes

• length/thickness =10, 20, 50, 75, 100

Test 2 is done using plate elements with the following thickness values:

• length/thickness =10, t=0.1





Boundary Conditions

Constraints – Test 1• Fully constrain the nodes on edges AB, A’B’, AD, and A’D’ in all translations and rota-

tions.

• Constrain the nodes on edge BC and B’C’ in the X translation and Y and Z rotations.

• Constrain the corner nodes at C and C’ in all translations and rotations except for the Z translation.

• Constrain the nodes on edge DC and D’C’ in the Y translation and X and Z rotations.

Constraints – Test 2• Fully constrain the nodes on edges AB and AD in all translations and rotations.

• Constrain the nodes on edge BC in the X translation and Y and Z rotations.

• Constrain the corner nodes at C in all translations and rotations except for the Z transla-tion.

• Constrain the nodes on edge DC in the Y translation and X and Z rotations.

Loads• Load case 1:

Elemental pressure p = 1E6 Pascals in -Z direction

• Load case 2: Point C

Nodal force F = 2.5E5 N in -Z direction

Boundary conditions for Test 1:

Boundary conditions for Test 2:


Results

Test Case 1 (T3 Translation at location C)

File NameLength/Thick-

nessLoad Case

Node#

AnalyticalReference

FEMFEMAP


sslv09a10 10 Pressure 242 -.6552E-4 -.76231E-4 -.735942E-4 12.32%sslv09a10 10 Force 242 -.29146E-3 -.42995E-3 -.426662E-3 46.38%sslv09a20 20 Pressure 242 -.52416E-3 -.53833E-3 -.523376E-3 0.15%sslv09a20 20 Force 242 -.23317E-2 -.25352E-2 -.242500E-2 4.00%sslv09a50 50 Pressure 242 -.81900E-2 -.80286E-2 -.778247E-2 4.98%sslv09a50 50 Force 242 -.36433E-1 -.35738E-1 -.346276E-1 4.96%sslv09a75 75 Pressure 242 -.27641E-1 -.26861E-1 -.259820E-1 6.00%sslv09a75 75 Force 242 -.12296 -.11837 -.114411 6.95%sslv09a100 100 Pressure 242 -.65520E-1 -.63389E-1 -.612191E-1 6.56%sslv09a100 100 Force 242 -.29146 -.27794 -.268120 8.00%

Test Case 2 (T3 Translation at location C)



Part Name

Length/Thick-

nessLoad Case

Node#

AnalyticalReference

FEMFEMAP


sslv09b10 10 Pressure 1 -.6552E-4 -.78661E-4 -.797294E-4 21.69%sslv09b10 10 Force 1 -.29146E-3 -.41087E-3 -.395973E-3 35.86%sslv09b20 20 Pressure 36 -.52416E-3 -.55574E-3 -.564973E-3 8.69%sslv09b20 20 Force 36 -.23317E-2 -.25946E-2 -.260199E-2 11.59%sslv09b50 50 Pressure 36 -.81900E-2 -.83480E-2 -.849953E-2 3.78%sslv09b50 50 Force 36 -.36433E-1 -.37454E-1 -.381471E-1 4.70%sslv09b75 75 Pressure 36 -.27641E-1 -.28053E-1 -.285676E-1 3.35%sslv09b75 75 Force 36 -.12296 -.12525 -.127845 3.97%sslv09b100

100 Pressure 1 -.65520E-1 -.66390E-1 -.676175E-1 3.20%

sslv09b100

100 Force 1 -.29146 -.29579 -.302292 3.72%

Mechanical Structures - Normal Modes/Eigenvalue Analysis

The normal modes/eigevanlues test cases from the Societe Francaise des Mecaniciens include:

• "Lumped Mass-Spring System"

• "Short Beam on Simple Supports"

• "Axial Loading on a Rod"

• "Thin Circular Ring"

• "Cantilever Beam with a Variable Rectangular Section"

• "Thin Circular Ring Clamped at Two Points"

• "Vibration Modes of a Thin Pipe Elbow"

• "Cantilever Beam with Eccentric Lumped Mass"

• "Thin Square Plate (Clamped or Free)"

• "Simply-Supported Rectangular Plate"

• "Thin Ring Plate Clamped on a Hub"

• "Vane of a Compressor - Clamped-free Thin Shell"

• "Bending of a Symmetric Truss"

• "Hovgaard’s Problem - Pipes with Flexible Elbows"

• "Rectangular Plates"

Lumped Mass-Spring SystemThe complete model and results for this test case are in file sdld02.neu.

This test is a normal modes/eigenvalue analysis of an elastic link with lumped mass. It pro-vides the input data and results for benchmark test SDLD02/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material PropertiesSpring constant

Finite Element Modeling• 8 mass elements

• 9 DOF springs

• 8 nodes


Boundary Conditions

Constraints• Constrain all the nodes (1-8) in all translations and rotations except for the X translation.



ResultsThe mode shapes results are exact. The multiplication coefficient is 0.4642 for mode 1 and -0.4642 for mode 8.

Frequency Results:

Normal Mode

BenchValue(Hz)

FEMAP Structural(Hz)

Difference

1 5.5274 5.5274 0.00%2 10.8868 10.8868 0.00%3 15.9155 15.9155 0.00%4 20.4606 20.4606 0.00%5 24.3840 24.3840 0.00%6 27.5664 27.5664 0.00%7 29.9113 29.9113 0.00%8 31.3474 31.3474 0.00%

Mode Shapes Results:


structures, (Paris, Afnor Technique,1990.) Test No. SDLD02/89, p. 178.

NormalMode

PointBenchValue

FEMAP Structural

1 P1 0.1612 0.34731 P2 0.3030 0.65271 P3 0.4082 0.87941 P4 0.4642 1.00001 P5 0.4642 1.00001 P6 0.4082 0.87941 P7 0.3030 0.65271 P8 0.1612 0.34738 P1 0.1612 -0.34738 P2 -0.3030 0.65278 P3 0.4082 -0.87948 P4 -0.4642 1.00008 P5 0.4642 -1.00008 P6 -0.4082 0.87948 P7 0.3030 -0.65278 P8 -0.1612 0.3473

Short Beam on Simple Supports The complete model and results for this test case are in the following files:

• sdll01a.neu

• sdll01b.neu

This test is a modal analysis of a straight short beam with simple supports both inline and off-set. It provides the input data and results for benchmark test SDLL01/89 from “Guide de vali-dation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Problem 1 (sdll01a)• 10 bar elements

• 11 nodes

E 2x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Problem 2 (sdll01b)• 10 bar elements

• 2 rigid elements (master node 4 to slave node 2; master node 3 to slave node 1)

Boundary Conditions

Constraints• Node 1: Constrain in all directions and rotations, except the Z rotation.

• Node 2: Constrain in all directions and rotations, except for the X translation and Z rota-tion.

• Constrain all other nodes in the Z translation and the X and Y rotations.

Loads• no load case

The boundary conditions for both problems are shown in the following figure:


Results

Problem 1: Frequency Results

Problem 2: Frequency Results


structures, (Paris, Afnor Technique,1990.) Test No. SDLL01/89.

Normal Mode

BenchValue(Hz)


Difference

Bending 1 431.555 431.555 0.03%Tension 1 1265.924 1267.226 0.10%Bending 2 1498.295 1503.171 0.33%Bending 3 2870.661 2904.096 1.16%Tension 2 3797.773 3833.003 0.93%Bending 4 4377.837 4493.912 2.65%

Modenumber

BenchValue(Hz)


Difference

1 392.8 394.3 0.38%2 902.2 922.4 2.24%3 1591.9 1641.0 3.08%4 2629.2 2800.0 6.50%5 3126.2 3291.2 5.28%

Axial Loading on a RodThe complete model and results for this test case are in the following file:

• sdll05a.neu

• sdll05b.neu

This test is a modal analysis of a simply–supported beam with stress stiffening. It provides the input data and results for benchmark test SDLL05/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 11 nodes


E 2x1011

Pa=

ρ 7800kgm3--------=

Boundary Conditions

Problem 1 (sdll05a):• Node 1: Leave the Z rotation free and constrain the node in all other translations and

rotations.

• Node 2 : Leave the X translation and Z rotation free and constrain in all other translations and rotations.

Problem 2 (sdll05b):• Node 1: Leave the Z rotation free and constrain the node in all other translations and

rotations.

• Node 2: Leave the X translation and Z rotation free and constrain the node in all other translations and rotations.

• Load Set 1 (node 2): Define a nodal force = to 1E5N in the -X direction. Ensure that Stress Stiffening is turned on in the analysis set.


Results

Frequency Results:



NormalMode

BenchValue(Hz)

FEMAP Structural

(Hz)Difference

sdll05a Mode 1 28.702 28.672 0.10%sdll05a Mode 3 114.807 114.351 0.40%sdll05b Mode 1 22.434 22.399 0.16%sdll05b Mode 3 109.080 108.61 0.43%

Cantilever Beam with a Variable Rectangular Section

The complete model and results for this test case are in the following file: sdll09a.neu

This test is a modal analysis of a straight cantilever beam with a variable section. It provides the input data and results for benchmark test SDLL09/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 10 beam elements (tapered)

• 11 nodes

b0

b1

β b0b1------=

E 2x1011

Pa=

ρ 7800kgm3--------=


Boundary Conditions• Constrain node 1 in all directions.

• Constrain all other nodes in the Z translation and X and Y rotations only.

• no load case



Results

Frequency Results



NormalMode

BenchValue(Hz)

FEMAP Structural

(Hz)Difference

4 1 54.18 54.13 0.09%2 171.94 171.36 0.34%3 384.40 381.70 0.70%4 697.24 688.89 1.20%5 1112.28 1092.92 1.74%

β

Thin Circular RingThe complete model and results for this test case are in file sdll11.neu.

This test is a modal analysis of a thin curved beam. It provides the input data and results for benchmark test SDLL11/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 36 nodes


E 7.2x1010

Pa=

ν 0.3=

ρ 2700kg

m3

-------=

Boundary Conditions

Constraints• Unconstrained (free) conditions

• Create 1 constraint set (Kinematic DOF set) to fully constrain the 3 nodes shown below (nodes 7, 21, 30).




Results

Frequency Results

NormalMode

BenchValue(Hz)


Difference

Modes 1-6 0 0 0.00%Modes 7, 8 318.36 318.99 0.20%Modes 9, 10 511 508 0.59%Modes 11, 12 900.46 900.19 0.03%Modes 13, 14 1590 1569 1.32%



Modes 15, 16 1726.55 1721.56 0.29%Modes 17, 18 2792.21 2774.91 0.62%Modes 19, 20 3184 3116 2.14%

Thin Circular Ring Clamped at Two Points

The complete model and results for this test case are in file sdll12.neu.

This test is a modal analysis of a thin curved beam. It provides the input data and results for benchmark test SDLL12/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 29 nodes


E 7.2x1010

Pa=

ν 0.3=

ρ 2700kg

m3

-------=

Boundary Conditions• Points A and B (nodes 1 and 2): Fully constrained in all directions

• All other nodes: Constrained the Z translation and X and Y rotations only.

• no load case



Results

Frequency Results

NormalMode

BenchValue(Hz)


Difference

1 235.3 235.9 0.25%2 575.3 575.1 0.03%3 1105.7 1102.7 0.27%4 1405.6 1398.0 0.54%5 1751.1 1740.8 0.59%6 2557.0 2536.6 0.80%7 2801.5 2723.0 2.80%

Vibration Modes of a Thin Pipe Elbow


• sdll014a.neu

• sdll014b.neu

• sdll014c.neu

This test is a modal analysis of a straight cantilever beam, and a thin curved beam. It provides the input data and results for benchmark test SDLL14/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

A

D

C

BL

L

Material Properties


Problem 1 (sdll14a) where L=0 and Problem 2 (sdll14b) where L=0.6: • 18 bar elements

• 19 nodes

Problem 3 (sdll14c) where L=2:• 28 bar elements

• 29 nodes

The FE model is shown below:

E 2.1x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Boundary Conditions

Problem 1 (sdll14a):• Fully constrain points C and D (nodes 1 and 2) in all translations and rotations.

Problem 2 (sdll14b) and Problem 3 (sdll14c):• Fully constrain points C and D (nodes 1 and 4) in all translations and rotations.

• Constrain point B (node 2) in the X and Z translations.

• Constrain point C (node 3) in the Y and Z translations.


Results

Problem 1 (sdll14a) Frequency Results:

Problem 2 (sdll14b) Frequency Results:

Problem 3 (sdll14c) Frequency Results:



LNormalMode

BenchValue(Hz)


Difference

0 1 44.23 44.11 0.27%2 119 119 0.00%3 125 126 0.80%4 227 225 0.88%

LNormalMode

BenchValue(Hz)


Difference

0.6 1 33.4 33.3 0.30%2 94 94 0.00%3 100 99 1.00%4 180 184 2.22%

LNormalMode

BenchValue(Hz)


Difference

2 1 17.9 17.7 1.12%2 24.8 24.4 1.61%3 25.3 24.9 1.58%4 27 26.67 0.01%

Cantilever Beam with Eccentric Lumped Mass


• sdll15a.neu

• sdll15b.neu

This test is a modal analysis of a straight cantilever beam and a mass element. It provides the input data and results for benchmark test SDLL15/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


Problem 1 (sdll15a)• 10 bar elements

• 1 mass element at point B

• 11 nodes

E 2.1x1011

Pa=

ρ 7800kg

m3

-------=

A

B

Problem 2 (sdll15b)• 10 bar elements

• 1 rigid element from point B to point C

• 1 mass element at point C

• 12 nodes

Boundary Conditions

Constraints:• Fully constrain point A (node 1) in all translations and rotations.

Solution TypeNormal Modes/Eigenvalue - SVI

A

B

C

Results

Frequency Results:

Mode Shapes Results:

• wc=T3 translation at point C

• wb= T3 translation at point B

• uc=T1 translation at point C

• vb= T2 translation at point B

yc NormalMode

BenchValue(Hz)

FEMAP Structural

(Hz)Difference

0 1,2 1.65 1.65 0.00%3,4 16.07 15.91 1.00%5,6 50.02 48.75 2.54%7 76.47 76.48 0.01%8 80.47 80.84 0.46%9,10 103.20 98.53 4.53%

1 1 1.636 1.635 0.06%2 1.642 1.640 0.12%3 13.46 13.37 0.67%4 13.59 13.52 0.52%5 28.90 28.68 0.76%6 31.96 31.54 1.31%7 61.61 59.97 2.66%8 63.93 61.82 3.30%

ycNormalMode

ModalDisplacement

BenchValue

FEMAP Structural

Difference

1 1 wc/wb 1.030 1.030 0.00%2 uc/vb 0.148 0.148 0.00%3 uc/vb 2.882 2.845 1.28%4 wc/wb -0.922 -0.956 3.69%

Thin Square Plate (Clamped or Free)


• sdls01a.neu

• sdls01b.neu

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate. It provides the input data and results for benchmark test SDLS01/89 from “Guide de valida-tion des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 121 nodes


E 2.1x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

A D

B C

Boundary Conditions• Problem 1 (sdls01a): Constrain the nodes along side BC in all translations and rotations.

• Problem 2 (sdls01b) : Free plate; Create a constraint set (Kinematic DOF set) to con-strain the three nodes shown below (nodes 1, 11, and 111) in all translations and rota-tions.


Results

Problem 1 (sdls01a) Frequency Results:

Normal ModeBench Value

(Hz)FEMAP Structural

(Hz)Difference

1 8.7266 8.6719 0.63%2 21.3042 21.1474 0.74%3 53.5542 53.9586 0.76%

Problem 2 (sdls01b) Frequency Results:


structures, (Paris, Afnor Technique,1990.) Test No. SDLS01/89.

4 68.2984 68.4467 0.21%5 77.7448 77.7814 0.05%6 136.0471 135.783 0.19%



(Hz)Difference

7 33.7119 32.9104 2.38%8 49.4558 47.4165 4.12%9 61.0513 59.1873 3.05%10,11 87.5160 83.0785 5.07%

Simply-Supported Rectangular Plate

The complete model and results for this test case are in file sdls03.neu.

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate. It provides the input data and results for benchmark test SDLS03/89 from “Guide de valida-tion des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 176 nodes


E 2.1x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Boundary Conditions• Constrain the Z translation of the nodes on all sides of the plate.

• Create a constraint set to define the Master (ASET) DOFs on nodes 47, 55, 119. Con-strain these nodes in all directions except for the Z translation.

• no load case



Results

Frequency Results:

NormalMode

BenchValue(Hz)


Difference

4 35.63 35.21 1.18%5 68.51 67.21 1.90%6 109.62 108.96 0.60%7 123.32 121.13 1.78%8 142.51 138.30 2.95%9 197.32 187.94 4.75%

Thin Ring Plate Clamped on a HubThe complete model and results for this test case are in file sdls04.neu.

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of an annular thin plate. It provides the input data and results for benchmark test SDLS04/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingMapped meshing


• 440 nodes


E 2.1x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Boundary Conditions

Constraints• Fully constrain all the nodes on the inner ring as shown below.



Solution TypeNormal Modes/Eigenvalue – SVI

Results

Frequency Results:



(Hz)Difference

1 79.26 79.41 0.19%2, 3 81.09 81.05 0.05%4, 5 89.63 89.64 0.01%6, 7 112.79 113.45 0.58%8, 9 not available 158.3810, 11 not available 226.0212, 13 not available 317.0414, 15 not available 433.0416, 17 not available 527.5118 518.85 532.19 2.57%



19, 20 528.61 561.91 6.30%21, 22 559.09 576.90 3.18%23 609.70 612.63 0.48%

Vane of a Compressor - Clamped-free Thin Shell


• sdls05a.neu (linear quadrilateral, coarse mesh)

• slds05b.neu (linear quadrilateral, fine mesh)

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a cylindrical thin shell. It provides the input data and results for benchmark test SDLS05/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling - Coarse MeshMapped meshing


• 121 nodes

E 2.0685x1011

Pa=

ν 0.3=

ρ 7857.2kg

m3

-------=

The coarse mesh is shown in the following figure:

Finite Element Modeling - Fine MeshMapped Meshing


• 256 nodes

The fine mesh is shown in the following figure:

Boundary ConditionsFully constrain the nodes on one side as shown in the following figure:


Results

Frequency Results:

Reference Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de struc-tures, (Paris, Afnor Technique,1990.) Test No. SDLS05/89.

Normal ModeBenchValue(Hz)

FEMAP Structuralcoarse mesh

(Hz)

FEMAP Structuralfine mesh

(Hz)

1 85.6 85.6 85.72 134.5 138.2 138.33 259.0 249.8 248.04 351.0 345.9 343.75 395.0 386.5 386.06 531.0 549.8 537.7

Bending of a Symmetric TrussThe complete model and results for this test case are in file sdlx01.neu.

This test is a normal modes/eigenvalue analysis (plane problem) of a straight cantilever beam structure. It provides the input data and results for benchmark test SDLX01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 24 nodes


E 2.1x1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Boundary Conditions

Constraints• Fully constrain nodes 1 and 4 in all translations and rotations.

• Constrain nodes 2-3 and 5-24 in the Z translation and X and Y rotations.


Solution TypeNormal Modes/Eigenvalue – SVI

Results

Frequency Results:

NormalMode

BenchValue(Hz)


Difference

1 8.8 8.8 0.00%2 29.4 29.4 0.00%3 43.8 43.8 0.00%4 56.3 56.3 0.00%5 96.2 96.2 0.00%6 102.6 102.7 0.10%7 147.1 147.4 0.20%8 174.8 175.3 0.29%9 178.8 179.3 0.28%


structures, (Paris, Afnor Technique,1990.) Test No. SDLX01/89.

10 206.0 206.9 0.44%11 266.4 268.1 0.64%12 320.0 322.4 0.75%13 335.0 338.7 1.10%

Hovgaard’s Problem - Pipes with Flexible Elbows

The complete model and results for this test case are in file sdlx02.neu.

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a straight, thin curved cantilever beam. It provides the input data and results for benchmark test SDLX02/89 from “Guide de validation des progiciels de calcul de structures.”


Material Properties

UnitsSI


• 26 nodes


E 1.658x· 1011

Pa=

ν 0.3=

ρ 13404.106kg

m3

-------=

Boundary Conditions• Fully constrain nodes 1 and 6 in all translations and rotations.



Results

Frequency Results:



NormalMode

BenchValue(Hz)


Difference

1 10.18 10.40 2.16%2 19.54 19.87 1.69%3 25.47 25.36 0.43%4 48.09 47.71 0.79%5 52.86 51.80 2.01%6 75.94 82.84 9.09%7 80.11 85.20 6.35%8 122.34 125.53 2.61%9 123.15 127.64 3.65%

Rectangular PlatesThe complete model and results for this test case are in file sdlx03.neu.

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate with rigid body modes. It provides the input data and results for benchmark test SDLX03/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties


• 320 nodes


E 2.1x· 1011

Pa=

ν 0.3=

ρ 7800kg

m3

-------=

Boundary Conditions

Constraints• Constraint Set 1 (Kinematic DOF Set): Fully constrain nodes 2, 69, and 84 in all transla-

tions and rotations.



Results

Frequency Results:



NormalMode

BenchValue(Hz)


Difference

7 584 586 0.34%8 826 824 0.24%9 855 854 0.11%10 911 904 0.76%11 1113 1072 3.68%12 1136 1140 0.35%

Stationary Thermal Tests - Steady State Heat Transfer Analysis

The stationary thermal test cases for steady-state heat transfer analysis from the Societe Francaise des Mecaniciens include:

• "Hollow Cylinder - Fixed Temperatures"

• "Hollow Cylinder - Convection"

• "Cylindrical Rod - Flux Density"

• "Hollow Cylinder with Two Materials - Convection"

• "Wall - Fixed Temperatures"

• "Wall - Convection"

• "Hollow Sphere - Fixed Temperatures, Convection"

• "L-Plate"

• "Hollow Sphere with Two Materials -Convection"

Hollow Cylinder - Fixed Tempera-tures

The complete model and results for this test case are in file htpla01.neu.

This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with fixed tem-peratures. It provides the input data and results for benchmark test TPLA01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingTwo tests:

• Test 1 - 5 linear quadrilateral axisymmetric solid elements

• Test 2 - 5 parabolic quadrilateral axisymmetric solid elements

The meshes are shown in the following figure:

Boundary Conditions• One temperature set:

λ 1Wm-----°C=

Internal temperature

External temperature

Solution TypeSteady–State Heat Transfer

Ti 100°C=

Te 20°C=

Results

Temperature Results (0 degrees Celsius):

Total Heat Flux Results (W/m**2):


structures, (Paris, Afnor Technique,1990.) Test No. TPLA01/89.

Radius(m)BenchValue

FEMAP Structural 5 linear quads.

FEMAP Structural5 parabolic quads.

0.30 100.00 100.00 100.000.31 82.98 82.98 82.980.32 66.51 66.51 66.510.33 50.54 50.54 50.540.34 35.04 35.04 35.040.35 20.00 20.00 20.00

Radius(m)

BenchValue

FEMAP Structural

5 linear quads.

FEMAP Structural

5 parabolic quads.

0.30 1729.91 1701.69 1701.700.31 1674.11 1674.68 1674.690.32 1621.79 1622.32 1622.320.33 1572.64 1573.13 1573.130.34 1526.39 1526.84 1526.830.35 1482.78 1504.39 1504.38

Hollow Cylinder - ConvectionThe complete model and results for this test case are in file htpla03.neu.

This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with convec-tion. It provides the input data and results for benchmark test TPLA03/89 from “Guide de val-idation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingThree tests:

• Test 1 - 10 linear axisymmetric quadrilateral solid elements

• Test 2 - 2 linear axisymmetric quadrilateral solid elements

• Test 3 - 2 parabolic axisymmetric quadrilateral solid elements

The meshes are shown in the following figure:

λ 40Wm-----°C=

Boundary ConditionsElemental Convection

• Convection on internal surface (nodes 3, 14, 16):

• Convection on external surface (nodes 12, 15, 17):


Results

Temperatureand

Element Total Heat Flux

BenchValue

FEMAP Structural

10 linear quads.

FEMAP Structural

2 linear quads.

FEMAP Structural2 parabolic

quads.

Ti (°C) 272.27 272.35 272.17 272.35

hi 150.0W

m2

-------°C=

Ti 500°C=

he 142.0W

m2

-------°C=

Ti 20°C=

So:



Te (°C) 205.05 204.51 204.66 204.5134160.00 33637.10 31746.69 31792.7

26276.90 26508.40 27824.15 27853.8

ϕiW

m2

-------

ϕeW

m2

-------

ϕL--- ϕ2πR=

ϕL--- 34173.82 2 π 0.300⋅ ⋅ ⋅ 64416.13

Wm-----= =

Cylindrical Rod - Flux DensityThe complete model and results for this test case are in file htpla05.neu.

This test is a steady–state heat transfer analysis of a 2D axisymmetric rod with fixed tempera-tures and flux density. It provides the input data and results for benchmark test TPLA05/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 20 linear quadrilateral axisymmetric solid elements

• 42 nodes


λ 33.33Wm-----°C=

Boundary Conditions

Nodal Temperatures• z = 0 (nodes 1 and 3):

• z = 1 (nodes 2 and 4):

Elemental Heat Flux• Cylindrical surface (elements 1-20):



Set temperature to 0°C


Set flux ϕ to 200Wm2--------–

Results

Temperature Results (degrees C):

Results are post–processed on the internal surface.



Node # z (m)BenchValue

FEMAP Structural

Difference

Node 3 0.0 0.00 0.00 0.00%Node 41 0.1 -4.00 -4.02 0.50%Node 39 0.2 4.00 3.98 0.50%Node 37 0.3 24.00 23.97 0.13%Node 35 0.4 56.00 55.97 0.05%Node 33 0.5 100.00 99.97 0.03%Node 31 0.6 156.00 155.97 0.02%Node 29 0.7 224.00 223.97 ~0.00%Node 27 0.8 304.00 303.98 ~0.00%Node 25 0.9 396.00 395.98 0.01%Node 4 1.0 500.00 500.00 0.00%

Hollow Cylinder with Two Materials - Convection

The complete model and results for this test case are in file htpla08.neu.

This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with two mate-rials and convection. It provides the input data and results for benchmark test TPLA08/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties• Material 1:

• Material 2:

Finite Element Modeling• 7 linear quadrilateral axisymmetric solid elements

• 16 nodes

λ1 40.0Wm-----°C=

λ2 20.0Wm-----°C=

The mesh is shown in the following figure.

Boundary Conditions

Elemental Convection• Convection on internal surface:

• Convection on external surface:

hi 150.0W

m2

-------°C=

Ti 70°C=

hi 200.0W

m2

-------°C=

Ti 15°–( )C=


Results

Node #Temperature/

Element X Heat Flux

BenchValue

FEMAP Structural

Difference

Node 9 Ti (°C) 25.42 25.42 0.00%Node 14 Tm (°C) 17.69 17.69 0.00%Node 16 Te (°C) 12.11 12.11 0.00%Node 9 6687.44 6577.88 1.64%

Node 14 5732.09 5733.33 0.02%

Node 16 5422.25 5496.59 1.37%

ϕiW

m2

-------

ϕmW

m2

-------

ϕeW

m2

-------

ϕL--- ϕ2πR=

So:



ϕL--- 5733.33 2 π 0.35⋅ ⋅ ⋅ 12608.25

Wm-----= =

Wall - ConvectionThe complete model and results for this test case are in file htpl03.neu.

This test is a steady–state heat transfer analysis of a 1D wall with fixed convection. It provides the input data and results for benchmark test TPLL03/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 1 linear quadrilateral plate element

• 4 nodes

The plate element thickness is set to 1m.


λ 1.0Wm-----°C=

Boundary Conditions



• Convection coefficient is defined as

energy / (length*time*temperature) in the current system of units.



hA 20.0W

m2

-------°C=

TA 20.0°C–=

hB 10.0W

m2

-------°C=

TB 500°C=

AB

Results

Temperature Results (Degrees Celsius):


structures, (Paris, Afnor Technique,1990.) Test No. TPLL03/89.

Node #Temperature

FluxBenchValue

FEMAP Structural

Difference

Node 1(Temp)

TA (°C) 21.71 21.71 0.00%

Node 4(Temp)

TB (°C) 416.58 416.57 ∼0.00%

Node 1(Flux)

ϕ (W/m**2) 834.2 834.3 0.01%

Wall - Fixed TemperaturesThe complete model and results for this test case are in file htpl01.neu.

This test is a steady–state heat transfer analysis of a 1D wall with fixed temperatures. It pro-vides the input data and results for benchmark test TPLL01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and InformationThe mesh is shown in the following figure:

UnitsSI

Material Properties

Finite Element Modeling• 5 beam (line 2) elements

• 6 nodes

λ 0.75Wm-----°C=

Boundary Conditions

Nodal Temperatures• Internal temperature

• External temperature



Results


Node #Length:

x (m)BenchValue


Node 1 0.00 100.0 100.0 0.00%Node 2 0.01 84.0 84.0 0.00%Node 3 0.02 68.0 68.0 0.00%Node 4 0.03 52.0 52.0 0.00%Node 5 0.04 36.0 36.0 0.00%Node 6 0.05 20.0 20.0 0.00%

Ti 100°C node 1( )=

Te 20°C node 6( )=

The flux calculated with the software is exact:


structures, (Paris, Afnor Technique,1990.) Test No. TPLL01/89.

ϕ 1200Ωµ2------=

L-PlateThe complete model and results for this test case are in the following files:

• htpp01a.neu (linear quadrilateral)

• htpp01b.neu (parabolic quadrilateral)

This test is a steady–state heat transfer analysis of a 2D L–plate with fixed temperatures. It provides the input data and results for benchmark test TPLP01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element ModelingTwo tests:

• 21 nodes, 12 linear quadrilateral plate elements

• 53 nodes, 12 parabolic quadrilateral plate elements


λ 1.0Wm-----°C=

Boundary ConditionsNodal Temperatures

• AF side:

• DE side:



Results


Node Bench ValuesFEMAP

Structurallinear quads.

%Difference

FEMAP Structuralparabolic

quads.

%Difference

8 7.869 7.861 1.10 7.883 0.189 5.495 5.502 0.13 5.519 0.4310 2.816 2.845 1.03 2.834 0.64



A B

C D

EF


structures, (Paris, Afnor Technique,1990.) Test No. TPLP01/89.

19 8.018 8.026 0.10 8.015 0.0418 5.680 5.669 0.19 5.666 0.2520 2.881 2.959 2.71 2.877 0.1417 8.514 8.505 0.11 8.519 0.066 6.667 6.667 0.00 6.667 0.0016 2.972 2.990 0.61 2.963 0.3021 9.001 9.015 0.16 9.108 1.2015 8.640 8.661 0.24 8.669 0.3414 9.316 9.294 0.24 9.283 0.355 9.009 8.996 0.14 8.961 0.53

Hollow Sphere - Fixed Tempera-tures, Convection

The complete model and results for this test case are in file htpv02.neu.

This test is a steady–state heat transfer analysis of a 3D sphere with fixed temperatures and convection. It provides the input data and results for benchmark test TPLV02/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 500 solid (brick and wedge) elements

• 666 nodes

The test is executed on 1/8 of a mapped meshed sphere.


λ 1.0Wm-----°C=

Boundary Conditions


Nodal Temperature• Set external surface temperature



hi 30W

m2

-------°C=

Ti 100°C(elements 401-500)=

Te to 20°C(nodes 1-111)

Results

Temperature results (Degrees C):

Element X Heat Flux results (W/m**2):


structures, (Paris, Afnor Technique,1990.) Test No. TPLV02/89.

Radiusr (m)

Node #BenchValue

FEMAP Structural

Difference

0.3 566 65.00 64.87 0.20%0.31 455 54.84 54.74 0.18%0.32 344 45.31 45.24 0.15%0.33 233 36.36 36.32 0.11%0.34 122 27.94 27.92 0.07%0.35 11 20.00 20.00 0.00%

Radiusr (m)

Node #BenchValue

FEMAP Structural

Difference

0.3 566 1050.00 1019.34 2.92%0.31 455 983.35 987.57 0.43%0.32 344 922.85 926.90 0.43%0.33 233 867.47 871.65 0.48%0.34 122 817.47 821.21 0.45%0.35 11 771.43 797.11 3.32%

Hollow Sphere with Two Materials -Convection


• htpv04a.neu (linear brick)

• htpv04b.neu (parabolic tetrahedron)

• htpv04c.neu (axisymmetric solid)

This test is a steady–state heat transfer analysis of a 3D sphere with two materials and convec-tion. It provides the input data and results for benchmark test TPLV04/89 from “Guide de val-idation des progiciels de calcul de structures.”


UnitsSI

Material Properties• Material 1:

• Material 2:

Finite Element ModelingThree tests:

λ1 40.0Wm-----°C=

λ2 20.0Wm-----°C=

• Test 1 - 888 nodes, 700 solid (brick and wedge) elements

• Test 2 - 3818 nodes, 2192 solid parabolic tetrahedron elements

• Test 3 - 23 nodes, 4 axisymmetric solid parabolic quadrilateral elements

The test is executed on 1/8 of a mapped meshed sphere.

Boundary Conditions





hi 150.0W

m2

-------°C=

Ti 70°C=

he 200.0W

m2

-------°C=

Te 9°–( )C=

Results



structures, (Paris, Afnor Technique,1990.) Test No. TPLV04/89.

TemperatureBenchValue

FEMAP Structural

linear brick(htpv04a)

FEMAP Structural parabolic

tetrahedron(htpv04b)

FEMAP Structural

axisymmetric solid

(htpv04c)

Ti (C°) 25.06 N1 25.03 N19 25.06 N2 25.01Tm (C°) 17.84 N556 17.84 N9 17.84 N6 17.75Te (C°) 13.16 N778 13.18 N5 13.15 N5 13.17

Thermo-mechanical Test - Linear Statics Analysis

The stationary thermal-mechanical test cases for linear statics analysis from the Societe Francaise des Mecaniciens include:

• "Thermal Gradient on a Thin Pipe"

Thermal Gradient on a Thin PipeThe complete model and results for this test case are in file hsla01.neu.

This test is a thermo–mechanical linear statics analysis of a thin pipe with thermal gradient and plane strain. It provides the input data and results for benchmark test HSLA01/89 from “Guide de validation des progiciels de calcul de structures.”


UnitsSI

Material Properties

Finite Element Modeling• 500 axisymmetric (linear quadrilateral solid) elements

• 561 nodes


E 1x· 1011

Pa=

ν 0.3=

Coefficient of expansion: α 1x10

5–

C°-----------=

Boundary Conditions

Constraints• Constrain nodes 1-11 in the X and Z translations.

Nodal Temperature• Radial temperature



T Ti1 r Ri–( )–( )

Re Ri–( )-------------------------------- with Ti=100°C⋅=

Results

Post Processing

Point StressBenchValue

FEMAP Structural

Difference

r = Ri 0 -0.85E6

-74.07E6 -74.20E6 0.18%

r=(Re+Ri)/2 -3.95E6 -3.89E6 1.52%

1.306E6 1.40E6 1.22%

r=Re 0 -0.65E6

68.78E6 68.53E6 0.36%

ValueDefinition

= the axisymmetric C1 radial stress at node 265

= the axisymmetric C4 Azimuth stress at node 265

=the axisymmetric C1 radial stress at node 270

=the axisymmetric C1 Azimuth stress at node 270

= the axisymmetric C1 radial stress at node 275

σr Pa( )

σθ Pa( )

σr Pa( )

σθ Pa( )

σr Pa( )

σθ Pa( )

σr

σθ

σr

σθ

σr


structures, (Paris, Afnor Technique,1990.) Test No. HSLA01/89.

= the axisymmetric C2 Azimuth stress at node 275

ValueDefinition

σθ

IndexAAnnular membrane 152Annular plate 117, 140, 171, 182Anti-symmetric modes 108Articulated plane truss 203Articulated rod truss 201Articulated supports 192Axial distributed load 6Axial loading 291Axisymmetric solid elements 165, 168,255, 268, 331, 334, 337, 340Axisymmetric vibration 165, 171BBar elements 76, 78, 83, 92, 95, 98, 101,192, 194, 196, 199, 203, 206, 288, 291,297, 300, 303, 307, 323, 326Beam 4, 6, 9, 12, 15, 18, 95, 101, 174,186, 192, 194, 206, 288, 294, 307Beam elements 294, 347Bending 27, 196, 199, 274, 323Bending load 210CCantilever 92Cantilever beam 4, 9, 12, 76, 101, 294,307Cantilever mass 78Cantilevered plate 105Cantilevered solid beam 186Cantilevered square membrane 144Cantilevered tapered membrane 148Cantilevered thin square plate 124, 156,161Circular hole 212Circular plate 215Circular ring 98Clamped beams 194

Clamped thick rhombic plate 136Clamped thin rhombic plate 121Clamped-free thin shell 320Compressor 320Convection 334, 340, 344, 353, 356Curved beam elements 196Curved pipe 196Cylindrical rod 337Cylindrical shell 39, 42, 221DDeep simply-supported beam 95Deep simply-supported solid beam 174Displacement 15Distorted mesh 124Distributed loads 9, 18EElastic foundation 206Elliptic membrane 34FFixed temperatures 331, 347, 353Flux density 337Free annular membrane 152Free cylinder 165GGravity loading 232HHeated beam 15Hemisphere point loads 44Hollow cylinder 331, 334, 340Hollow sphere 353, 356Hovgaard’s Problem 326hsla01.neu 361htpl01.neu 347htpl03.neu 344htpla01.neu 331htpla03.neu 334htpla05.neu 337htpla08.neu 340htpp01a.neu 350

htpp01b.neu 350htpv02.neu 353htpv04a.neu 356htpv04b.neu 356htpv04c.neu 356Hydrostatic pressure 229IInfinite plate 212In-plane vibrations 83, 98Internal pressure 221, 261, 268KKirchhoff formulation 251Lle1001.neu 53le1002.neu 53le1003.neu 53le101.neu 34le102.neu 34le103.neu 34le1101a.neu 58le1101b.neu 58le1102a.neu 58le1102b.neu 58le1103a.neu 58le1103b.neu 58le1104a.neu 58le1104b.neu 58le1105a.neu 58le1105b.neu 58le1106a.neu 58le1106b.neu 58le201a.neu 39, 42le201b.neu 39, 42le202a.neu 39, 42le202b.neu 39, 42le301.neu 44le302.neu 44le303.neu 44le304.neu 44

le501.neu 47le502.neu 47le601.neu 49le602.neu 49Linear beam 6, 18Linear Statics 4, 6, 9, 12, 15, 18, 21, 24,27, 30, 34, 39, 42, 44, 49, 53, 58, 192,194, 196, 199, 201, 203, 210, 212, 215,218, 221, 225, 229, 232, 236, 239, 242,247, 251, 261, 268, 274, 361L-Plate 350Lumped mass 285, 307MMass elements 65, 68, 71, 78, 285, 307Membrane 21Membrane loads 21Mindlin formulation 251Moment load 12mstv1001.neu 4mstv1002.neu 6mstv1003.neu 9mstv1004.neu 12mstv1007.neu 15mstv1008.neu 18mstv1009.neu 21mstv1014.neu 24mstv1015.neu 27mstv1016.neu 30mstvn002.neu 65mstvn003.neu 68mstvn004.neu 71mstvn005.neu 73mstvn006.neu 76mstvn007.neu 78NNatural frequency 78ne014ll.neu 117nf001ac.neu 83nf002ac.neu 86

nf003ac.neu 89nf004a.neu 92nf005ac.neu 95nf006ac.neu 98nf011alc.neu 105nf011all.neu 105nf011apc.neu 105nf011apl.neu 105nf011blc.neu 108nf011bll.neu 108nf011bpc.neu 108nf011bpl.neu 108nf0121c.neu 111nf012ll.neu 111nf012pc.neu 111nf012pl.neu 111nf013lc.neu 114nf013ll.neu 114nf013pc.neu 114nf013pl.neu 114nf014lc.neu 117nf014pc.neu 117nf014pl.neu 117nf015lc.neu 121nf015ll.neu 121nf015pc.neu 121nf015pl.neu 121nf021alc.neu 129nf021all.neu 129nf021apc.neu 129nf021apl.neu 129nf021blc.neu 133nf021bll.neu 133nf021bpc.neu 133nf021bpl.neu 133nf0221c.neu 136nf022ll.neu 136nf022pc.neu 136nf022pl.neu 136

nf023lc.neu 140nf023ll.neu 140nf023pc.neu 140nf023pl.neu 140nf031ll.neu 144nf031llc.neu 144nf031pc.neu 144nf031pl.neu 144nf032lc.neu 148nf032ll.neu 148nf032pc.neu 148nf032pl.neu 148nf033lc.neu 152nf033ll.neu 152nf033pc.neu 152nf033pl.neu 152nf041lc.neu 165nf041ll.neu 165nf041pc.neu 165nf041pl.neu 165nf042lc.neu 168nf042ll.neu 168nf042pc.neu 168nf042pl.neu 168nf043lc.neu 171nf043ll.neu 171nf043pc.neu 171nf043pl.neu 171nf051lc.neu 174nf051ll.neu 174nf051pc.neu 174nf051pl.neu 174nf052lc.neu 178nf052ll.neu 178nf052pc.neu 178nf052pl.neu 178nf053lc.neu 182nf053ll.neu 182nf053pc.neu 182

nf053pl.neu 182nf071a.neu 101nf071b.neu 101nf071c.neu 101nf072ac.neu 186nf072al.neu 186nf072bc.neu 186nf072bl.neu 186nf073ac.neu 156nf073al.neu 156nf073bc.neu 156nf073bl.neu 156nf073cc.neu 156nf073cl.neu 156nf073dc.neu 156nf073dl.neu 156nf074c.neu 161nf074l.neu 161Nodal loads 4, 201Normal Modes/Eigenvalue 65, 68, 71,76, 78, 83, 92, 95, 98, 101, 105, 108,114, 117, 121, 124, 129, 133, 136, 140,144, 148, 152, 156, 161, 165, 168, 171,174, 178, 182, 186, 285, 291, 294, 297,300, 303, 307, 311, 314, 317, 320, 323,326, 328OOff-center point masses 92Out-of-plane vibration 98PPatch test 39, 42Pinched cylindrical shell 236Pin-ended cross 83Pipes 326Plane bending 199, 210Plane strain elements 34Plane truss 203Plate elements 34, 39, 42, 44, 49, 105,108, 114, 117, 121, 124, 129, 133, 136,

140, 148, 152, 156, 161, 210, 212, 215,218, 221, 225, 229, 232, 236, 239, 242,247, 251, 279, 311, 314, 317, 320, 328,344, 350plate elements 144Pressure 53, 221, 229, 268Prismatic rod 274Pure bending 27, 274Pure tension 24, 255RRectangular plates 328Rhombic plate 121, 136Rhomboid plate 247Rigid elements 65, 194Rod elements 201Ssdld02.neu 285sdll014a.neu 303sdll014b.neu 303sdll014c.neu 303sdll01a.neu 288sdll01b.neu 288sdll05a.neu 291sdll05b.neu 291sdll09a.neu 294sdll11.neu 297sdll12.neu 300sdll15a.neu 307sdll15b.neu 307sdls01a.neu 311sdls01b.neu 311sdls03.neu 314sdls04.neu 317sdls05a.neu 320sdls05b.neu 320sdlx01.neu 323sdlx02.neu 326sdlx03.neu 328Shear loading 251

Short beam 192, 288Simply-supported annular plate 117, 171Simply-supported rectangular plate 242,314Simply-supported rhomboid plate 247Simply-supported solid annular plate182Simply-supported solid square plate 178Simply-supported thick annular plate140Simply-supported thick square plate 133Simply-supported thin square plate 114Single DOF 65Skew plate normal pressure 49Solid cylinder 58, 255Solid elements 53, 58, 174, 178, 182,186, 255, 268, 274, 279, 353, 356, 361Solid sphere 58Solid square plate 178Solid taper 58Spherical shell 239Spring elements 65, 68, 71, 206, 285Square tube 218ssll02.neu 192ssll05.neu 194ssll07a.neu 196ssll07b.neu 196ssll08.neu 199ssll11.neu 201ssll14a.neu 203ssll14b.neu 203ssll16.neu 206sslp01.neu 210sslp02.neu 212ssls03a.neu 215ssls03b.neu 215ssls05.neu 218ssls06a.neu 221ssls06b.neu 221

ssls07a.neu 225ssls07b.neu 225ssls08.neu 229ssls09.neu 232ssls20a.neu 236ssls20b.neu 236ssls21a.neu 239ssls21b.neu 239ssls21c.neu 239ssls24a.neu 242ssls24b.neu 242ssls24c.neu 242ssls25a.neu 247ssls25b.neu 247ssls27a.neu 251ssls27b.neu 251ssls27c.neu 251sslv01a.neu 255sslv01b.neu 255sslv01c.neu 255sslv01d.neu 255sslv03a.neu 261sslv03b.neu 261sslv03c.neu 261sslv03d.neu 261sslv04a.neu 268sslv04b.neu 268sslv04c.neu 268sslv04d.neu 268sslv08a.neu 274sslv08b.neu 274sslv08c.neu 274sslv08d.neu 274sslv09a10.neu 279sslv09a100.neu 279sslv09a20.neu 279sslv09a50.neu 279sslv09a75.neu 279sslv09b10.neu 279

sslv09b100.neu 279sslv09b20.neu 279sslv09b50.neu 279sslv09b75.neu 279Steady-State Heat Transfer 331, 334,337, 340, 344, 347, 350, 353, 356Strain energy 30Stress 15Symmetric modes 105Symmetric truss 323TTapered beam elements 294Tapered membrane 148Temperatures 58, 331, 347, 353Tension 24Thermal gradient 361Thermal strain 15Thick annular plate 140Thick hollow sphere 168Thick plate 279Thick plate pressure 53Thick square plate 129, 133Thick-walled infinite cylinder 268Thick-walled spherical container 261Thin arc 199Thin circular ring 297, 300Thin pipe 361Thin pipe elbow 303Thin ring plate 317Thin shell 320Thin shell beam wall 27Thin square cantilevered plate 105, 108Thin square plate 124, 156, 161, 311Thin wall cylinder 24, 225, 229, 232Three DOF 71Torque loading 218Torsional system 71Transverse bending 196Truss 30

Two DOF 68UUndamped free vibration 65, 68Undamped free vibrations 76Uniform axial load 225Uniform radial vibration 168Uniformly distributed load 215, 242,247VVibrations 65, 68, 76, 83, 98, 165, 168,171, 303WWall 344, 347

femap structural - verification guide

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