Chapter 3 2D Simulations 1
Chapter 32D Simulations3.1 Step-by-Step: Triangular Plate
3.2 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
Chapter 3 2D Simulations Section 3.1 Triangular Plate 2
Section 3.1Triangular 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 3.2Threaded 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
[1] The 2D simulation
model.
[6] Frictionless support.
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 3.3More Details
Plane-Stress Problems
• Plane stress condition:
σ
Z= 0, τ
ZY= 0, τ
ZX= 0
• The Hook's law becomes
εX=σ
X
E−ν
σY
E
εY=σ
Y
E−ν
σX
E
εZ= −ν
σX
E−ν
σY
E
γXY
=τ
XY
G, γ
YZ= 0, γ
ZX= 0
• A problem may assume the
plane-stress condition if its
thickness direction is not
restrained and thus free to
expand or contract.
σ
X σ
X
σ
Y
τ
XY
τ
XY
τ
XY
τ
XY
X
Y Z
σ
Y Stress state at a point of a zero thickness
plate, subject to in-plane forces.
Chapter 3 2D Simulations Section 3.3 More Details 8
Plane-Strain Problems
[2] Strain state at a point of a plane-strain structure.
X
Y
Z
ε
Y
ε
X
γ
XY
ε
X
ε
Y
γ
XY
• Plane strain condition:
ε
Z= 0, γ
ZX= 0, γ
ZY= 0
• The Hook's law becomes
σX= E
(1+ ν )(1− 2ν )(1−ν )ε
X+ νε
Y⎡⎣ ⎤⎦
σY= E
(1+ ν )(1− 2ν )(1−ν )ε
Y+ νε
X⎡⎣ ⎤⎦
σZ= E
(1+ ν )(1− 2ν )νε
X+ νε
Y⎡⎣ ⎤⎦
τXY
= GγXY
, τYZ
= 0, τZX
= 0
• A problem may assume the 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
ε
R
ε
R
ε
Z
ε
Z
εθ
εθ
γ
RZ γ
RZ
σ
R
σ
R
σ
Z
σ
Z
σθ
σθ
τ
RZ τ
RZ
[1] Strain state at a point of a
axisymmetric structure.
[2] Stress state at a point of a
axisymmetric structure.
Axisymmetric Problems
• If the geometry, supports, and
loading of a structure are
axisymmetric about the Z-axis,
then all response quantities are
independent of θ coordinate.
• In such a case,
γ θR
= 0, γ θZ= 0
τθR
= 0, τθZ= 0
• both σθ andεθ are generally not
zero. They are termed hoop
stress and hoop strain respectively.
Chapter 3 2D Simulations Section 3.3 More Details 10
Mechanical GUI
• Pull-down Menus
and Toolbars
• Outline of Project
Tree
• Details View
• Geometry
• Graph
• Tabular Data
• Status Bar
• Separators
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 won't be
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-Defined 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 3.4Spur Gears
Problem Description
[2] And the bending stress here.
[1] What we are concerned most is 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
100 100
100
50
R15
50 kN 50 kN
Section 3.5Filleted Bar
Problem Description
[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
Displacement field 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.
[4] By default, stresses are averaged on the nodes, and the stress field is recalculated. That
way, the stress field 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 finer 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
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
Dis
plac
emen
t (m
m)
Number of Nodes
Part C. Finite Element Convergence
[1] Quadrilateral element.
[2] Triangular element.
[3] Increasing nodes.
Chapter 3 2D Simulations Section 3.5 Filleted Bar 22
Part D. Stress Concentration
[1] To accurately evaluate the
concentrated stress, finer mesh is needed,
particularly around the corner.
[2] Stress concentration.
Chapter 3 2D Simulations Section 3.5 Filleted Bar 23
Part E. Stress Sigularity
The stress in this zero-radius fillet is 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.