Femap Structural - Verification Guide

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Version 8.2

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Proprietary and Restricted Rights Notice

Overview

Linear Statics Verification Using Theoretical Solutions

Nodal Loads on a Cantilever Beam ...4

Axial Distributed Load on a Linear Beam ...6

Distributed Loads on a Cantilever Beam ...9

Moment Load on a Cantilever Beam ...12

Thermal Strain, Displacement, and Stress on Heated Beam ...15

Uniformly Distributed Load on Linear Beam ...18

Membrane Loads on a Plate ...21

Thin Wall Cylinder in Pure Tension ...24

Thin Shell Beam Wall in Pure Bending ...27

Strain Energy of a Truss ...30

Linear Statics Verification Using Standard NAFEMS Benchmarks

Elliptic Membrane ...34

Cylindrical Shell Patch Test ...39

Laminate Strip ...42

Hemisphere-Point Loads ...44

Z–Section Cantilever ...47

Skew Plate Normal Pressure ...49

Thick Plate Pressure ...53

Solid Cylinder/Taper/Sphere–Temperature ...58

Normal Modes/Eigenvalue Verification Using Theoretical Solutions

Undamped Free Vibration - Single Degree of Freedom ...65

Two Degrees of Freedom Undamped Free Vibration - Principle Modes ...68

Three Degrees of Freedom Torsional System ...71

Two Degrees of Freedom Vehicle Suspension System ...73

Cantilever Beam Undamped Free Vibrations ...76

Natural Frequency of a Cantilevered Mass ...78

Normal Modes/Eigenvalue Verification Using Standard NAFEMS

Bench-marks

Bar Element Test Cases ...82

Pin-ended Cross - In-plane Vibration ...83

Pin-ended Double Cross - In-plane Vibration ...86

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Cantilever with Off-Center Point Masses ... 92

Deep Simply-Supported Beam ... 95

Circular Ring - In-plane and Out-of-plane Vibration ... 98

Cantilevered Beam ... 101

Plate Element Test Cases ... 104

Thin Square Cantilevered Plate -Symmetric Modes ... 105

Thin Square Cantilevered Plate - Anti-symmetric Modes ... 108

Free Thin Square Plate ... 111

Simply-Supported Thin Square Plate ... 114

Simply-Supported Thin Annular Plate ... 117

Clamped Thin Rhombic Plate ... 121

Cantilevered Thin Square Plate with Distorted Mesh ... 124

Simply-Supported Thick Square Plate, Test A ... 129

Simply-Supported Thick Square Plate, Test B ... 133

Clamped Thick Rhombic Plate ... 136

Simply-Supported Thick Annular Plate ... 140

Cantilevered Square Membrane ... 144

Cantilevered Tapered Membrane ... 148

Free Annular Membrane ... 152

Cantilevered Thin Square Plate ... 156

Cantilevered Thin Square Plate #2 ... 161

Axisymmetric Solid and Solid Element Test Cases ... 164

Free Cylinder - Axisymmetric Vibration ... 165

Thick Hollow Sphere - Uniform Radial Vibration ... 168

Simply-Supported Annular Plate -Axisymmetric Vibration ... 171

Deep Simply-Supported Solid Beam ... 174

Simply-Supported Solid Square Plate ... 178

Simply-Supported Solid Annular Plate ... 182

Cantilevered Solid Beam ... 186

Verification Test Cases from the Societe Francaise des Mechaniciens

Mechanical Structures - Linear Statics Analysis with Bar or Rod Elements ... 191

Short Beam on Two Articulated Supports ... 192

Clamped Beams Linked by a Rigid Element ... 194

Transverse Bending of a Curved Pipe ... 196

Plane Bending Load on a Thin Arc ... 199

Nodal Load on an Articulated Rod Truss ... 201

Articulated Plane Truss ... 203

Beam on an Elastic Foundation ... 206

Mechanical Structures - Linear Statics Analysis with Plate Elements ... 209

Plane Shear and Bending Load on a Plate ... 210

Infinite Plate with a Circular Hole ... 212

Uniformly Distributed Load on a Circular Plate ... 215

Torque Loading on a Square Tube ... 218

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Uniform Axial Load on a Thin Wall Cylinder ...225

Hydrostatic Pressure on a Thin Wall Cylinder ...229

Gravity Loading on a Thin Wall Cylinder ...232

Pinched Cylindrical Shell ...236

Spherical Shell with a Hole ...239

Uniformly Distributed Load on a Simply-Supported Rectangular Plate ...242

Uniformly Distributed Load on a Simply-Supported Rhomboid Plate ...247

Shear Loading on a Plate ...251

Mechanical Structures - Linear Statics Analysis with Solid Elements ...254

Solid Cylinder in Pure Tension ...255

Internal Pressure on a Thick-Walled Spherical Container ...261

Internal Pressure on a Thick-Walled Infinite Cylinder ...268

Prismatic Rod in Pure Bending ...274

Thick Plate Clamped at Edges ...279

Mechanical Structures - Normal Modes/Eigenvalue Analysis ...284

Lumped Mass-Spring System ...285

Short Beam on Simple Supports ...288

Axial Loading on a Rod ...291

Cantilever Beam with a Variable Rectangular Section ...294

Thin Circular Ring ...297

Thin Circular Ring Clamped at Two Points ...300

Vibration Modes of a Thin Pipe Elbow ...303

Cantilever Beam with Eccentric Lumped Mass ...307

Thin Square Plate (Clamped or Free) ...311

Simply-Supported Rectangular Plate ...314

Thin Ring Plate Clamped on a Hub ...317

Vane of a Compressor - Clamped-free Thin Shell ...320

Bending of a Symmetric Truss ...323

Hovgaard’s Problem - Pipes with Flexible Elbows ...326

Rectangular Plates ...328

Stationary Thermal Tests - Steady State Heat Transfer Analysis ...330

Hollow Cylinder - Fixed Temperatures ...331

Hollow Cylinder - Convection ...334

Cylindrical Rod - Flux Density ...337

Hollow Cylinder with Two Materials - Convection ...340

Wall - Convection ...344

Wall - Fixed Temperatures ...347

L-Plate ...350

Hollow Sphere - Fixed Temperatures, Convection ...353

Hollow Sphere with Two Materials -Convection ...356

Thermo-mechanical Test - Linear Statics Analysis ...360

Thermal Gradient on a Thin Pipe ...361

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Overview

This 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

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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 Format

Each 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.

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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.

References

The 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.)

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Nodal Loads on a Cantilever Beam

The 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 Geometry

Length=480 in

Cross Sectional Properties

• Area = 30 x 30 in

• I_y =I_z = 67500 in4

Material Properties

• E = 30 E+06 psi

Finite Element Modeling

• 5 nodes

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Boundary Conditions

Constraints

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

Loads

Set 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%

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Axial Distributed Load on a Linear

Beam

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

Test Case Data and Information

Element Type

bar

Units

Inch

Model Geometry

Length = 300 in

Cross Sectional Properties

• Area = 9 in2

• square cross section (3 in x 3 in)

• I = 6.75 in4

Material Properties

E = 30E+6 psi

Finite Element Modeling

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

Boundary Conditions

Constraints

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

Loads

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

Solution Type

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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

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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.

Test Case Data and Information

Element Type

bar

Units

Inch

Model Geometry

• Length = 480 in

Cross Sectional Properties

• Area = 900 in2

• Iy = Iz = 67500 in4

Material Properties

E = 30 E+06 psi

Finite Element Modeling

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• 8 successive bar elements along the X axis

Boundary Conditions

Constraints

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

Loads

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

Solution Type

Statics

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

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* Includes shear deformation which is neglected in theoretical value.

Reference

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Moment Load on a Cantilever Beam

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.

Test Case Data and Information

Element Type

bar

Units

Inch

Model Geometry

Length = 480 in

Cross Sectional Properties

• Area = 900 in2

• I_y = I_z = 67500 in4

Material Properties

E = 30 E+06 psi

Finite Element Modeling

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• 8 successive bar elements along the X axis.

Boundary Conditions

Constraints

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

Loads

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

Solution Type

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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

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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.

Test Case Data and Information

Element Type

bar

Units

SI - meter

Model Geometry

Length = 1 m

Cross Sectional Properties

Area = 0.01 m2

Material Properties

• E = 2.068E+11 PA

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

• v = 0.3

Finite Element Modeling

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• 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.

Solution Type

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Results

Case: One Fixed End

Case: Both Ends Fixed

Reference

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

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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.

Test Case Data and Information

Element Type

bar

Units

Inch

Model Geometry

Length = 480 in

Cross Sectional Properties

• Rectangular cross section (1.17 in x 43.24 in)

• Iz = 7892 in4

Material Properties

• E = 30E6 psi

Finite Element Modeling

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• 4 successive bar elements that are each 10 feet long

Boundary Conditions

Constraints

Constrain 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.

Solution Type

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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

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Membrane Loads on a Plate

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.

Test Case Data and Information

Element Types

plate

Units

Inch

Model Geometry

• Length = 15 in • 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

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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

Constraints

Constrain 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.

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Solution Type

Statics

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

Reference

T1 Translation (in) T3 Translation (in) Bench Value 4.8E-03 14.4E-03

FEMAP Structural 4.8E-03 14.4E-03

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Thin Wall Cylinder in Pure Tension

The 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.

Test Case Data and Information

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.

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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

Solution Type

Statics

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

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Reference

• Roark, R. and Young, W., Formulas for Stress and Strain, 6th Edition, (New York: McGraw–Hill Book Company, 1989.) p. 518, Case 1a.

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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.

Test Case Data and Information

Element Type

linear quadrilateral plate

Units

Inch

Model Geometry

• Length = 30 in • Width = 5 in • Thickness = 0.1 in

Material Properties

• E = 30E6 psi • v = 0.03

Finite Element Modeling

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• 6 linear quadrilateral plate elements

Boundary Conditions

Constraints

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

Out–of–plane Loads

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

Solution Type

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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

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Strain Energy of a Truss

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.

Test Case Data and Information

Element Type

rod

Units

Inch

Model Geometry

• Length = 10 in

Cross Sectional Properties

Cross sectional area (A) = 0.01 in2

Material Properties

E = 30E6 psi

Finite Element Modeling

• 4 nodes

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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

Solution Type

Statics

Results

Reference

Total Strain Energy (lb in) Bench Value 5.846

FEMAP Structural 5.846 Difference 0

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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.

Understanding the Test Case Format

- finite element modeling information - boundary conditions (loads and constraints) - solution type

• results

• reference

References

The 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.

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• 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).

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Elliptic Membrane

The 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:

Test Case Data and Information

Physical and Material Properties

• Thickness = 0.1 m • Isotropic material • E = 210 x 103_MPa A B C D X Y Ellipse AC: x 2 ---   2 y2 + = 1 Ellipse BD: x 3.25 ---   2 _y 2.75 ---   2 + = 1

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• v = 0.3

Units

SI

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

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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

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Loads

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

• Inner curved edge AC is unloaded

Solution Type

Statics

Results

Output - Plate Mid Y Normal Stress at point D

Node # Element Type & Mesh

NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa) Plane Strain Elements with a Plane

Strain Formulation (le101):

Node 4 linear quad - coarse mesh 92.7 62.8 Node 204 linear quad - fine mesh 92.7 80.3 Node 104 parabolic quad - coarse mesh 92.7 88.3 Node 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 54.2 Node 204 linear triangle - fine mesh 92.7 72.0 Node 104 parabolic triangle - coarse mesh 92.7 93.0 Node 304 parabolic triangle – fine mesh 92.7 94.0

(44)

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 66.4 Node 204 linear quad - fine mesh 92.7 82.3 Node 104 parabolic quad - coarse mesh 92.7 88.6 Node 304 parabolic quad - fine mesh 92.7 91.7

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Cylindrical Shell Patch Test

• 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.

Test Case Data and Information

Physical and Material Properties

• Thickness = 0.01 m • Isotropic material • E = 210 x 103_MPa • v = 0.3

Units

SI

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

(46)

Boundary Conditions

Constraints

Fully 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

(47)

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

Solution Type

Statics

Results

Output - 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.

References

Plate Element & Loading

NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa)

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

(48)

Laminate Strip

The 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.

Test Case Data and Information

Geometry

Material Properties

Laminate material: 0° fiber direction 10 15 15 10 X Y X Z 1 E 10N/mm A B 0° 90° 0° 90° 0° 90° 0° 0.1 0.1 0.1 0.4 0.1 0.1 0.1 C E D F

E = 1.0E5 MPa ν₁₂ = 0.4 E₂ = 5.0E3 MPa ν₁₂ E₁ --- ν21

E₂ ---=

(49)

Units

SI

Finite Element Modeling

8 x 40 4-noded shells (quarter model)

Boundary Conditions

Constraints

The one quarter model is:

• simply supported at A (Z=0)

• reflective symmetry about X=25 and Y=5

Loads

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

Solution Type

Statics

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 NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa) Z deflection at E -1.06 -1.06 Bending stress at E 683.9 *668 Bending stress at F - 601 Interlaminar shear stress at D -4.1 **-4.1 Shear stress at F - -2.2

(50)

Hemisphere-Point Loads

The 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.

Test Case Data and Information

Physical and Material Properties

• Thickness = 0.04 m • Isotropic material • E = 68.25 x 103_MPa • v = 0.3

Units

SI

Finite Element Modeling

plate - linear & parabolic quadrilaterals - coarse & fine mesh equally spaced nodes on AC, CE, EA

Point G at X = Y = Z = 10 3 1 2 --- ---        Node 7

(51)

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 A C E F G D B F G D B

(52)

• Concentrated radial load inward at C = 2KN

Solution Type

Statics

Results

References

Test Case

Number Plate Element & Mesh

NAFEMS Bench Value(m) FEMAP Structural Result at node 1 (point A) T1 Translation (m)

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

(53)

Z–Section Cantilever

• 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.

Test Case Data and Information

Physical and Material Properties

• Thickness = 0.1 m • Isotropic material • E = 210 x 103_MPa • v = 0.3

Units

SI

Finite Element Modeling

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

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

Boundary Conditions

Constraints

(54)

Loads

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

Solution Type

Statics

Results

Output - Plate Top Von Mises Stress (σ_xx), point A, node 30 (compression)

References

Plate Element & Loading NAFEMS Bench Value (MPa)

FEMAP Structural Result

(MPa)

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

B1 B2 B3

(55)

Skew Plate Normal Pressure

• 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.

Test Case Data and Information

Physical and Material Properties

• Thickness = 0.01m • Isotropic material • E = 210 x 103_MPa • v = 0.3

Units

SI A B C D E 150o 30o 10m

(56)

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.

(57)

• 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

(58)

Solution Type

Statics

Results

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

References

Test Case

Name Node # Plate Element & Mesh

NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa)

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

(59)

Thick Plate Pressure

• 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:

Test Case Data and Information

Physical and Material Properties

• Isotropic material • E=210x103_MPa • v = 0.3 A B C D A B C D A’ B’ C’ D’ Ellipse AD: x 2 ---   2 y2 + = 1 Ellipse BC: x 3.25 ---   2 _y 2.75 ---   2 + = 1

(60)

Units

SI

Finite Element Modeling

• Solid brick

• Solid wedge

• Solid tetrahedron

Solid Brick

Linear and parabolic, coarse and fine mesh.

Solid Wedge

(61)

Solid Tetrahdron

(62)

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

Solution Type

(63)

Results

Output - Solid Y normal stress at point D3σ yy

References

• 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

NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa)

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

(64)

Solid

Cylinder/Taper/Sphere–Tem-perature

• 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.

Test Case Data and Information

Physical and Material Properties

• Isotropic material • E = 210 x 103_MPa • v = 0.3 • a = 2.3 x 10-4_/o_C

Units

SI

(65)

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 Brick

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.

(66)

Loads

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

T°C (X2+Y2) 1 2 ---Z + =

(67)

Solution Type

Statics

Results

Output - Solid Y Normal Stress at point A.

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

References

Case Node # at

Point A Element Type & Mesh

NAFEMS Bench Value (MPa) FEMAP Structural Result (MPa)

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

(68)

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Normal Modes/Eigenvalue

Verifica-tion Using Theoretical SoluVerifica-tions

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

Understanding the Test Case Format

- solution type - element type

- boundary conditions (loads and 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.

(70)

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.

References

The 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.)

(71)

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.

Test Case Data and Information

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.

(72)

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

Boundary Conditions

Constraints

Constrain 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%

(73)

Reference

• Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, Theory and Applications, (Boston: Allyn and Bacon, Inc., 1978.) p. 75.

(74)

Two Degrees of Freedom

Undamped Free Vibration -

Princi-ple Modes

The complete model and results for this test case are in file mstvn003.neu.

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

Test Case Data and Information

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.

(75)

• 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 Type

(76)

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 0.2756644 FEMAP Structural 0.159155 0.2756644 Difference 0.00% 0.00%

(77)

Three Degrees of Freedom

Tor-sional System

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

Test Case Data and Information

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.

(78)

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.

Solution Type

Normal Modes/Eigenvalue – Guyan method

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%

(79)

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.

Test Case Data and Information

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)

Femap Structural - Verification Guide

Version 8.2

)(0$36WUXFWXUDO

Proprietary and Restricted Rights Notice

Table of Contents

Proprietary and Restricted Rights Notice

Overview

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

Bench-marks

72&

Verification Test Cases from the Societe Francaise des Mechaniciens

72& 

Overview

Linear Statics Verification Using

Theoretical Solutions

Understanding the Test Case Format



Understanding Comparisons with Theoretical

Solutions

Nodal Loads on a Cantilever Beam

Test Case Data and Information

Element Types

Units

Model Geometry

Cross Sectional Properties

Material Properties

Finite Element Modeling



Boundary Conditions

Constraints

Loads

Solution Type

Results

Axial Distributed Load on a Linear

Beam

Test Case Data and Information

Element Type

Units

Model Geometry

Cross Sectional Properties

Material Properties

Finite Element Modeling



Boundary Conditions

Constraints

Loads

Solution Type



Results

Distributed Loads on a Cantilever

Beam

Test Case Data and Information

Element Type

Units

Model Geometry

Cross Sectional Properties

Material Properties

Finite Element Modeling



Boundary Conditions

Constraints

Loads

Solution Type

Results



Moment Load on a Cantilever Beam

Test Case Data and Information

Element Type

Units

Model Geometry

Cross Sectional Properties

Material Properties

Finite Element Modeling



Boundary Conditions

Constraints

Loads

72&

72&