Chapter 3 2D Simulations 1

Chapter 3 2D Simulations3.1 Triangular Plate3.2 Threaded Bolt-and-Nut3.3 More Details3.4 Spur Gears3.5 Structural Error, FE Convergence, and Stress Singularity3.6 Review

Chapter 3 2D Simulations Section 3.1 Triangular Plate 2

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.

Chapter 3 2D Simulations Section 3.1 Triangular Plate 3

Techniques/Concepts Project Schematic

Concepts>Surface From Sketches Analysis Type (2D) Plane Stress Problems Generate 2D Mesh 2D Solid Elements Relevance Center and

Relevance

Loads>Pressure Weak Springs Solution>Total Deformation Solution>Equivalent Stress Tools>Symmetry Coordinate System

Chapter 3 2D Simulations Section 3.2 Threaded Bolt-and-Nut 4

Section 3.2 Threaded Bolt-and-Nut

Problem Description

[1] Bolt. [2] Nut.

[3] Plates.[4] Section

view.

Chapter 3 2D Simulations Section 3.2 Threaded Bolt-and-Nut 5

The plane of symmetry

The axis of sym

metry

17 mm

[5] The 2D simulation

model.

[6] Frictionless support.

Problem Description

Chapter 3 2D Simulations Section 3.2 Threaded Bolt-and-Nut 6

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

Chapter 3 2D Simulations Section 3.3 More Details 7

Section 3.3 More Details

Plane-Stress Problems

Z= 0,

ZY= 0,

ZX= 0

X=

X

E Y

E

Y=

Y

E X

E

Z= X

E Y

E

XY=

XY

G,

YZ= 0,

ZX= 0

X

X

Y

XY

XY

XY

XY

X

Y Z

Y

Stress state at a point in plane stress condition.

Plane stress condition

The Hookes law becomes

A problem may assume plane-stress condition if its thickness

direction is not restrained and

thus free to expand or contract.

Chapter 3 2D Simulations Section 3.3 More Details 8

Plane-Strain Problems

[2] Strain state at a point in plane strain condition.

Z= 0,

ZX= 0,

ZY= 0

X= E

(1+ )(1 2 ) (1 )X + Y

Y= E

(1+ )(1 2 ) (1 )Y + X

Z= E

(1+ )(1 2 ) X + Y

XY= G

XY,

YZ= 0,

ZX= 0

X

Y

Z

Y

X

XY

X

Y

XY

Plane strain condition

The Hookes law becomes

A problem may assume plane-strain condition 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 plane.

Chapter 3 2D Simulations Section 3.3 More Details 9

R

R

Z

Z

RZ

RZ

R

R

Z

Z

RZ

RZ

[1] Strain state at a point in

axisymmetric condition.

[2] Stress state at a point in

axisymmetric condition.

Axisymmetric Problems

R = 0, Z = 0

R = 0, Z = 0

If the geometry, supports, and loading of a structure are

axisymmetric about the Z-axis,

then all response quantities are

independent of coordinate.

In such a case,

both and are generally not zero. They are termed hoop stress

and hoop strain respectively.

Chapter 3 2D Simulations Section 3.3 More Details 10

Pull-down Menus and Toolbars

Outline of Project Tree Details View Geometry Graph Tabular Data Status Bar Separators

Mechanical GUI

Chapter 3 2D Simulations Section 3.3 More Details 11

Project Tree

A project tree may contain one or more simulation models.

A simulation model may contain one or more Environment branches, along with other objects.

Default name for the Environment branch is the

name of the analysis system.

An Environment branch contains Analysis Settings, environment conditions, and a Solution

branch.

A Solution branch contains Solution Information and several results objects.

Chapter 3 2D Simulations Section 3.3 More Details 12

Unit Systems[1] Built-in unit

systems.

[2] Unit system for current

project.

[3] Default project unit

system.

[4] Checked unit systems

are not available in the

pull-down menu.

[5] These, along with the SI, are consistent unit systems.

Consistent versus Inconsistent Unit Systems.

Built-in versus User-Dened Unit Systems.

Project Unit System. Length Unit in DesignModeler. Unit System in Mechanical. Internal Consistent Unit System.

Chapter 3 2D Simulations Section 3.3 More Details 13

Environment Conditions

Chapter 3 2D Simulations Section 3.3 More Details 14

Results Objects

View Results

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

maximum/minimum.

[2] Click to turn on/off the probe.

[4] You may select the scale of deformation.

[5] You can control how the contour displays.

[6] Some results can display with vectors.

[3] Label.

Chapter 3 2D Simulations Section 3.4 Spur Gears 15

Section 3.4 Spur Gears

Problem Description

[2] And the bending stress here.

[1] W are concerned with the contact stress here.

Chapter 3 2D Simulations Section 3.4 Spur Gears 16

Techniques/Concepts

Copy bodies (Translate) Contacts

Frictionless Symmetric (Contact/Target) Adjust to Touch

Loads>Moment True Scale

Chapter 3 2D Simulations Section 3.5 Filleted Bar 17

Section 3.5 Structural Error, FE Convergence, and Stress Singularity

Problem Description

100 100

1 00

50

R15

50 kN 50 kN

[2] The bar has a thickness of

10 mm.

[1] The bar is made of steel.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 18

Part A. Stress Discontinuity

[1] Displacement eld is continuous

over the entire body.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 19

[2] Original calculated stresses (unaveraged) are not continuous across element boundaries, i.e.,

stress at boundary has multiple values.

[3] By default, stresses are

averaged on the nodes, and the stress eld is

recalculated. That way, the stress eld is continuous over

the body.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 20

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 is

called Structural Error of the element.

The ner the mesh, the smaller the structural error. Thus, the structural error can be used as an indicator of mesh adequacy.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 21

D

i sp l

a ce m

e nt

( mm

)

0.0776

0.0777

0.0779

0.0780

0.0782

0.0783

0.0784

0.0786

0.0787

Number of Nodes

0 2000 4000 6000 8000 10000 12000 14000

Part C. Finite Element Convergence

[1] Quadrilateral element.

[2] Triangular element.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 22

Part D. Stress Concentration

[1] To accurately evaluate the concentrated stress, ner mesh is needed, particularly around

the corner.

[2] Stress concentration.

Chapter 3 2D Simulations Section 3.5 Filleted Bar 23

Part E. Stress Singularity

Stress singularity is not limited

to sharp corners. Any locations that have stress

of innity are called singular

points. Besides a concave llet of zero

radius, a point of concentrated

forces is also a singular point.

The stress in a sharp concave

corner is theoretically

innite.