Section2 Module3 InstructorNotes r3

8/13/2019 Section2 Module3 InstructorNotes r3

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Autodesk Simulation Workshop

Section 2: Linear Analysis

This section presents the theory and methods used to

perform linear analyses using Autodesk Simulation. This is

an important section since the bulk of the problems solved

by engineers on a daily basis fall into this classification. One

definition of a linear analysis is that a plot of response

parameters versus input parameters is a straight line. In this

section a linear analysis is one that involves linear materials.

Some of the analyses may contain non-linear responses

such as large displacements, stress stiffening, or buckling.

However, in all cases the material will be an isotropic-linear-

elastic material.

The first three modules present the underlying theory

associated with the finite element method. Since geometricnon-linear phenomena are included in subsequent modules,

the theory will cover non-linear solution methods. The

starting point is the differential equation of equilibrium.

Equilibrium and the numerical methods used to ensure that a solution satisfies it is a central

theme of the first three sections. The remaining modules, four through nine, provide

additional theory and information needed to set up a specific type of problem, execute it, and

interpret the results for one of the analysis types contained in Autodesk Simulation.

The intent is to present the material in the first three modules in such a way that a senior level

undergraduate engineering student or a practicing engineer could use the modules as a

learning resource. Standard matrix notation is used instead of the more concise indicial

notation taught in graduate school. The content is presented at a depth that an interestedstudent can work through the mathematics and obtain insight into the methods and workings

of a commercial program. Each subsequent module contains just enough theory to teach a

student or practicing engineer how to setup and perform each analysis type. It is possible for

someone to start at Module 4 without first mastering Modules 1 through 3.

In Modules 4 through 9 a cantilevered beam modeled using 3D brick elements with mid-side

nodes serves as the example problem. The cantilevered beam is familiar to undergraduate

students, and for some analysis types students can check results using manual calculation.

Modules Contained in Section 2

1. Virtual Work Equation

2. Newton-Raphson Equations

3. Truss Element Equations

4. Static Analysis

5. Natural Frequency Analysis

6. Transient Stress (Modal

Superposition)

7. Transient Stress (Direct

Integration)

8. Load Factor Analysis9. Fatigue Analysis


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Important Note on Archived Datasets

The datasets associated with each module in Section 2 have been Archived to facilitate

downloading. An Archived dataset is a compressed file created by Autodesk Simulation

Multiphysics to reduce the overall size of the file. The Archived files do not contain solution

results, and it will be necessary to execute the analysis in order to obtain the results. In mostcases there is a Begin and an End dataset file. The Beginfile contains the problem data

prior to setting the analysis parameters, and will not run properly. It provides a starting point

for the module. The videos show how to change the analysis parameters in the Begin

datasets to those in the Enddatasets. The Enddataset can be executed to obtain the

analysis results.

Some datasets do not have a Begin and End file (i.e. Modules 1, 2, 3, and 9). Modules 1, 2,

and 3 do not involve executing the software to solve a problem. Module 9 on Fatigue

requires that a static analysis be executed prior to performing the fatigue analysis. The

dataset for Module 9 is set up to run the static analysis when the file is retrieved.

An Archived dataset can be retrieved by selecting the Autodesk Simulation Icon in the upperleft corner of the screen, selecting Archivein the drop down menu, and then selecting

Retrieve.

Autodesk Inventor Files

The Inventor part or assembly files are also included as part of the datasets. In some cases

the Inventor files are used as in the videos or PowerPoint presentations. For example, the

intent of the video may be to show how to go from Autodesk Inventor into Autodesk

Simulation Multiphysics. In other cases the Inventor files are not part of the videos or

PowerPoint presentations, but having them available will enable instructors or students todevelop additional problems or exercises.


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Table of ContentsClick below to jump to the current Module:

1. Module 1: Virtual Work Equation ........................................................................... 4

2. Module 2: Newton-Raphson Equations ................................................................. 4

3. Module 3: Truss Element Equations .................................................................. 6

4. Module 4: Static Analysis ...................................................................................... 7

5. Module 5: Natural Frequency Analysis ................................................................ 11

6. Module 6: Transient Stress (Modal Superposition) .............................................. 13

7. Module 7: Transient Stress (Direct Integration) ................................................... 16

8. Module 8: Load Factor Analysis .......................................................................... 18

9. Module 9: Fatigue Analysis ................................................................................. 19


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1. Module 1: Virtual Work Equation

Introduct ion

In this module the differential equations of equilibrium for stresses are used to develop thevirtual work equation that represents an integral statement of equilibrium. The equation is

first developed with respect to equilibrium in the deformed shape and is then stated in terms

of conjugate stress and strain measures written with respect to a reference configuration.

Execut ion

1) The linear analysis types available in Autodesk Simulation can be seen by selecting

Analysis Types in the browser, right clicking and selecting Linear. A list of linear

analysis types is shown. These include: 1) Static Stress with Linear Material Models,

2) Natural Frequency (Modal), 3) Natural Frequency (Modal) with Load Stiffening, 4)

Response Spectrum, 5) Random Vibration, 6)Frequency Response, 7) Transient

Stress (Direct Integration), 8) Transient Stress (Modal Superposition), and 9) CriticalBuckling Load Factor.

2) The modules in Section II that show how to perform a specific type of analysis come

from this list. They are: 1) Module 4Static Analysis, 2) Module 5Natural

Frequency Analysis, 3) Module 6Modal Superposition, 4) Module 7Direct

Integration, 5) Module 6 - Load Factor Analysis, and 6) Module 9 - Fatigue. Module 5

addresses how to compute natural frequencies with and without load stiffening.

Module 9 on fatigue requires that a stress analysis first be performed.

3) Breaking the object to be analyzed into small parts called elements is a fundamental

part of the finite element method. One of the primary reasons for breaking the object

into parts is that the integrals associated with the integral form of the virtual work

equation can be carried out more easily. Another is that the displacement variation

within the elements can be approximated by simple polynomials.

4) In addition to breaking the object into small elements, boundary conditions and loads

must be defined. One end of the beam used in the modules is clamped using

displacement boundary conditions that do not allow the nodes on the end surface to

move. Loads are applied either on the free end or are applied as pressures on the

top surface.

5) The objective of the various analysis types is usually to find the displacements and

stresses within the elements that make them in equilibrium with the applied loads and

boundary condition constraints.

2. Module 2: Newton-Raphson Equations

Introduct ion

In this module the Newton-Raphson method for solving non-linear equations is used to

develop an incremental form of the virtual work equation presented in Module 1. This

incremental form gives rise to the various components of the tangent stiffness matrix and


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residual load that are central to the types of analyses available in Autodesk Simulation. Later

modules covering the change in natural frequencies due to stress stiffening and linear

buckling or load factor analysis require specific components of the tangent stiffness matrix.

Various equilibrium iteration methods available in Autodesk Simulation are based on the

theory contained in this section.

Execut ion

1) The linear analysis types that use the stress-dependent contribution to the tangent

stiffness matrix can be seen by selecting Analysis Types, right clicking and

selecting Set Current Analysis Type, and selecting Linear. Linear analysis types

that include stress stiffening are Natural Frequency (Modal) with Load Stiffening and

Critical Buckling Load. The stiffness matrix used in both of these methods changes

as the loads acting on the system being analyzed change. They require that loads

be applied to the structure and that a static analysis be performed before the natural

frequencies or buckling load factor are computed. This static analysis is

automatically performed by Autodesk Simulation.

2) Non-linear problems require that equilibrium iterations be performed to obtain a

solution to the non-linear equations. Module 4 presents a non-linear analysis of a

cantilevered beam that experiences large displacements when subjected to a

pressure load on its top surface. This simple problem is non-linear for two reasons.

First, the deflections are large and the stiffness matrix changes due to the change in

geometry. The final geometry is considerably different in shape than the original

geometry. This is known as a geometric non-linearity. Second, the pressures remain

normal to the surface and the direction of the surface normal vectors change as the

beam deflects.

3) The equilibrium iteration options available for a non-linear problem can be seen by

selecting Analysis Type in the browser for a Static Analysis with Non -linear

Materials, right clicking and selecting Edit Analysis Parameters, and selecting the

Advanced button at the bottom of the Analysis Parameters data entry box.Although the title of the analysis type includes the words Non-linear Materials, this

analysis type is also applicable to linear material models. Linear materials are the

default type and a non-linear material must be selected to activate the non-linear

material model. A linear material model is used if no non-linear material is specified.

This analysis type must be used any time a static analysis is performed that needs to

account for large displacements. This is the case regardless of the type of material

being used.

4) The types of equilibrium iterations available include combined Newton with and

without line search, Newton with and without line search, and modified Newton with

and without line search. The theoretical background for these types of equilibrium

iterations is included in the Power Point slides for Module 2 and additional theory is

included in Module 4. It is interesting to note that not all of these methods will

converge for this relatively simple problem while others converge significantly slower

than others. The reason that all of these methods are made available is that there is

no single method that works best all of the time. The ability to select the best method

for a particular problem requires experience, an understanding of the theory, and

often some trial and error. An experienced analyst will often try different iteration

methods in the course of an analysis to see which performs the best. This is

particularly important when large problems are being solved that require a significant

amount of time.


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3. Module 3: Truss Element Equations

Introduct ion

In this module the components of the tangent stiffness matrix and residual load are

developed for a truss element to demonstrate how to go from the equations in Module 2 to

the element level equations implemented in a production finite element program such as

Simulation Mechanical. A truss element is chosen because the integrations required by the

virtual work equation can be carried out by hand. More complicated elements, such as a 3D

brick, use Quassian quadrature methods to evaluate the integrals. Numerical integration

adds another level of sophistication that is not necessary to understand the basic steps.

Execut ion

1) The element types available in Autodesk Simulation can be seen by selecting an

Element Type for a particular part in the browser. A list appears that contains

Truss, Beam, Membrane, Brick, Plate, Tetrahedron, Spring, Rigid, Gap, Thin

Composite, and Thin Composite (Sandwich). This list may change depending on the

type of analysis being performed.

2) Although these element types have unique characteristics that are accounted for in

their mathematical formulations, all of the element types are based on the

incremental form of the equations of equilibrium.

3) Interpolation or shape functions are used to describe the variation of a parameter

within an element. In the case of solid elements the interpolation functions can be

used to describe the shape of an element and the variation of the displacement

within the element. When the same functions are used to describe both the shapeand displacement variation, the element is called an isoparametric element. The

temperature variation within an element is described using interpolation functions

when a heat transfer analysis is being performed. The interpolation functions for a

truss element are used in the PowerPoint slides for Module 3. These same

interpolation functions are used for spring and other line elements.

4) A graphical user interface makes the application of finite element methods extremely

powerful. When the finite element method was first being developed, only engineers

with specialized education and advanced degrees used the method. The analyses

took long periods of time to perform because meshes were developed manually and

the post-processing graphics tools were crude by todays standards. Now engineers

at all experience and educational levels use finite element programs and most of theunderlying mathematics is hidden from the user. However, it is still important to

remember that all numerical methods have issues associated with accuracy,

convergence, and stability. Users of the software must still ensure that their answers

have converged and that the analysis being performed is sufficient to capture the

underlying physics or mechanics of the problem. This more often than not requires

that multiple analyses be performed with different meshes and mesh densities in high

stress areas. The graphical user interfaces and speed of modern computers make

this iterative process relatively quick.


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5) Students or inexperienced users always have questions about what time step or

element size to use. There is no universal answer to this question. The more that is

understood about the underlying theory, the better a user will be able to judge what a

good starting value might be. However, even the educated and experienced user

will still perform multiple analyses using different parameters to determine the

sensitivity of the results to selected input data.

4. Module 4: Static Analysis

Introduct ion

A static analysis involves loads that are applied so slowly that they do not create a dynamic

response, or steady state loads that exist after the dynamic transients have been damped

out. A static analysis is probably the most common type of analysis performed to evaluatethe strength of a mechanical component or system. The failure mechanism most usually

associated with a static analysis is yielding of the material and stress results are depicted

using the von Mises stress that can be compared directly to the yield strength. A factor of

safety can be computed that provides a measure of how close the stresses at a point are to

causing permanent deformation due to exceeding the yield strength.

The problem analyzed in this module is a cantilevered beam that has a pressure loading that

follows the surface of the beam as it deforms. The beam experiences large displacements

and equilibrium iterations are required to obtain accurate results. Although a relatively simple

problem, the large deformations and associated pressure load that follows the deformation

causes some iteration methods to fail or converge slowly. The videos associated with this

module are divided into four parts. The first part addresses meshing the model and applying

loads. The second part shows how to set up the problem by editing the analysis parameters.

The third part explains the information that is created and displayed on the computer screen

during the analysis. The fourth part addresses tools and methods available for analyzing the

results. The Execution section that follows is also divided into these four parts.

Execut ion

Note that the Analysis Type displayed in the FEA Editor is Static Stress with Nonlinear

Material Models. Although Nonlinear Material Models are available in this analysis type,

linear elastic materials are also available. In this case we are using material properties for

the aluminum alloy 6061-T6.

Video 4A: Meshing th e model

1) With the selection tab set on surfaces, select the surface on the end of the beam that

is to be fixed.

2) Right click and select Add. Select Surface Boundary Condition in the pop-up

menu. Click the Tx, Ty, and Tz boxes. Click Ok.

3) Select the top surface that is to have the pressure load applied.


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4) Right click and select Add. Select Surface Pressure/Traction in the pop-up menu.

Select the Pressure button and then add 2 in the Magnitude field. Check the box

Follows Displacement. Click Ok. Checking the box Follows Displacement is

important for problems that experience large displacements. If the box is not

checked, the pressure will always act normal to the original surface orientation. If the

box is checked, the pressure will act normal to the surface of the last computed

equilibrium configuration. You are encouraged to run the problem with and without

the box being checked to see the difference in the computed displacements and

stresses.

5) Select Element Definition in the FEA Editor browser. Right click and select Edit

Element Definition. At the bottom of the pop-up menu beside Midside Nodes, select

the arrow to obtain the Included option. Click Ok. This is an important option for

thin sections that experience bending. Standard 8-noded brick elements do not have

sufficient flexibility to model bending. Two options exist: 1) add a large number of

elements through the thickness, or 2) increase the flexibility of the brick elements by

adding mid-sized nodes. The two elements through the thickness are insufficient to

accurately model the bending through the thin cross section, so mid-sized nodes are

used. The mid-side nodes permit a quadratic displacement variation within theelement. A quadratic displacement variation will result in a linear strain variation.

The linear strain variation will result in a linear stress distribution.

6) Select 3D Mesh Settings in the ribbon menu across the top of the screen. SelectOptions. Click on Model. Uncheck the top box, Use automatic geometry-based

mesh size function.

7) Select the Surface icon. Click on the down arrow head beside percent of automaticto obtain the Absolute mesh size option. Enter 0.0625 in the Size field. Click Ok.

8) Select Generate 3D Mesh in the ribbon menu across the top of the screen togenerate the mesh on the outside surfaces of the beam. Note that the pressure and

displacement constraints are automatically applied to the nodes of elements

associated with the designated surfaces.

Video 4B: Problem Setup

1) Select Analysis Type in the FEA Editor browser, then select Edit Analysis

Parameters.

2) Set the Event Duration to 1 second. Time is used with a static analysis because

some materials have time dependent properties. Examples include viscoelastic

models for polymers and creep models for high temperature metal applications. In

our problem we can use any time value.

3) Set the Capture rate to 1. This indicates that the load will be applied in 1 increment

and that the stress and displacement data will be saved for later analysis at each

increment. If a value of 2 were used, then the load would be applied in 2 increments(50% and 100%) and the stress and displacement data would be saved for each load

increment.

4) Set the first row of the Time column for the Load Curve table to 0 and the first row

of the Multiplier column to 0. Set the second row of the Time column to 1 and the

second row of the Multiplier column to 1. Note that the time value of 1 entered in

the second row agrees with the Event Duration entered in step 2.


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5) Select View Plot to see a plot of the load curve. It appears as a ramp which means

that the load will be increased linearly from a value of 0% at time 0 to 100% at time 1.

Click Ok.

6) Select the Gravity/Acceleration tab. Set the X-multiplier to 0, the Y-multiplier to -1,

and the Z-multiplier to 0. This will cause the gravitational force to be applied in the

negative-y direction. Set the acceleration due to body force to 386.4. If a problem is

modeled that has a constant acceleration or deceleration, this same location can be

used to enter the information. Click Apply.

7) Select the Output tab. Uncheck the box Nodal forces due to contact. Since the

problem being solved does not have contact, unchecking this box will not have any

effect. It is being unchecked to make the user familiar with the option.

8) Click on the Advanced button at the lower right hand corner of the Output menu.

This provides access to several addition options that can have a significant effect on

the solution of a non-linear problem.

9) Select the down-arrow associated with the Nonlinear iterative solution method.

There are six equilibrium iteration methods available. Which of these methods is

best for a particular problem is not always clear and a trial-and-error process may be

required. A Full Newton w/o line search works best for this problem. The modifiedNewton options will not converge and the Full Newton w/line search converges

slowly. Select Full Newton w/o line search. The user is encouraged to select the

other iteration methods to see what happens.

Video 4C: Running the Simulat ion

1) Click on the traffic light icon, Run Simulation, to start the analysis.

2) The first thing that occurs during an analysis is the generation of the nodes and

elements within the interior volume of the part. For large systems this can take

considerable time.

3) As the analysis proceeds, load step status information will be displayed in tabular

form. The columns are Time, Percent, DT, L, Iter, and Residual.a. Time gives the instance in time for which ASM is seeking a solution.

b. Percent is the percent load being applied.

c.

d. L is the parameter used to decrease the size of the load increment if

convergence is not being obtained fast enough. DT and L are related

through the formula

)1(2

11

LeCaptureRat

DT

e. Iter gives the current number of the equilibrium iteration being performed

(incremented sequentially by ASM).

f. Residual is the convergence parameter. Convergence is achieved when thisnumber becomes less than the value specified in the analysis parameters.

4) The program will iterate until the specified convergence tolerance is achieved.

Convergence is indicated by placing a small c immediately following the value of the

Residual. If convergence is proceeding too slowly, the L parameter will be increased

and a new time step will be computed.


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Video 4D: Stat ic A nalysis Resul ts

1) ASM will automatically display a contour plot of the von Mises stress. The von Mises

stress can be compared directly to the yield stress of ductile materials. It is also very

easy to see the neutral axis of beams in bending using the von Mises stress. Since

the stress at the neutral axis is zero, the von Mises stress will appear as a blue line in

the contour plot.

2) The pressure and boundary conditions can be turned off to improve the visibility of

the contours. Select the Results Options tab and then click on Loads and

Constraints.

3) A safety factor defined as the yield stress divided by the von Mises stress can also be

displayed as a contour plot. The safety factor defined by the following formula

provides a factor that indicates how much the load can be increased before yielding

occurs:

eff

ysFS

.. ,

where ys is the yield strength of the material and eff is the von Mises effective

stress.

If the Safety Factor is less than or equal to one, the material will yield. The material

will not yield if the Safety Factor is greater than one. The Safety Factor contour plot

is obtained by first loading the yield strength of the material. Select the Results

Contours tab and then click on the down arrow beside the Safety Factor icon.

Select Set Allowable Stress Values and click on Load Yield Stress. Next click on

the down arrow beside the Safety Factor icon and select View. A contour plot of

the safety factor will be displayed.

4) In many cases an engineer has a good idea how a part will deform when it is

subjected to loads and boundary conditions. Verifying that the deformations are

consistent with expectations is an important way to check that loads and boundaryconditions have been applied correctly. The displacements often have to be scaled

to enable the part deformation to be seen on the screen. This is accomplished by

going to Results Contours and selecting the Show Displaced on the top ribbon

menu. Select Displaced Options to obtain a set of controls. These controls enable

the displacements to be scaled as a percentage of the model size or as a multiple of

the actual value. The cantilevered beam has large displacements that can be seen

using a scale factor of one times the actual displacements. Select As an Absolute

Value and enter a scale factorof 1 and close the box. The displacements can be

turned off using this set of controls if no deformation is wanted in the plot.

5) Principal stress components and the six tensor components of stress can also be

displayed as a contour plot. Tensor components are related to the Cartesian

coordinate system used to model the part. The icons for selection of the Principal

stress and the tensor components of stress are located on the ribbon menu of

Results Contours. The zz component of stress can be used to view the bending

stress through the thickness of the beam. Select Results Contours and then select

the down arrow beside the Tensor icon. Select the ZZ Component. Note that a

tensile stress occurs on the top surface while a compressive stress occurs on the

bottom surface. The two values are essentially the same, which is consistent with

beam theory.


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5. Module 5: Natural Frequency Analysis

Introduct ion

The natural frequencies and mode shapes of a system are important parameters that affect

the dynamic response of the system. The natural frequency of a system is sometimes

computed simply to compare it to the frequency of excitation to see if a resonance situation

will exist. In other cases, the natural frequencies and mode shapes must be computed prior

to performing a transient analysis. This is the case for a Modal Superposition analysis

covered in Module 6. The time step used to perform a dynamic response analysis (Modules

6 and 7) is also based on the period of the highest natural frequency of interest. In addition,

practicing engineers often compute the natural frequencies and mode shapes of a system to

make sure that a system is constrained and responds as intended. Considerable insight can

be gained about a system by studying its mode shapes.

Accurate mode shapes are important to obtaining accurate dynamic stresses. Natural

frequencies are less sensitive to modeling technique than mode shapes since they are based

on the square root of the ratio of stiffness divided by mass. The stiffness can be off by a

factor of two and the natural frequency will only have a 40% error. However, strains are

based on the derivative of displacements and displacements are directly related to the mode

shapes. Errors in mode shapes lead directly to errors in strains and associated stresses.

The problem analyzed in this module is a cantilevered beam. The natural frequencies and

mode shapes are first calculated without any static loads (video for Module 5A). The natural

frequencies and mode shapes are then calculated when the beam is subjected to a tension

force acting along its length (video for Module 5B). The shift in frequencies due to stress

stiffening is very apparent.

Execut ion

Video 5A: Natural Frequenc y (Modal)

The cantilevered beam used in Module 4 will also be used in this module. Once the beam is

meshed, the Analysis Parameters must be entered. Right click on Analysis Type and select

Edit Analysis Parameters.

1) First set the Number of frequencies/modes to calculate to 5. The number of modes

to be calculated depends on what their use is going to be. In our case, we will usethe natural frequencies and mode shapes to compute the transient response of the

cantilever in Module 6. Several runs were made with each using a different number

of natural frequencies and mode shapes. Five modes were determined to provide a

sufficiently accurate result.

2) Next set the Lower cut-off frequency to zero and the Upper cut -off frequency to

1000. This establishes a range within which the algorithm will search for five

frequencies. If it doesnt find five natural frequencies, the algorithm will only return

the number found within this range. Zero is a common value for the Lower cut-off


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frequency. Some trial and errormay be required to find a suitable Upper cut-off

frequency. If the value you enter doesnt result in the desired number of modes, you

will need to re-run the analysis using a higher number.

3) Make sure that the Rigid body modes are expected box is not checked. The

numerical algorithm will use a frequency shift more suited to extracting natural

frequencies of zero Hz if this box is selected. Close the Analysis Parameters box.

4) The natural frequencies and mode shapes are computed by selecting the Run

Simulation icon under the Analysis tab located at the top of the screen.

5) After the analysis is complete the first mode will be displayed in the results section.

The frequency associated with the first mode is displayed along with the mode

number in the lower left hand corner of the screen.

6) The scale factor used to display the mode shape can be changed to enhance the

visibility of the mode shape. Select Show Displaced on the Results Contours

menu. Select Displaced Options. The scale factor can be specified as a

percentage of the model size or as an absolute value. As an example, set the scale

factor to 10% of the model size, press enter, and close the box. Notice the increased

amplitude of the mode shape.

7) Other modes can be displayed by selecting them using the Load Case options.Each mode shape is treated as a load case and the arrows allow you to move up or

down to obtain a specific mode shape. Note that the natural frequency for each

mode shape is displayed in the lower left hand portion of the screen.

8) Each of the mode shapes can be animated to aid in understanding the motion. The

animation start, pause, and stop buttons are located at the top of the Captures area

on the top menu.

Video 5B: Natural Frequency (Modal) wi th Load Sti ffening

The major difference between this type of analysis and a standard natural frequency and

mode shape analysis discussed in Video 5A is that a static analysis is first performed by ASM

before computing the natural frequencies and mode shapes. The stresses computed in thestatic analysis are used to form the stress-dependent contribution to the tangent stiffness

matrix. Tensile stresses will stiffen the beam and raise the natural frequencies. Compressive

stresses will decrease the stiffness of the beam and lower the natural frequencies.

The same cantilevered beam used in Video 5A will be used in this demonstration. The

process used to mesh the beam is shown in Video 4A.

1) The analysis type for this analysis should be Natural Frequency (Modal) with Load

Stiffening. If it is not, select Analysis Type, right click and select Set Current

Analysis Type, select Linear then Natural Frequency (Modal) with Load Stiffening.

2) With the meshed model in the FEA Editor, select the end surface on the free end of

the beam, then right click and select Add Surface Force. Enter a magnitude of 100and select the z-direction button to apply a 100 lb load acting in the z-direction. This

creates a tensile zz stress of 1,600 psi in the beam.

3) Select the Analysis Type, right click and select Edit Analysis Parameters. The

same parameters will be used as in Video 5A. Set the Number of

frequencies/modes to calculate to 5, Lower cut-off frequency to 0, and Upper cut-

off frequency to 1,000 Hz.

4) We are not going to use the Linear Modal results as a starting point for the current

eigenvalue extraction so the Begin with Linear Modal results box is not checked.


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Using this option could decrease the analysis time and is recommended for large

problems. Our problem is small, so it is not necessary.

5) Select the Gravity/Acceleration tab and set the X multiplier, Y multiplier, and Z

multiplier to zero. Gravity would normally be set to an appropriate direction.

However, in this problem we want to see what effect the axial load by itself will have,

so gravity is ignored. Close the Analysis Parameters box.

6) Select the Analysis tab and then select the Run Simulation icon.

7) The tools available for analyzing the results are the same as those shown in Video

5A.

8) A comparison of the natural frequencies obtained without and with load stiffening

shows that even a relatively small stress can have a big effect on some of the modes.

The results are compared in the following table.

Mode NaturalFrequency

(Linear)[Hz]

Natural Frequency(Load Stiffening)

[Hz]

1 28.1 54.82 175.7 220.4

3 222.0 228.04 491.9 533.9

5 615.6 616.7

6. Module 6: Transient Stress (Modal Superposition)

Introduct ion

Modal Superposition is the first of two methods available in ASM for determining the dynamic

response of a system to time-dependent loads. Modal Superposition is limited to linear

systems having constant natural frequencies and mode shapes. Problems involving contact

or other mechanisms that result in changes in the natural frequencies must use direct

integration methods covered in Module 7. Modal superposition uses natural frequencies and

mode shapes as input data and a natural frequency analysis must be performed prior to

performing a modal superposition analysis.

Computing the dynamic response of a system involves solving the equations of motion using

numerical methods. The solutions are more time consuming and require more computer

resources (disk space for storing results) than static analyses. The modal superpositionmethod offers a relatively quick method for computing the dynamic response of systems in

which the response is controlled by a few low frequency modes. In these cases the

computation time can be significantly reduced.

A cantilevered beam subjected to a step input at its free end is used as an example problem

to demonstrate how to setup and perform a modal superposition analysis. The same

problem is used in Module 7. The results for both methods are the same.


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Execut ion

The results from Module 5A are used as a starting point for this analysis. The natural

frequency analysis is contained in Design Scenario 1.

1) Select Design Scenario 1 in the browser, right click and select copy. This creates

a new design scenario with the same information that is in Design Scenario 1.

2) Select Analysis Type in the browser, right click and select Set Current Analysis

Type, select Linear, and then select Transient Stress (Modal Superposition).

3) We must determine the identification numbers of the nodes where the loads will be

applied. The loads will be applied along the top edge of the free end of the beam.

With the selection mode set to vertices (select the Selection tab and select

Vertices), select each node where a load will be applied, right click and sele ct

Inquire. The node number and coordinates of the node will be displayed. Write

the node numbers down, they will be needed in the next step. Select only the

corner nodes. Loads are typically not applied to the mid-side nodes.

4) We are now ready to setup the analysis parameters required to perform a modal

superposition analysis.

5) Select Analysis Type in the browser, then right click on Edit Analysis Parameters.

A box appears that contains information required to perform the analysis.

6) Load information is entered in two places: 1) the Dynamic Load Data table and 2)

in the Load Curves. The Load Curves contain multipliers that are a function of

time. The multipliers are used to scale the vector components of the load that are

entered into the Dynamic Load Data table. The vector components are entered in

the X scale, Y scale, and Z scale columns. We will first define the applied loads by

filling in the Dynamic Load Data table.

7) A row of the Dynamic Load Data table must be filled in for each node along the top

edge of the free end of the beam. The node numbers found in Step 3 weredetermined for this purpose. The Add Row button is used to add rows to the table.

Note that the Index in the first column is automatically incremented. The node

numbers are entered in the second column. It does not matter what order is used to

enter the node numbers. The Load Curve associated with each node will be

entered in the next column. In this problem we will only use one load curve, so a 1

should be placed in each row of the Load Curve column. The Type column is

used to tell ASM whether the vector components are for a force or a moment. Type

0 is a force and Type 1 is a moment. We will only be entering forces, so each row

of the table should contain a 0 in the Type column. The X Scale, Y Scale, and Z

Scale columns contain the vector components of the force to be applied to each

node. In our problem a 5 lb force will be applied in the negative y-direction. This

force will be divided equally among the 17 nodes located at the corner of each

element. This gives a Y Scale factor of -0.294 lb in each row of the table. The X

Scale and Z Scale columns should contain zeros. The last column contains an

Activation Time. The activation time can be used to specify a time at which the

vector components become active. They are zero before becoming active. In this

problem we will use the activation time to simulate a step function that is zero until

0.05 seconds. At all times greater than 0.05 seconds, the vector components will be

as entered in the X Scale, Y Scale, and Z Scale columns. The 0.05 seconds is an


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arbitrary number used in this example. The step function could also be specified

using the Load Curves. This will be done in Module 7.

8) Select the Load Curves button to open the Load Curve data box.

9) Click the Add Rows button three times to add three rows to the table.

10) Add the following data into the load curve table:

Time(sec) Factor

0.0 0

0.05 0

0.051 1

2.0 1

Note that the plot of the load curve that has a step at 0.05 seconds is automatically

updated. Click Ok to exit the Load Curves data box.

11) The time step will be based on the highest natural frequency for a weak axis

bending mode computed in Module 5A. The third weak axis bending mode has a

frequency of 491.9 Hz. The period of this frequency is 0.00203 seconds. Common

practice is to use a time step of 1/8 -1/10 of this period. This works out to a time

step of 0.000203 seconds.

12) The lowest weak axis bending mode has a frequency of 28.1 Hz or a period of

0.0356 seconds. If we capture three complete cycles of this mode it will take 0.1068

seconds (3 * 0.0356 sec). We must also wait for the step function to start at 0.05

seconds. Adding 0.05 to 0.1068 seconds gives a total analysis time of 0.1568

seconds. A time step of 0.000203 seconds will require 772 time steps. Based on

these calculations well set the number of time steps to 1000. This will give us

slightly more than three cycles of the lowest weak axis bending mode.

13) The data stored when a transient analysis is performed can be quite large. We will

set the Output Interval to 2, which will cause the results associated with every

other time step to be saved. Setting this value to 1 would cause the resultsassociated with every time step to be saved.

14) The Damping Factor is set to 0.005 or 0.5% of critical. This damping value is

typical of that found in a structure not having any connections.

15) All of the analysis parameters have been entered and we are ready to submit the

analysis. Select the Analysis tab in the top menu, and press the Run Simulation

icon.

16) After the analysis is complete, the program will switch to the Results mode.

17) All of the tools available for static stress analysis are also available for each load

case of dynamic analysis. Each time step is considered to be a load case and

different load cases can be selected using the Load Case arrows located at the top

of the screen.

18) The ability to plot stress or displacement data as a function of time is a veryimportant tool for the analysis of data from a transient analysis.

19) Click on a node in the model mesh. The default selection mode in Results is Vertex.

If you cannot select a node, select Selection at the top of the screen and then

select Vertex.

20) Right click and select Edit New Graph. A time series graph of the von Mises stress

at the node you selected will appear. You can change the parameter being plotted

by selecting a stress tensor component or displacement component at the top of the

screen.


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7. Module 7: Transient Stress (Direct Integration)

Introduct ion

The second method available in Autodesk Simulation to determine the response of a system

to time-dependent loads is Direct Integration. This method gains its name from the fact that it

is a direct approach that does not attempt to reduce the size of the problem through a

transformation process, as is done in a modal superposition analysis. Instead it uses

numerical methods that integrate the equations of motion containing the full size and form of

the stiffness and mass matrices.

Although this method will be applied to linear problems in this module, it can handle a wide

range of non-linear phenomena and forms the basis for the methods used for non-linear

problems. The direct integration method presented in this module is an implicit method that

requires the formation of the stiffness and mass matrices. Explicit integration methods (not

covered in this Section) are used for problems requiring very small time steps to capture highfrequency content.

The same cantilevered beam used in Module 7 is used for this module so that the results

from both methods can be compared.

Execut ion

1) The first step is to apply a five lb force acting in the negative y-direction to the top

edge of the free end of the beam. This is accomplished by changing the selection

mode to Edge, selecting the top edge of the free end, right clicking and selecting

Add edge load. After the data entry box appears, enter a Magnitude of -5, and set

the direction cosine for the y-direction to 1 by selecting the y button. Closeout thebox by selecting Ok.

2) The next step involves setting up the analysis controls by editing the analysis

parameters. Select Analysis Type in the browser, then right click and select Edit

Analysis Parameters.

3) The Analysis Parameters data entry box contains areas for defining the Event,

Damping, Load Curve, and Dynamic Load Data. The Dynamic Load Data table could

be used to define the loads acting on the structure. This table was used in

conjunction with the Modal Superposition Analysis in Module 6. Since we entered

the 5 lb load using the FEA Editor, we dont need to use this table.

4) Next we will define the Event information. This includes the number of time steps,

time step size and output interval. Determining the time step can be a trial and error

process. In our case we know the natural frequencies of the beam from Module 5.

The time step will be based on the highest natural frequency of a weak axis bending

mode. The third weak axis bending mode has a frequency of 491.9 Hz. The period

of this frequency is 0.00203 seconds. Common practice is to use a time step of 1/8 -

1/10 of this period. This works out to a time step of 0.000203 seconds.

5) The lowest weak axis bending mode has a frequency of 28.1 Hz or a period of

0.0356 seconds. If we capture three complete cycles of this mode, it will take 0.1068

seconds (3 * 0.0356 sec). We must also wait for the step function to start at 0.05


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seconds. Adding 0.05 to 0.1068 seconds gives a total analysis time of 0.1568

seconds. A time step of 0.000203 seconds will require 772 time steps. Based on

these calculations well set the number of time steps to 1000. This will give us

slightly more than three cycles of the lowest weak axis bending mode.

6) The data stored when a transient analysis is performed can be quite large. We will

set the Output Interval to 2, which will cause the results associated with every other

time step to be saved. Setting this value to 1 would cause the results associated with

every time step to be saved.

7) Next well define the load multiplier that specifies how the 5 lb load will vary with time.

The magnitude of the load is 5 lb and the load multiplier ranges from 0 to 1. In our

case we will define a step function that goes from 0 to 1 at 0.05 seconds. The load

multiplier information is entered in the table and is automatically plotted in the graph

window. We will need to add three rows to the table. Press the Insert Row button

three times. Next enter the following information into the table: (0,0), (0,0.05),

(0.0501,1), and (2,1). Note the step function plotted in the graph. It should be noted

that the step function could be started at time t=0 by defining a constant load curve of

1. The 0.05 second delay is for demonstration purposes only. Close out the box by

selecting Ok.8) Next, the Rayleigh Damping coefficients Alpha and Beta will be specified. These

coefficients must be estimated based on the modal damping expected for each

mode. We have used a 0.5% of critical modal damping value for the first three weak

axis bending modes. Using the procedure outlined in the PowerPoint slides, we have

determined that the values of Alpha and Beta are 4.768 and 0 respectively. These

values are entered. Close the Analysis Parameters box.

9) Next we will submit the analysis by selecting the Analysis tab and then picking the

Run Simulation icon. After the analysis has completed, the results can be reviewed

using the tools available in the Results section. These tools are discussed and

illustrated in Video 6B.


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8. Module 8: Load Factor Analysis

Introduct ion

Buckling is an important failure mechanism for compressively loaded components. Bucklingloads can be estimated using a Load Factor Analysis. Some systems will buckle after the

loads have caused yielding to occur. A Load Factor Analysis is restricted to systems that are

in the elastic region when buckling occurs.

A load factor analysis uses the small-deformation and stress-dependent components of the

tangent stiffness matrix to set up an eigenvalue problem. The eigenvalue problem returns a

load factor that indicates how much the applied load must be changed to reach the buckling

load. The stress-dependent component of the tangent stiffness matrix is a function of the

stress in the system and a static analysis must be performed prior to setting up the

eigenvalue problem. The static analysis is automatically performed by ASM as part of the

analysis.

A cantilevered beam subjected to a compressive axial load is used as an example problem

for this module. The numerical results can be compared directly to the finite element results.

Execut ion

1) A 5 lb load acting in the negative z-direction is added to the end of the cantilevered

beam using the FE Editor.

2) Next we edit the analysis parameters prior to executing the analysis. Select

Analysis Type in the browser, then right click and select Edit Analysis Parameters.

3) The Load Multipliers can be used to change the loads or combinations of loads

acting on the structure without having to change them in the FE Editor. This allows

the analysis to be done multiple times using baseline loads. Our demonstration willnot use these since we have only one load.

4) Next we will set gravity to zero so that the results can be compared directly to

calculations made using hand calculations. The formulas used in hand calculations

do not generally include gravitational effects. Select the Gravity/Acceleration tab.

Set the Acceleration due to body force to zero. Close out the Analysis Parameters

box.

5) The analysis is executed by selecting the Analysis tab followed by picking the Run

Simulation icon.

6) Once the analysis is complete, Simulation automatically switches to the Results

section and displays a deformed shape of the lowest buckling mode. The Buckling

Load Multiplier is displayed in the lower-left-hand corner of the graphics window.

The load multiplier of 5.6 times the applied load of 5 lb gives a buckling load of 28

lbs. This agrees with a 28.1 lb load computed using the buckling equation for fixed-

free beam.

7) The various buckled shapes and load factors can be seen by toggling through the

five modes computed by using the Load Case Options arrows located at the top of

the screen. These higher order buckling modes are generally of little practical use

since the design is controlled by the lowest load factor.


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9. Module 9: Fatigue Analysis

Introduct ion

Fatigue is a failure mechanism that results when cyclic loads cause internal dislocations in

the microstructure to move to the surface. The molecular-level dislocations lead to surface

cracks and subsequent failure. There are two methods used to determine the fatigue life of a

component: 1) stress-life, and 2) strain-life. The stress-life approach is used for high-cycle

fatigue situations that have stresses significantly below the yield strength of the material. The

strain-life approach is applicable to both low and high-cycle fatigue and is recommended

when stresses are near or exceed the yield strength.

The stress-life approach uses an S-N curve, mean stress interaction curve, and a cumulative

damage law to provide a framework for evaluating whether a part is susceptible to fatigue

crack initiation and failure. The strain-life approach uses cyclic-strain data, mean stress

correlations, and a cumulative damage law to estimate the fatigue life. Both methods requirea technique for counting the number of cycles or strain reversals when variable amplitude

loading is considered.

The Fatigue Wizard in Autodesk Simulation can estimate the life of a component using either

the stress-life or strain-life approach. The cumulative damage rule used in each case is

Miners rule. The rain flow counting method is used to determine the stress/strain range and

associated mean values for variable amplitude loading.

In this module a cantilevered beam subjected to cyclic loads applied to its free end is used to

demonstrate how to set up and perform a fatigue analysis.

Execut ion

A fatigue analysis is performed after a stress analysis has been performed. The Fatigue

Wizard is accessed in the Results Section using the Analysis tab located at the top of the

screen.

1) After entering the Results Section, select the Analysis tab then select the Fatigue

Analysis icon.

2) The Fatigue Wizard guides you through the seven steps needed to perform a fatigue

analysis. The first step is to select the type of fatigue analysis that will be used. The

two options are Strain Based or Stressed Based. Note the warning on the screen

that the Stress Based approach is only suitable for high cycle fatigue calculations. In

this problem we will perform a Strain-Life analysis, so the radial button beside StrainBased (EN) is selected. We are using the Strain-Life method because we expect to

be in the low cycle fatigue regime based on how close the peak stress is to the yield

stress, and by how much it exceeds the endurance limit of the 6061-T6 aluminum

used in the cantilevered beam.

3) In the next step we have the opportunity to modify the material properties used in the

fatigue analysis. We will use the default values for the elastic modulus and tensile

strength contained in the Simulations material library. We must enter Poissons ratio

in the yellow field. A value of 0.3 is entered. The strain-life method requires material


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properties that define the slope and y-intercepts of cyclic stress-range and strain-

range data plotted against reversals or (2N) using log-log scales. This information

can be entered by selecting the Advanced button. It is possible to estimate these

quantities from information such as the fracture strain, ultimate strength, etc. In this

problem we will use the estimated values by selecting the Approximate button. We

must also uncheck the My material is steel button since our material is aluminum.

4) The next allows a stress concentration factor to be added. If the stresses in the area

that controls the fatigue life are modeled accurately enough to capture a stress

concentration, then we can enter a value of 1. If the model is too coarse to

accurately estimate the stress, then a stress concentration factor can be used to

raise the stress or strain to a more realistic value. In a linear analysis where the

stress exceeds the yield strength, the program will estimate the actual stress and

strain using the Neuber method. We can also select a surface finish life reduction

factor based on values contained in the Specimen Data Modifiers area. Click the

Surface Finish box and select the Machined option. Other life reduction factors

can be specified by selecting the Miscellaneous box. In this case a number is

entered to account for other factors.

5) In the next step we define a load multiplier curve that specifies how the load variesduring a cycle. In our problem we will define a curve that starts at zero, ramps

linearly to a value of 1, and then ramps down to a value of zero. A more complicated

load multiplier can be specified to accommodate variable amplitude loads. The Rain

Flow Counting method is used to determine the amplitudes and mean values. Click

on the Load Curve box. A box containing a load curve table and graph area is

displayed. Additional rows can be added to the table by right clicking on the

Multiplier box and selecting Insert row. We will need a total of three rows to enter

our load multiplier data. Enter the following data pairs (0 sec, 0), (0.5 sec, 1), and

(1.0 Sec, 0). Close the box by selecting Ok.

6) In the next step we specify the type of information that we want from the fatigue

analysis. We can select either Safety Factor or Cycles to Failure. Since we can

easily calculate a safety factor if we know the life, well select Cycles to Failure. We

must also enter a Desired Life. This information will be used for comparison

purposes only. If the life is computed to be less than this desired life, the Fatigue

Wizard will tell us that the design is unsafe. If it is greater, then Fatigue Wizard will

tell us that it is safe. The value entered does not affect that computed fatigue life.

7) In the next step we submit the analysis by selecting the stop light icon.

8) In the final step we see the fatigue analysis results. Based on our desired life of

100,000 cycles, our design is considered to be unsafe because we have a computed

life of 71,200 cycles. The columns in the table contain Log(N) at different nodes

using various mean stress corrections. The second column has no mean stress

correction; the third column is based on the Morrow correction; and the fourth column

is based on the Smith-Watson-Topper correction. The Morrow correction is generallyconsidered to be the most correct for compressive mean stresses while the Smith-

Watson-Topper correction is considered to be the most correct when the mean

stresses are in tension. In our problem the top surface has a tensile alternating

stress and a tension mean stress. Therefore the last column data provides the most

appropriate estimate.

Section2 Module3 InstructorNotes r3

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