c01 Ansys Wb 14 Eval

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Chapter 1

Introduction to FEA

Learning Objectives

After completing this chapter, you will be able to:

• Understand the basic concepts and general working of FEA.• Understand the advantages and limitations of FEA.• Understand the types of analysis.• Understand important terms and denitions in FEA.



1-2 ANSYS Workbench 14.0: A Tutorial Approach

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IntroductIon to FEA The inite element analysis (FEA) is a computing technique that is used to obtainapproximate solutions to boundary value problems. It uses a numerical method called niteelement method (FEM). FEA involves the computer model o a design that is loaded andanalyzed or specic results, such as stress, deormation, defection, natural requencies, modeshapes, temperature distributions, and so on.

The concept o FEA can be explained through a basic example involving measurement o the perimeter o a circle. To measure the perimeter o a circle without using the conventional

ormula, divide the circle into equal segments, as shown in Figure 1-1. Next, join the startpoint and the endpoint o each o these segments by a straight line. Now, you can measure thelength o straight line very easily, and thus, the perimeter o the circle by adding the lengtho these straight lines.

Figure 1-1 The circle divided into small equal segments

I you divide the circle into our segments only, you will not get accurate results. For accuracy,

divide the circle into more number o segments. However, with more segments, the timerequired or getting the accurate result will also increase. The same concept can be appliedto FEA also, and thereore, there is always a compromise between accuracy and speed whileusing this method. This compromise between accuracy and speed makes it an approximatemethod.

The FEA was rst developed to be used in the aerospace and nuclear industries, where thesaety o structures is critical. Today, even the simplest o products rely on FEA or design

evaluation.

The FEA simulates the loading conditions o a design and determines the design response inthose conditions. It can be used in new product design as well as in existing product renement. A model is divided into a nite number o regions/divisions called elements. These elementscan be o predened shapes, such as triangular, quadrilateral, hexahedron, tetrahedron, andso on. The predened shape o an element helps dene the equations that describe how theelement will respond to certain loads. The sum o the responses o all elements in a modelgives the total response o the complete model.



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Geeal Wkig FEA

A better knowledge o FEA helps in building more accurate models. Also, it helps inunderstanding the backend working o ANSYS. Here, a simple model is discussed to give youa brie overview o the working o FEA.

Figure 1-2 shows a spring assembly that represents a simple two-spring element model. Inthis model, two springs are connected in series and one o the springs is xed at the let mostendpoint, reer to Figure 1-2. In this gure, the stiness o the springs has been representedby the spring constants K

1and K

2. The movement o endpoints o each spring is restricted to

the X direction only. The change in position rom the undeormed state o each endpoint canbe dened by the variables X

1and X

2. The two orces acting on the end points o the springs

are represented by F1

and F2.

Figure 1-2 Representation of a two-spring assembly

To develop a model that can predict the state o this spring assembly, you can use the linearspring equation given below:

F = KXwhere,

F = orce applied,X = displacement, andK = spring constant

I you use the spring parameters dened above and assume a state o equilibrium, the ollowingequations can be written or the state o each endpoint:

F1

- X1K

1+ (X

2- X

1)K

2= 0

F2

- (X2

- X1

)K 2

= 0

Thereore,

F1

= (K 1

+ K 2)X

1+ (-K

2)X

2

F2

= (-K 2)X

1+ K

2X

2

I the set o equation is written in matrix orm, it will be represented as ollows:

K 1 + K 2 -K 2 X1

-K 2

K 2

X2

=F1F

2





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In the above mathematical model, i the spring constants (K 1

and K 2) are known and the

deormed shapes (X1 and X2) are dened, then the resulting orces (F 1 and F2) can bedetermined. Alternatively, i the spring constants (K 1and K

2) are known and the orces (F

1and

F2) are dened, then the resulting deormed shape (X

1and X

2) can be determined.

Various terminologies that are used in the previous example are discussed next.

Siess Maix In the previous equation, the ollowing part represents the stiness matrix (K):

K 1 + K 2 -K 2-K

2K

2

This matrix is relatively simple because it comprises only one pair o springs, but it turnscomplex when the number o springs increases.

degees FeemDegrees o reedom is dened as the least number o independent cordinates required to

dene the conguration o a system in space. In the previous example, you are only concerned with the displacement and orces. By making one endpoint xed, you will restrict all degreeso reedom or that particular node. Which means that, there will be no translational orrotational degrees o reedom or that node. But, there are two nodes still have some degreeso reedom. As these two nodes are allowed to translate along the X axis only, they have 1degree o reedom each considering that no rotational degree o reedom exist in them.The number o the degrees o reedom on ree nodes in a model determines the number o equations required to solve a mathematical model.

Ba ciisThe boundary conditions are used to eliminate the unknowns in the system. A set o equationsthat is solvable is meaningless without the input. In the previous example, the boundarycondition X

0= 0, and the input orces are F1 and F2. In either ways, the displacements could

have been specied in place o orces as boundary conditions and the mathematical modelcould have been solved or the orces. In other words, the boundary conditions help you reduceor eliminate the unknowns in the system.

neThe solutions generated by using FEA are always approximate.

Elemes a Eleme ShapesBeore proceeding urther, you must be amiliar with the concepts o elements and elementshapes, because these are the building blocks o FEM. These concepts are discussed next.

Elemes

Element is an entity into which the system under study is divided. An element shape isspecied by nodes. The shape (area, length, and volume) o an element depends on the nodes with which it is made. An element (triangular shaped) is shown in Figure 1-3.





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Figure 1-3 A node and an element

Eleme ShapesThere are many types o element shapes that are urther divided into various classes, dependingon their uses. The ollowing are some basic element shapes:

Lie Eleme

A line element has the shape o a line or a curve. Thereore, a minimum o two nodes arerequired to dene it. There can be higher order elements that have additional nodes (at

the middle o the edge o an element). An element that does not have a node in betweenits edges is called a linear element. The elements that have nodes in between edges arecalled quadratic or second order elements. Figure 1-4 shows some line elements.

Figure 1-4 Line elements

Aea Eleme

An area element has the shape o a triangle or a quadrilateral; thereore, it requiresa minimum o three or our nodes to dene it. Some area elements are shown inFigure 1-5.

Figure 1-5 The area elements





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Vlme Eleme

A volume element has the shape o a hexahedron (8 nodes) , wedge (6 nodes),tetrahedron (4 nodes), or a pyramid (5 nodes). Some o the volume elements are shownin Figure 1-6.

Figure 1-6 The volume elements

Geeal Pee c Fiie Eleme AalsisTo conduct the nite element analysis, you need to ollow certain steps that are given next.1. Set the type o analysis to be used.2. Create model.

3. Dene the element type.4. Divide the given geometry into nodes and elements (mesh the model).5. Apply material properties and boundary conditions.6. Derive element matrices and equations.7. Assemble element equations.8. Solve the unknown parameters at nodes.9. Interpret the results.

Figure 1-7 FEA through software

The general process o FEA by using

sotware is divided into three main phases:preprocessing, solution, and postprocessing,reer to Figure 1-7.

PepessThe preprocessor is a phase that processesinput data to produce output, which is usedas input in the subsequent phase (solution).Following are the input data that need to begiven to the preprocessor:

1. Type o analysis (structural or thermal,static or dynamic, and linear or nonlinear)

2. Element type3. Real constants or elements (Cross-sectional area, Moment o Inertia, Shell thickness, and

so on)4. Material properties (Young’s Modulus, Poisson’s ratio, Spring Constant, Thermal

Conductivity, Coecient o Thermal Expansion, and so on)5. Geometric model (either created in the FEA sotware or imported rom other CAD

packages)





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6. FEA model (discretizing the geometric model into small elements)

7. Loading and boundary conditions (dening loads, pressures, moments, temperature,conductivity, convection, constraints (xed, pinned, or rictionless/symmetrical), and soon.

The input data are preprocessed or the output data and the preprocessor generates the datales automatically with the help o users. These data les are used in the subsequent phase(solution), reer to Figure 1-7.

Sli

The solution phase is completely automatic. The FEA sotware generates element matrices,computes nodal values and derivatives, and stores the result data in les. These les are urtherused in the subsequent phase (postprocessor) to review and analyze the results through thegraphic display and tabular listings, reer to Figure 1-7.

PspessThe output rom the solution phase (result data les) is in the numerical orm and consists o nodal values o the eld variable and its derivatives. For example, in structural analysis, the

output o the postprocessor is nodal displacement and stress in elements. The postprocessorprocesses the result data and displays them in graphical orm to check or analyze the result.The graphical output gives the detailed inormation about the required result data. Thepostprocessor phase is automatic and generates graphical output in the specied orm, reerto Figure 1-7.

FEA SoFtWArEThere are a variety o commercial FEA sotware packages available in the market. Every CAE

sotware provides various modules or various analysis requirements. Depending on yourrequirement, you can select a required module or your analysis. Some rms use one or moreCAE sotware and others develop customized version o commercial sotware to meet theirrequirements.

Avaages a Limiais FEA SwaeFollowing are some o the advantages and limitations o FEA sotware:

Avaages1. It reduces the amount o prototype testing, thereby saving the cost and time.2. It gives the graphical representation o the result o analysis.3. The nite element modeling and analysis are perormed in the preprocessor and

solution phases, which i done manually would consume a lot o time and in some cases,might be impossible to perorm.

4. Variables such as stress and temperature can be measured at any desired point o the model.5. It helps optimize a design.6. It is used to simulate the designs that are not suitable or prototype testing.

7. It helps you create more reliable, high quality, and competitive designs.





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Limiais

1. It does not provide exact solutions.2. FEA packages are costly.3. An inexperienced user can deliver incorrect answers, upon which expensive decisions will

be based4. Results give solutions but not remedies.5. Features such as bolts, welded joints, and so on cannot be accommodated to a model. This

may lead to approximation and errors in the result.6. For more accurate results, more hard disk space, RAM, and time are required.

KEy ASSuMPtIonS In FEA There are our types o key assumptions that must be considered while perorming the niteelement analysis. These assumptions are not comprehensive, but cover a wide variety o situationsapplicable to the problem. Moreover, by no means do all the ollowing assumptions apply toall situations. Thereore, you need to consider only those assumptions that are applicable or your analysis problem.

Assmpis relae Geme 1. Displacement values will be small so that a linear solution is valid.2. Stress behavior outside the area o interest is not important. Thereore, geometric

simplications in those areas do not aect the outcome.3. Only internal llets in the area o interest will be included in the solution.4. Local behavior at the corners, joints, and intersection o geometries is o primary

interest, thereore, no special modeling o these areas is required.5. Decorative external eatures will be assumed insigniicant or the stiness and

perormance o the part and these external eatures will be omitted rom the model.

6. Variation in the mass due to suppressed eatures is negligible.

Assmpis relae Maeial Ppeies1. Material properties will remain in the linear region and the nonlinear behavior o the

material property cannot be accepted.2. Material properties are not aected by the load rate.3. The component is ree rom surace imperections that can produce stress

concentration.

4. All simulations will assume room temperature, unless otherwise specied.5. The eects o relative humidity or water absorption on the material used will beneglected.

6. No compensation will be made to account or the eect o chemicals, corrosives, wears,or other actors that may have an impact on the long term structural integrity.

Assmpis relae Ba ciis1. Displacements will be small so that the magnitude, orientation, and distribution o

the load remains constant throughout the process o deormation.2. Frictional loss in the system is considered to be negligible.3. All interacing components will be assumed rigid.





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4. The portion o the structure being studied is assumed as a separate part rom the rest o

the system, so that any reaction or input rom adjacent eatures is neglected.

Assmpis relae Fasees1. Residual stresses due to abrication, pre loading on bolts, welding, or other manuacturing

or assembly processes will be neglected.2. All welds between components will be considered as ideal and continuous.3. The ailure o asteners will not be considered.4. The load on the threaded portion o the part is supposed to be evenly distributed among

the engaged threads.5. The stiness o bearings, both in radial and in axial directions, will be considered as

innite or rigid.

tyPES oF EnGInEErInG AnALySES You can perorm dierent types o analyses using FEA sotware and these are discussesnext.

Sal AalsisIn structural analysis, rst the nodal degrees o reedom (displacement) are calculated andthen the stress, strains, and reaction orces are calculated rom nodal displacements. Theclassication o structural analysis is shown in Figure 1-8.

Figure 1-8 Types of structural analysis

Sai AalsisIn static analysis, the load or eld conditions do not vary with respect to time, and thereore,

it is assumed that the load or eld conditions are applied gradually, not suddenly. The systemunder this analysis can be linear or nonlinear. The inertia and damping eects are ignored instructural analysis. In structural analysis, the ollowing matrices are solved:

[K] x [X ]= [F]

Where,K = Stiness MatrixX = Displacement Matrix

F = Load Matrix





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The above equation is called the orce balance equation or the linear system. I the elements

o matrix [K] are the unction o [X], the system is known as the nonlinear system. Nonlinearsystems include large deormation, plasticity, creep, and so on. The loadings that can beapplied in a static analysis include:

1. Externally applied orces and pressures2. Steady-state inertial orces (such as gravity or rotational velocity)3. Imposed (non-zero) displacements4. Temperatures (or thermal strain)5. Fluences (or nuclear swelling)

The outputs that can be expected rom the FEA sotware are given next.

1. Displacements2. Strains3. Stresses4. Reaction orces

dami AalsisIn dynamic analysis, the load or eld conditions vary with the time and are applied suddenly.The system can be linear or nonlinear. The dynamic load includes oscillating loads, impacts,collisions, and random loads. The dynamic analysis is classied into the ollowing three maincategories:

Mal Aalsis

It is used to calculate the natural requency and mode shape o a structure.

Hami AalsisIt is used to calculate the response o a structure to harmonically time varying loads.

tasie dami Aalsis

It is used to calculate the response o a structure to arbitrary time varying loads.

In dynamic analysis, the ollowing matrices are solved:

For the system without any external load:

[M] x Double Derivative o [X] + [K] x [X]= 0

Where,M = Mass MatrixK = Stiness MatrixX = Displacement Matrix

For the system with external load:

[M] x Double Derivative o [X] + [K] x [X]= [F]





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Where,

K = Stiness MatrixX = Displacement MatrixF = Load Matrix

The above equations are called the orce balance equations or a dynamic system. By solvingthe above set o equations, you can extract the natural requencies o a system. The load typesapplied in a dynamic analysis are the same as that in a static analysis. The outputs that canbe expected rom a sotware are Natural requencies, Mode shapes, Displacements, Strains,Stresses, and Reaction orces. All these outputs can also be obtained with respect to time.

Spem AalsisThis is an extension o the modal analysis and is used to calculate stress and strain due to theresponse o the spectrum (random vibrations). For example, you can use it to analyze how well a structure will perorm and survive in an earthquake.

Bklig AalsisThis type o analysis is used to calculate the buckling load and the buckling mode shape. Slender

structures (that is thin and long structures) when loaded in the axial direction, buckle underrelatively small loads. For such structures, the buckling load becomes a critical design actor.

Explii dami AalsisThis type o structural analysis is available only in the ANSYS LS-Dyna program and is used toget ast solutions or large deormation dynamics and complex contact problems, or example,explosions, aircrat crash worthiness, and so on.

themal AalsisThe thermal analysis is used to determine the temperature distribution and related thermalquantities such as: Thermal distribution, Amount o heat loss or gain, Thermal gradients,and Thermal fuxes.

All primary heat transer modes such as conduction, convection, and radiation can be simulated. You can perorm two types o thermal analysis, steady-state and transient.

Sea Sae themal Aalsis

In this analysis, the system is studied under steady thermal loads with respect to time.

tasie themal AalsisIn this analysis, the system is studied under varying thermal loads with respect to time.

Fli Flw AalsisThis analysis is used to determine the fow distribution and temperature o a fuid. The ANSYS/FLOWTRAN program is used to simulate the laminar and turbulent fow, compressible and

electronic packaging, automotive design, and so on. The outputs that can be expected romthe fuid fow analysis are Velocities, Pressures, Temperatures, and Film coecients





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Elemagei Fiel Aalsis

This type o analysis is conducted to determine the magnetic elds in electromagnetic devices.The types o electromagnetic analyses are Static analysis, Harmonic analysis, and Transientanalysis

cple Fiel AalsisThis type o analysis considers the mutual interaction between two or more elds. It isimpossible to solve elds separately because they are interdependent. Thereore, you need aprogram that can solve both the problems by combining them.

For example, i a component is exposed to heat, you may rst require to study the thermalcharacteristics o the component and then the eect o the thermal heating on the structuralstability.

Alternatively, i a component is bent in dierent shapes using one o the metal ormingprocesses and then subjected to heating, the thermal characteristics o the component willdepend on the new shape o the component. Thereore, rst the shape o the component hasto be predicted through structural simulations. This is called as the coupled eld analysis.

IMPortAnt tErMS And dEFInItIonSSome o the important terms and denitions used in FEA are discussed next.

Segh When a material is subjected to an external load, the system undergoes a deormation. Thematerial, in turn, oers resistance against this deormation. This resistance is oered by virtueo the strength o the material.

LaThe external orce acting on a body is called load.

SessThe orce o resistance oered by a body against the deormation is called stress. The stressis induced in the body while the load is being applied on the body. The stress is calculated asload per unit area. p = F/A

Where,p = Stress in N/mm2

F = Applied Force in NewtonA = Cross-Sectional Area in mm2

The material can undergo various types o stresses, which are discussed next.





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tesile Sess

I the resistance oered by a body is against the increase in the length, the body is said to beunder tensile stress.

cmpessive SessI the resistance oered by a body is against the decrease in the length, the body is said to beunder compressive stress. Compressive stress is just the reverse o tensile stress.

Shea SessThe shear stress exists when two materials tend to slide across each other in any typical planeo shear on the application o orce parallel to that plane.

Shear Stress = Shear resistance (R) / Shear area (A)

Sai When a body is subjected to a load (orce), its length changes. The ratio o change in thelength o the body to its original length is called strain. I the body returns to its original shapeon removing the load, the strain is called elastic strain. I the body remains distorted ater

removing the load, the strain is called plastic strain. The strain can be o three types, tensile,compressive, and shear strain.

Strain (e) = Change in Length (dl) / Original Length (l)

Elasi LimiThe maximum stress that can be applied to a material without producing the permanentdeormation is known as the elastic limit o the material. I the stress is within the elastic limit,

the material returns to its original shape and dimension on removing the external stress.

Hke’s Law It states that the stress is directly proportional to the strain within the elastic limit.

Stress / Strain = Constant (within the elastic limit)

yg’s Mls Mls Elasii

In case o axial loading, the ratio o intensity o the tensile or compressive stress to thecorresponding strain is constant. This ratio is called Young’s modulus, and is denoted by E.

E = p/e

Shea Mls Mls rigii In case o shear loading, the ratio o shear stress to the corresponding shear strain is constant.This ratio is called Shear modulus, and it is denoted by C, N, or G.

ulimae SeghThe maximum stress that a material withstands when subjected to an applied load is calledits ultimate strength.





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Fa Sae

The ratio o the ultimate strength to the estimated maximum stress in ordinary use (designstress) is known as actor o saety. It is necessary that the design stress is well below theelastic limit, and to achieve this condition, the ultimate stress should be divided by a ‘actoro saety’.

Laeal SaiI a cylindrical rod is subjected to an axial tensile load, the length (l) o the rod will increase( dl) and the diameter (Ø ) o the rod will decrease ( dØ). In short, the longitudinal stress will

not only produce a strain in its own direction, but will also produce a lateral strain. The ratio dl/l is called the longitudinal strain or the linear strain, and the ratio dØ / Ø is called the lateralstrain.

Piss’s raiThe ratio o the lateral strain to the longitudinal strain is constant within the elastic limit. Thisratio is called the Poisson’s ratio and is denoted by 1/m. For most o the metals, the value o the ‘ m’ lies between 3 and 4.

Poisson’s ratio = Lateral Strain / Longitudinal Strain = 1/m

Blk MlsI a body is subjected to equal stresses along the three mutually perpendicular directions, theratio o the direct stresses to the corresponding volumetric strain is ound to be constant ora given material, when the deormation is within a certain limit. This ratio is called the bulkmodulus and is denoted by K.

Sess ceaiThe value o stress changes abruptly in the regions where the cross-section or prole o astructural member changes abruptly. The phenomenon o this abrupt change in stress isknown as stress concentration and the region o the structural member that is aected by stressconcentration is known as the region o stress concentration. The region o stress concentrationneeds to be meshed densely to get accurate results.

Beig When a non-axial orce is applied on a structural member, the structural member startsdeorming. This phenomenon is known as bending. In case o bending, strains vary linearlyrom the centerline o a beam to the circumerence. In case o pure bending, the value o strain is zero at the centerline. The plane section o the beam is assumed to remain plaineven ater the bending.

Beig Sess When a non-axial orce is applied on a structural member, some compressive and tensile

stresses are developed in the member. These stresses are known as bending stresses.





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ceep

At elevated temperature and constant load, many materials continue to deorm but at a slowrate. This behavior o materials is called creep. At a constant stress and temperature, therate o creep is approximately constant or a long period o time. Ater a certain amount o deormation, the rate o creep increases, thereby causing racture in the material. The rate o creep depends highly on both the stress and the temperature.

classifai MaeialsMaterials are classied into three main categories: elastic, plastic, and rigid. In case o elasticmaterials, the deormation disappears on the removal o load. In plastic materials, thedeormation is permanent. A rigid material does not undergo deormation when subjected toan external load. However, in actual practice, no material is perectly elastic, plastic, or rigid.The structural members are designed such that they remain in the elastic conditions underthe action o working loads. All engineering materials are grouped into three categories thatare discussed next.

Ispi MaeialIn case o Isotropic materials, material properties do not vary with direction, which means

they have the same material properties in all directions. Material properties are dened by Young’s modulus and Poisson’s ratio.

ohpi MaeialIn case o orthotropic materials, material properties vary with direction and are speciedin three orthogonal directions. Such materials have three mutually perpendicular planes o material symmetry. Material properties are dened by separate Young’s modulus and Poisson’sratios along each axis.

Aispi MaeialIn case o Anisotropic materials, material properties vary with direction, but there is no planeo material symmetry. This means they do not behave in the same way in all directions.

Aspe rai Aspect ratio is dened as the ratio o the longest side to the smallest side o an element.

Axismme Model that can be dened by rotating its cross-section by 360-degrees about an axis is knownas axisymmetry model.

degees Feem (doF)Degrees o reedom is dened as the reedom o a given point to move in any direction inspace.

There are six DOFs or any point in 3-dimensional (3D) space:3 translational DOFs (one each in the X,Y, and Z directions) and3 rotational DOFs (one each about the X, Y, and Z axes).





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Sel-Evalai tes

Answer the following questions and then compare them to those given at the end of thischapter:

1. FEA simulates the loading conditions o a model and determines its response under thoseconditions. (T/F)

2. A linear line element has a maximum o two nodes. (T/F)

3. A quadratic line element has a node in the middle. (T/F)

4. An area element should always be triangular in shape. (T/F)

5. You cannot import an external geometry le into an FEA sotware. (T/F)

6. The nodes dene the shape o an element. (T/F)

7. A minimum o ____________ nodes are required to dene a line element.

8. A minimum o ____________ nodes are required to dene an area element.

9. A minimum o ____________ nodes are required to dene a volume element.

review Qesis

Answer the following questions:

1. The Finite Element Method gives exact solutions to problems. (T/F)

2. In FEM, the geometry is discretized into small parts, known as elements. (T/F)

3. In space, a rigid body has six degrees o system. (T/F)

4. In dynamic analysis, the boundary conditions are a unction o __________.

5. Modal analysis is used to calculate the _______________ requencies o a model.

6. Hooke’s law states that stress is directly proportional to ____________ within elasticlimit.

Aswes Sel-Evalai tes

1. T, 2. T, 3. T, 4. F, 5. F, 6. T, 7. two, 8. three, 9. our

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