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Chapter 3 2D Simulations

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Chapter 3

2D Simulations

3.1

Step-by-Step: Triangular Plate3.2 Step-by-Step: Threaded Bolt-and-Nut

3.3

More Details

3.4 More Exercise: Spur Gears

3.5

More Exercise: Filleted Bar

3.6 Review

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Chapter 3 2D Simulations

Section 3.1 Triangular Plate

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Section 3.1

Triangular Plate

Problem Description

The plate is made of steel and designed to

withstand a tensile force of 20,000 N on each

of its three side faces.

We are concerned about the deformations

and the stresses.

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Chapter 3 2D Simulations

Section 3.1 Triangular Plate

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Techniques/Concepts Project Schematic Concepts>Surface From Sketches

Analysis Type (2D)

Plane Stress Problems

Generate 2D Mesh

2D Solid Elements

and

Loads>Pressure

Weak Springs

Solution>Total Deformation

Solution>Equivalent Stress

Tools>Symmetry

Coordinate System

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Chapter 3 2D Simulations

Section 3.2 Threaded Bolt-and-Nut

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The plane of symmetry

Theaxisofsymmetry

17 mm

[1] The 2Dsimulation

model.

[6] Frictionlesssupport.

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Chapter 3 2D Simulations

Section 3.2 Threaded Bolt-and-Nut

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Techniques/Concepts

Hide/Show Sketches

Display Model/Plane

Add Material/Frozen

Axisymmetric Problems

Contact/Target

Frictional Contacts

Edge Sizing

Loads>Force

Supports>Frictionless Support

Solution>Normal Stress

Radial/Axial/Hoop Stresses

Nonlinear Simulations

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Chapter 3 2D Simulations

Section 3.3 More Details

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Section 3.3

More Details

Plane-Stress Problems

Plane stress condition:

Z

= 0, ZY

= 0, ZX

= 0

The Hook's law becomes

X

=

X

E

Y

E

Y =

Y

E

X

E

Z

=X

E

Y

E

XY

=

XY

G,

YZ = 0,

ZX = 0

A problem may assume theplane-stress condition if its

thicknessdirection is not

restrained and thus free to

expand or contract.

X

X

Y

XY

XY

XY

XY

X

YZ

Y Stress state at a pointof a zero thickness

plate, subject to in-plane

forces.

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Chapter 3 2D Simulations

Section 3.3 More Details

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Plane-Strain Problems

[2] Strain state ata point of a plane-strain structure.

X

Y

Z

Y

X

XY

X

Y

XY

Plane strain condition:

Z

= 0, ZX

= 0, ZY

= 0

The Hook's law becomes

X =

E

(1+ )(12)(1)

X +

Y

Y =

E

(1+)(12)(1)

Y +

X

Z =

E

(1+)(12)X +Y

XY

=GXY

, YZ =0,

ZX =0

A problem may assume the plane-straincondition if its Z-direction is restrained

from expansion or contraction, all cross-

sections perpendicular to the Z-direction

have the same geometry, and all

environment conditions are in the XY lane.

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Chapter 3 2D Simulations

Section 3.3 More Details

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R

R

Z

Z

RZ

RZ

R

R

Z

Z

RZ

RZ

[1] Strain

state at apoint of a

axisymmetricstructure.

[2] Stressstate at apoint of a

axisymmetricstructure.

Axisymmetric Problems

If the geometry, supports, and

loading of a structure are

axisymmetric about theZ-axis,

then all response quantities are

independent of coordinate.

In such a case,

R = 0,

Z = 0

R = 0,

Z = 0

both and are generally not

zero. They are termed hoop

stressand hoop strainrespectively.

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Chapter 3 2D Simulations

Section 3.3 More Details

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Mechanical GUI

Pull-down Menus

and Toolbars

Outline of Project

Tree

Details View Geometry

Graph

Tabular Data

Status Bar

Separators

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Chapter 3 2D Simulations

Section 3.3 More Details

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Project Tree

A project tree may contain one or more

simulation models.

A simulation model may contain one or more

branches, along with other

objects. Default name for the

branch is the name of the analysis system.

An branch contains , environment conditions, and a

branch.

A branch contains and several results objects.

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Chapter 3 2D Simulations

Section 3.3 More Details

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Unit Systems[1] Built-in unit

systems.

[2] Unit systemfor current

project.

[3] Defaultproject unit

system.

[4] Checked unitsystems won't be

available in the pull-down menu.

[5] These, along with theSI, are consistent unit

systems.

Consistent versus InconsistentUnit Systems.

Built-in versus User-Defined UnitSystems.

Project Unit System. Length Unit in .

Unit System in .

Internal Consistent Unit System.

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Section 3.3 More Details

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Environment Conditions

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Chapter 3 2D Simulations

Section 3.3 More Details

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Results Objects

View Results

[1] Click to turn on/off the label of

maximum/minimum.

[2] Click to turnon/off the probe.

[4] You mayselect the scaleof deformation.

[5] You can controlhow the contour

displays.

[6] Some resultscan display with

vectors.

[3] Label.

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Chapter 3 2D Simulations

Section 3.4 Spur Gears

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Section 3.4

Spur Gears

Problem Description

[2] And the bendingstress here.

[1] What we areconcerned most isthe contact stress

here.

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Chapter 3 2D Simulations

Section 3.4 Spur Gears

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Techniques/Concepts

Copy bodies (Translate)

Contacts

Frictionless

Symmetric (Contact/Target) Adjust to Touch

Loads>Moment

True Scale

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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100 100

100

50

R15

50 kN 50 kN

Section 3.5

Filleted Bar

Problem Description

[2] The bar hasa thickness of

10 mm.

[1] The bar ismade of steel.

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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Part A. Stress Discontinuity

Displacement field is

continuous over theentire body.

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Section 3.5 Filleted Bar

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[2] Originalcalculated stresses

(unaveraged) are not

continuous acrosselement boundaries,

i.e., stress at boundaryhas multiple values.

[4] By default, stresses areaveraged on the nodes, and thestress field is recalculated. That

way, the stress field is

continuous over the body.

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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Part B. Structural Error

For an element, strain energies calculated using averaged stresses and unaveraged

stresses respectively are different. The difference between these two energy values iscalled of the element.

The finer the mesh, the smaller the structural error. Thus, the structural error can be

used as an indicator of mesh adequacy.

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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0.0779

0.0780

0.0781

0.0782

0.0783

0.0784

0.0785

0.0786

0.0787

0 2000 4000 6000 8000 10000 12000 14000

Displacement(mm)

Number of Nodes

Part C. Finite Element Convergence

[1] Quadrilateralelement.

[2] Triangularelement.

[3] Increasingnodes.

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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Part D. Stress Concentration

[1] To accuratelyevaluate the

concentrated stress,finer mesh is needed,

particularly around thecorner.

[2] Stressconcentration.

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Chapter 3 2D Simulations

Section 3.5 Filleted Bar

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Part E. Stress Sigularity

The stress in thiszero-radius filletis theoretically

infinite.

Stress singularity is not limited

to sharp corners. Any locations that have stress

of infinity are called singular

points.

Besides a concave fillet of zero

radius, a point of concentrated

forces is also a singular point.