Introduction to Structural Mechanics Module

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VERSION 4.
3
Introduction to
Structural Mechanics Module
C o n t a c t I n f o r ma t i o n
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Part No. CM021105
I n t r o d u c t i o n t o t h e S t r u c t u r a l Me c h a n i c s Mo d u l e

 1998–2012 COMSOL
Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.
This Documentation and the Programs described herein are furnished under the COMSOL Software License
Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agree
-
ment.
COMSOL, COMSOL Desktop, COMSOL Multiphysics, and LiveLink are registered trademarks or trade
-
marks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their
respective holders.
Version:May 2012 COMSOL 4.3
 3
Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The Structural Mechanics Module Interfaces . . . . . . . . 8
Physics List by Space Dimension and Study Type . . . 12
Model Examples in this Guide . . . . . . . . . . . . . . . . . . . 13
Opening the Model Library. . . . . . . . . . . . . . . . . . . . . . 14
The Fundamentals: A Static Linear Analysis. . . . . . . . . . . 16
Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Modeling Techniques for Structural Mechanics . . . . . 39
Including Initial Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Modeling Thermal Expansion . . . . . . . . . . . . . . . . . . . . . . 45
4 
Introduction
 5
Introduction
The Structural Mechanics Module is tailor-made to model and simulate applications
and designs in the fields of structural and solid mechanics. Engineers and scientists
use it to design new structures and devices and to study the performance of existing
structures.
This module can model static and dynamic analyses in 2D, 2D axisymmetry, and 3D
coordinate systems for solids, shells (3D), plates (2D), trusses (2D, 3D), and beams
(2D, 3D). The material models include linear descriptions, such as linear viscoelastic
material models. Other capabilities are for thermal stress, geometrical nonlinearities
(large deformations), and structural contact.
Figure 1: Von Mises stresses caused by thermal expansion in a turbine stator model. From the Heat
Transfer Module Model Library: Thermal Stress Analysis of a Turbine Stator Blade (turbine_stator). This
model uses both the Structural Mechanics and Heat Transfer Modules.
The Structural Mechanics physics interfaces are the backbone of the module. These
have predefined formulations for the capabilities as described above. This guide gives
an overview of these interfaces as well as examples of the modeling procedures used
with these interfaces.
The Applications
Simulations in structural mechanics are used in a wide range of applications—from
the microscale in MEMS components to the geomechanics scale of civil engineering
6 
Introduction
structures. In addition, these types of simulations are also frequently used to study
the behavior of existing structures from microscopic biostructures to glaciers.
Structural mechanics was the first engineering field to use the concept of finite
elements as a standard tool. Over time, these verifiable and validated formulations
have been developed and are available for a wide range of materials. This also implies
that simulations can often replace experimental measurements. For example, finite
element simulations are used extensively in safety-critical applications within the
aerospace and nuclear industries.
A traditional use of structural analysis is shown below. The device being studied is a
pipe with a bolted flange, with the purpose of the study being two-fold: to estimate
the stress in the pipe and to evaluate the performance of the bolted joint.
Figure

2

shows the deformation (exaggerated) and the von Mises stresses in the pipe.
Figure 2: The deformation (exaggerated) and the von Mises stresses in the pipe. From the Structural
Mechanics Module Model Library: Prestressed Bolts in a Tube Connection (tube_connection). This model
also requires the CAD Import Module.
A less traditional application of mechanical design for a MEMS device is shown in
Figure

3
. The microactuator is subjected to controlled thermal expansion by applying
a current through parts of the structure, which induces Joule heating of the parts.
The thermal expansion causes the actuator to deflect. The simulation predicts the
deflection as a function of the operating conditions for different designs. In addition,
Introduction
 7
the simulation also reveals the limitations of the design because the device will not
work properly if the legs of the actuator make contact along the free faces.
Figure 3: The total displacement in a MEMS device. From the MEMS Module Model Library: Thermal
Actuator (thermal_actuator_tem).
Structural analysis is also important to areas outside the field of traditional structural
engineering, for example in the bioscience field.
Figure

4
shows the results of a
simulation of part of a vascular system in a young child. The purpose of the simulation
is to study what happens when surgery is performed on a child with a malformed
aorta. The aorta and its ramified blood vessels are embedded in biological tissue,
which is modeled using hyperelastic materials. The pressure from the moving fluid is
applied as a face load in the structural analysis. The Structural Mechanics Module
includes the Fluid-Structure Interaction interface, which is a predefined multiphysics
interface dedicated to these types of studies.
8 
Introduction
Figure 4: Displacements in the blood vessel using a hyperelastic model. From the Structural Mechanics
Module Model Library: Fluid-Structure Interaction in a Network of Blood Vessels (blood_vessel).
The Structural Mechanics Module Interfaces
The figure below shows the physics interfaces in the COMSOL Model Wizard and
included with this license. In addition to the structural mechanics capabilities, the
Introduction
 9
module also has substantial multiphysics capabilities, such as AC/DC, acoustics, CFD,
heat transfer, fluid-structural interactions, Joule heating, and piezoelectricity.
Figure 5: The 3D model physics list available with the Structural Mechanics Module. The Plate interface
is not shown but is available in 2D.
A short description of the Structural Mechanics interfaces is described next.
S
OL I D

ME CHANI CS
The Solid Mechanics interface (
) defines the quantities and features for stress
analysis and general linear and nonlinear solid mechanics, solving for the
displacements. The Linear Elastic Material node is the default material model. Other
material models are generic hyperelastic and linear viscoelastic material models. In
addition, the elastic material model can be extended with plasticity, thermal
expansion, damping, and initial stress and strain features. The description of elastic
materials in the module includes isotropic, orthotropic, and fully anisotropic
10 
Introduction
materials. A number of preset study types are available for solid mechanics as shown
below. Also see
“Physics List by Space Dimension and Study Type” on page 12
.
The Linear Viscoelastic Material node describes materials that exhibit both elastic
and viscous behavior when they deform. In the module, this description is based on
the generalized Maxwell model that describes the viscoelastic behavior as a series of
spring-dashpot pairs.
Geomechanics and Nonlinear Structural Materials
There are two additional modules available to enhance the Solid Mechanics
interface—the Geomechanics Module and the Nonlinear Structural Materials
Module. With the addition of the Geomechanics Module, you can also add Soil
Plasticity, Concrete, and Rocks features to the interface. With the Nonlinear
Structural Materials Module, add Hyperelastic Material, Plasticity, Creep, and
Viscoplasticity nodes.
S
HE L L S

AND
P
L ATE S
The Shell interface (
) is intended for the structural analysis of thin-walled
structures. The formulation used in the Shell interface is a Mindlin-Reissner type,
which means that transverse shear deformations are accounted for, and it can
therefore be used for rather thick shells. It is also possible to prescribe an offset in
the direction normal to a selected surface. The Shell interface also includes other
features such as damping, thermal expansion, and initial stresses and strains. The
preset studies available are the same as for the Solid Mechanics interface.
The Plate interface (
) is the 2D analogy to the 3D Shell interface. Plates are similar
to shells but act in a single plane and usually only with out-of-plane loads. The
formulation and features for this physics interface are similar to the ones for the Shell
interface.
Introduction
 11
B
E AMS
The Beam interface (
) is intended for the modeling of slender structures (beams)
that can be fully described by cross-section properties, such as area and moments of
inertia. The Beam interface defines stresses and strains using Hermitian elements and
Euler-Bernoulli theory. Beam elements are used to model frame structures, both
planar and three-dimensional. It is also suitable for modeling reinforcements of solid
and shell structures. The Beam interface includes a library for rectangular, box,
circular, pipe, H-profile, U-profile, and T-Profile beam sections. Additional features
include damping, thermal expansion, and initial stresses and strains. The preset
studies for this physics interface are almost the same as for the Solid Mechanics
interface, with two exceptions—it does not include the Linear Buckling or
Prestressed study types.
T
RUS S E S
The Truss interface (
) can be used to model slender structures that can only
sustain axial forces. Trusses are modeled using Lagrange shape functions, which allow
specification of small strains as well as Green-Lagrange strains for large deformations.
Examples of truss structures are truss works with straight edges and cables exposed
to gravity forces (sagging cables). Additional features include damping, thermal
expansion, and initial stresses and strains. The preset studies for this physics interface
are the same as for the Solid Mechanics interface.
M
E MB R ANE S
The Membrane interface (
) is used for membranes, which can be considered as
plane stress elements in 3D with a possibility to deform both in the in-plane and
out-of-plane directions. The difference between a shell and a membrane is that the
membrane does not have any bending stiffness. When a membrane is used by itself,
a tensile prestress is necessary in order to avoid singularity, since a membrane with
no stress or compressive stress has no transverse stiffness.
O
THE R
S
TRUCTUR AL
M
E CHANI CS
I
NTE RF ACE S
The Thermal Stress interface (
) is similar to the Solid Mechanics interface with the
addition of a thermal linear elastic material. It can be used in combination with the
various forms of Heat Transfer interfaces to couple the temperature field to a
structure’s (material) expansion.
The Joule Heating and Thermal Expansion multiphysics interface (
) combines the
Thermal Stress interface with the Joule Heating interface. It describes the conduction
12 
Introduction
of electric current in a structure, the subsequent electric heating caused by the
Ohmic losses in the structure, and the thermal stresses induced by the temperature
field.
The Piezoelectric Devices interface (
) combines the Solid Mechanics and
Electrostatics interfaces to model piezoelectric materials. The piezoelectric coupling
can be in stress-charge or strain-charge form. All solid mechanics and electrostatics
functionalities are also accessible through this interface, for example, to model the
surrounding linear elastic solids or air domains.
F
L UI D
F
L OW
The Fluid-Structure Interaction (FSI) interface (
), found under the Fluid Flow
branch in the Model Wizard, combines fluid flow with solid mechanics to capture the
interaction between the fluid and the solid structure. A Solid Mechanics interface and
a Laminar Flow interface model the solid and the fluid, respectively. The FSI couplings
appear on the boundaries between the fluid and the solid. The interface uses an
arbitrary Lagrangian-Eulerian (ALE) method to combine the fluid flow formulated
using an Eulerian description and a spatial frame with solid mechanics formulated
using a Lagrangian description and a material (reference) frame.
Physics List by Space Dimension and Study Type
The table below lists the interfaces available specifically with this module in addition
to the COMSOL Multiphysics basic license.
PHYSICS
ICON
TAG
SPACE DIMENSION
PRESET STUDIES
Fluid Flow
Fluid-Structure Interaction
fsi
3D, 2D, 2D
axisymmetric
stationary; time dependent
Structural Mechanics
Solid Mechanics*
solid
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency; prestressed
analysis, eigenfrequency; time
dependent; time dependent modal;
frequency domain; frequency-domain
modal; prestressed analysis, frequency
domain; linear buckling
Thermal Stress
ts
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency; frequency
domain; time dependent
Introduction
 13
Model Examples in this Guide
In this introduction guide, you use several versions of the bracket model to learn how
to set up a static analysis, perform a parametric study for analyzing a varying load,
include initial strains, and model thermal expansion.
Shell
shell
3D
stationary; eigenfrequency; prestressed
analysis, eigenfrequency; time
dependent; time dependent modal;
frequency domain; frequency-domain
modal; prestressed analysis, frequency
domain; linear buckling
Plate
plate
2D
stationary; eigenfrequency; prestressed
analysis, eigenfrequency; time
dependent; time dependent modal;
frequency domain; frequency-domain
modal; prestressed analysis, frequency
domain; linear buckling
Beam
beam
3D, 2D
stationary; eigenfrequency; frequency
domain; frequency-domain modal;
time dependent; time dependent
modal
Truss
truss
3D, 2D
stationary; eigenfrequency; prestressed
analysis, eigenfrequency; time
dependent; time dependent modal;
frequency domain; frequency-domain
modal; prestressed analysis, frequency
domain; linear buckling
Membrane
mem
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency; prestressed
analysis, eigenfrequency; time
dependent; time dependent modal;
frequency domain; frequency-domain
modal; prestressed analysis, frequency
domain
Joule Heating and Thermal
Expansion
tem
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency; time
dependent
Piezoelectric Devices
pzd
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency; time
dependent; time-dependent modal;
frequency domain; frequency domain
modal
* This is an enhanced interface, which is included with the base COMSOL package but has added
functionality for this module.
PHYSICS ICON TAG SPACE DIMENSION PRESET STUDIES
14 
Introduction
In the Structural Mechanics Module Model Library there are several more detailed
examples that use the bracket model to extend this tutorial. These models show, for
example, how to:

Model with special features such as rigid connectors and spring conditions
• Model thin structures using the Shell interface
• Perform structural dynamics analysis using eigenfrequency extraction, transient
analysis, and frequency response analysis
• Take effects of geometrical nonlinearity into account
• Use nonlinear materials

Perform a contact analysis
Opening the Model Library
To open any Structural Mechanics Module models, select
ViewModel Library


from
the main menu in COMSOL Multiphysics. In the Model Library window that opens,
expand the Structural Mechanics Module folder and browse or search the contents
of the folders. Click
Open Model and PDF
to open both the model in COMSOL
Multiphysics and a PDF to read background theory including the step-by-step
instructions to build it.
The Model Library is updated on a regular basis by COMSOL in order to add new
models and to improve existing models. Choose
ViewModel Library Update
(
) to
update the model library.
Introduction
 15
The MPH-files in the COMSOL model libraries can have two formats—Full MPH-files
or Compact MPH-files.

Full MPH-files, including all meshes and solutions. In the Model Library these
models appear with the
icon. If the MPH-file’s size exceeds 25MB, a tip with
the text “Large file” and the file size appears when you position the cursor at the
model’s node in the Model Library tree.

Compact MPH-files with all settings for the model but without built meshes and
solution data to save space on the DVD (a few MPH-files have no solutions for
other reasons). You can open these models to study the settings and to mesh and
re-solve the models. It is also possible to download the full versions—with meshes
and solutions—of most of these models through Model Library Update. In the
Model Library these models appear with the
icon. If you position the cursor
at a compact model in the Model Library window, a No solutions stored message
appears. If a full MPH-file is available for download, the corresponding node’s
context menu includes a Model Library Update item.
16 
The Fundamentals: A Static Linear Analysis
The Fundamentals: A Static Linear Analysis
This section summarizes the fundamentals to model structural mechanics problems
and how to apply them in COMSOL Multiphysics and the Structural Mechanics
Module. This includes creating a geometry as well as defining material properties and
boundary constraints and conditions. After the solution is computed, you will learn
how to analyze results and check the reaction forces.
The model used in this guide is an assembly of a bracket and its mounting bolts, which
are all made of steel. This type of bracket can be used to install an actuator that is
mounted on a pin placed between the two holes in the bracket arms. The geometry
is shown in
Figure

6
.
Figure 6: The geometry for the bracket assembly. The bracket is gray and the bolts are blue.
In this analysis, the mounting bolts are assumed to be fixed and securely bonded to
the bracket. To model the external load from the pin, specify a surface load
p
with a
sinusoidal distribution on the inner surfaces of the two holes:
where
P
0
is the peak pressure in the main direction of the load as defined by
, the
angle from the
y
-axis. To model the external load from the pin, specify a surface load
with a sinusoidal distribution on the inner surfaces of the two holes. The direction of
the load is given local coordinate system controlled by a parameter
theta0
. The type
of load distribution is shown in
Figure

7
:
p P
0
 
0
–  0  
0
–  sin=

0
The Fundamentals: A Static Linear Analysis
 17
Figure 7: The load distribution of the bracket.
MODEL WIZARD
The first step to build a model is to open COMSOL and then specify the type of
analysis you want to do—in this case, a stationary, solid mechanics analysis.
1 Open COMSOL Multiphysics. In the
Model Wizard
the
Space Dimension
defaults to
3D
.
Click
Next
.
2 On the
Add Physics
page, under
Structural Mechanics,
double-click
Solid Mechanics
(solid)
to add it to the
Selected physics
list. You can also click the
Add Selected

button or right-click and choose
Add Selected
.
3 Click
Next
.
4 On the
Studies
window under
Preset Studies
, click
Stationary
.
5 Click
Finish
.
18 
The Fundamentals: A Static Linear Analysis
GLOBAL DEFINITIONS - PARAMETERS
It is good modeling practice to gather the constants and parameters in one place so
that you can change and vary them easily. In this model, the parameter set includes
theta0
, the angle with which the maximum pressure from the pin is applied to the
holes—a parameter you may want to vary later on. In general, these constants and
parameters are valid throughout the model, and are therefore denoted as global
parameters.
These parameters are defined: the load orientation angle
theta0
, the maximum load
value
P0
, and the
y
- and
z
-coordinate of the center of the bracket holes
y0
and
z0
.
1 Right-click
Global Definitions
and select
Parameters
. In the
Parameters
settings
window under the
Parameters
section, enter
theta0
in the
Name
column and
180[deg]

in the
Expression
column.
2 Fill in the
Parameters
table with the other parameters and expressions as displayed in
the figure below.
IMPORTING THE GEOMETRY
The next step is to create your geometry, which also can be imported from an
external program. COMSOL Multiphysics supports a multitude of CAD programs
and file formats. In this example, import a file in the COMSOL Multiphysics geometry
file format (.mphbin). The file contains the assembly of both the bracket and
mounting bolts.
Note: The location of the files used in this exercise varies based on your installation. For
example, if the installation is on your hard drive, the file path might be similar to
C:\Program Files\COMSOL43\models\
.
1 Under the
Model 1 (mod1)
node, right-click
Geometry 1
and select
Import
.
The Fundamentals: A Static Linear Analysis
 19
2 In the
Import
settings window under
Import
, click
Browse
. Then browse to the folder
Structural_Mechanics_ModuleTutorial_Models
under the COMSOL installation directory
and double-click the file
bracket.mphbin
.
3 Click
Import
.
Form Union (fin)
1 In the
Model Builder
, right-click
Form Union (fin)
and select
Build Selected
.
The
Finalize
feature node determines how the parts of the assembly are considered
in the analysis. By using the default setting,
Form a union
, the mounting bolts are
automatically bonded to the bracket and the internal boundary assumes continuity.
If
Form an assembly
is selected instead, the mounting bolts would not be connected to
the bracket, and the structural contact between the bolts and the bracket could be
modeled. This first example assumes that the mounting bolt is bonded to the
bracket.
DEFINITIONS - SELECTIONS
Independent of whether the model is treated as a single entity or as an assembly, you
still may need to access different parts of it for definitions such as multiple
specifications of similar boundary conditions or solid domain specifications. The use
of arbitrary box selections is one way to do this. In this example, one selection
contains the bolt domains, while a second selection contains the area around the
20 
The Fundamentals: A Static Linear Analysis
load-bearing boundaries of the holes. The box selection is independent of the
geometry, and allows you to change the geometry topology while still keeping the
desired selection.
1 Under the
Model 1 (mod1)
node, right-click
Definitions
and choose
SelectionsBox
.
2 In the
Box
settings window enter:
-
0
for
x minimum
,
y minimum
, and
z minimum
-
0.2
as
x maximum
-
0.1
as
y maximum

-
0.11
as
z maximum
3 Under
Output Entities
in the
Include entity if
list,
select
Entity inside box
.
The Fundamentals: A Static Linear Analysis
 21
4 Click the
Wireframe Rendering
button on the
Graphics
toolbar to display the
selection as in the figure:
5 Add a second box selection, as described in
step 1 but with different values.
6 In the
Box
settings window, select
Boundary
from
the
Geometric entity level
list.
7 Under
Box Limits
, set the values as in the figure
to the right and below. In the:
-
x minimum
field enter
-0.01
-
x

maximum
field enter
0.22
-
y minimum
field enter
-0.23

-
y maximum
field enter
-0.17
-
z minimum
field enter
-0.02

-
z maximum
field enter
0.08
8 Under
Output Entities
, in the
Include entity if
list
select
Entity inside box
.
22 
The Fundamentals: A Static Linear Analysis
View the selection in the
Graphics
window. It should match this figure.
DEFINITIONS - VARIABLES AND COORDINATE SYSTEMS
This section defines the Variables and adds a Rotated Coordinate System.
Variables
When it comes to the selection for the load-bearing holes, you can take advantage
of local expressions, which are available only on selected entities in this selection.
Here, you want to define expressions for the load applied to the load-bearing holes
by first evaluating the radial angle based on the position along the global
z
-coordinate. Using a sinusoidal function, the second expression is then defined by
the pressure distribution.
1 In the
Model Builder
under
Model 1
, right-click
Definitions
and select
Variables
.
2 In the
Variables
settings window, enter
angle
in the
Name
column and
atan2((y-y0),(z-z0))
in the
Expression
column. Add a second variable
pressure
The Fundamentals: A Static Linear Analysis
 23
with the expression
P0*sin(angle-theta0)*(sin(angle-theta0)>0).
The last part
of the expression ensures that the load is applied only to one half of the hole.
Coordinate Systems
Create a rotated coordinate system, which defines the orientation of the load applied
to the bracket holes.
1 In the
Model Builder
under
Model 1
, right-click
Definitions
and select
Coordinate
SystemsRotated System
.
24 
The Fundamentals: A Static Linear Analysis
2 In the
Rotated System
settings window, enter
-theta0
as the value for beta.
MATERIALS
COMSOL Multiphysics is equipped with built-in material properties for a number of
common materials. Here, choose structural steel as the material for both the bracket
and the bolts. The material is automatically assigned to all domains.
1 Select
ViewMaterial Browser
from the main menu.
2 In the
Material Browser
under
MaterialsBuilt-In
,

right-click
Structural steel

and select
Add Material to Model
.
The Fundamentals: A Static Linear Analysis
 25
SOLID MECHANICS
Now define the physics for the bracket assembly. Initially, the analysis was specified
to be stationary using the classical equations associated with solid mechanics. In this
step, you are more specific concerning the different modeling domains.
By default, the Solid Mechanics interface assumes that the participating material
models are linear elastic, which is appropriate for this example. All that is left to do
is to set the constraints and loads acting on the structure, making use of the box
selections defined previously (see
“Definitions - Selections” on page 19
).
Fixed Constraint
Assume that the bolts are stiff and that the displacements are perfectly constrained.
1 In the
Model Builder
right-click
Solid Mechanics (solid)
and select
MoreFixed
Constraint
from the domain level in the context menu.
2 In the
Fixed Constraint
settings window under
Domain Selection
, select
Box 1
from the
Selection
list.
26 
The Fundamentals: A Static Linear Analysis
Boundary Load
Apply a boundary load to the bracket holes.
1 Right-click
Solid Mechanics (solid)
and from
the boundary level of the context menu, select
Boundary Load
.
2 In the
Boundary Load
settings window under
Boundary selection
, select
Box2
from the
Selection

list.
Use the rotated coordinate system to have the
load orientation change with a value of
theta0
.
3 Under
Coordinate System Selection
, select
Rotated
System 2
.
4 Under
Force
, enter
pressure
for the
x2

component.
The
Graphics
window shows the domain
selection including symbols to describe the type
of load applied to the selection.
The symbols indicate only the type of settings
applied to the model and not the magnitude. If
you are interested in visualizing the actual
applied pressure distribution, a solution must be computed first.
The Fundamentals: A Static Linear Analysis
 27
Note: To turn the physics symbols on, from the main menu select
OptionsPreferences
and click the
Graphics
section. Click to select the
Show physics symbols
check box. Click
Apply
and
OK
. Click anywhere in the
Model Builder
, then click the node again. The symbols
display in the geometry.
STUDY
You are now ready to compute the solution.
1 In the
Model Builder
, right-click
Study 1
and
select
Compute
.
The
Study
node automatically defines a solver
sequence for the simulation based on the selected
physics (solid mechanics) and study type
(stationary). Since a mesh is required, and it has
not been created yet, the Study node
automatically generates this at the same time as the solver sequence.
Note: In general, do not only rely on the default mesh settings. For most real problems
suitable meshing parameters should be set up under the
Mesh
node.
28 
The Fundamentals: A Static Linear Analysis
RESULTS
The default plot displays the von Mises stress distribution together with an
exaggerated (automatically scaled) picture of the deformation. As expected, the high
stress values are located around the holes and in the vicinity of the mounting bolts.
The maximum von Mises stress remains below the yield stress value for structural
steel (260

MPa) which validates the choice of a linear elastic material to analyze this
structure.
Now, add an arrow plot to this plot group to display the applied pressure.
1 In the
Model Builder
under
Results
, right-click
Stress (solid)
and select
Arrow Surface
.
2 In the
Arrow Surface
settings window under
Expression
, click
Replace Expression
.
Select
Solid MechanicsLoadLoad (Spatial)(solid.FperAreax,...,solid.FperAreaz)
from the
list.
3 Under
Coloring and Style
, enter
3000
in the
Number of arrows
field.
4 Click the
Plot
button .
From the
Graphics
window you can now check that the applied load is as intended.
Note that the arrows as a default are plotted on the undeformed structure.
The Fundamentals: A Static Linear Analysis
 29
Displacements
To study the displacement, add a second 3D plot group to the Results node.
1 Right-click the
3D Plot Group
node and select
Surface
. Click the
Plot
button .
30 
The Fundamentals: A Static Linear Analysis
In the figure below you can see that the bracket base remains fixed while only the
arms are deformed. The maximum total displacement is about 27


m, which is in
agreement with the assumption of small deformations.
Principal Stresses
Create another plot to display the principal stresses in the bracket.
The Fundamentals: A Static Linear Analysis
 31
1 Right click
Results
to add a
3D Plot Group

then right-click the node and select
More
PlotsPrincipal Stress Volume
.
2 In the
Principal Stress Volume
settings window
under
Positioning
, find the
X grid points

subsection. From the
Entry method
list, select
Coordinates
.
3 In the
Coordinates
field, enter
4e-3 15e-3 30e-3 1e-2
. A space between
each of the values is required.
4 For both
Y grid points
and
Z grid points
, enter
15
in the
Points
field.
5 Expand the
Coloring and Style
section and
select
Scale factor
. In the field enter
1.5e-9
.
6 Click the
Plot
button .
As the load is oriented along the negative
y

direction, the principal stress plot shows
tensile stress in the arm of the bracket.
32 
The Fundamentals: A Static Linear Analysis
Note: In the plot on the previous page, the red arrows show the largest principal stress,
the blue arrows show the smallest principal stress, and the green arrows show the
intermediate principal stress. The order is ‘RGB’, just as for the coordinate system
arrows. To view the green arrows, zoom into the red arrows in the
Graphics
window.
Reaction Forces
A final check is to compute the total reaction force along the
x
,
y
, and
z

directions.
Because the mounting bolts are fully constrained, use a volume integration.
1 In the
Model Builder
under
Results
, right-click

Derived Values
and select
IntegrationVolume
Integration
.
2 In the
Volume Integration
settings window
under
Selection
, select
Box 1
from the
Selection

list.
3 Under
Expression
, click
Replace Expression
.
Select
Solid MechanicsReactionsReaction force
(Spatial)Reaction force, x component (solid.RFx)

from the list (or enter
solid.RFx
in the
Expression
field). Click the
Evaluate
button .
4 Click
Replace Expression
. Select
Solid
MechanicsReactionsReaction
force(Spatial)Reaction force, y component
(solid.RFy)
from the list (or enter
solid.RFy
in
the
Expression
field). Click the
Evaluate

button .
The Fundamentals: A Static Linear Analysis
 33
5 Click
Replace Expression
. Select
Solid MechanicsReactionsReaction
force(Spatial)Reaction force, z component (solid.RFz)
from the list (or enter
solid.RFz

in the
Expression
field). Click the
Evaluate
button .
The results are available in the table located in the lower-right side of the desktop.
Or select
ViewResults

from the main menu to open the
Results
window. You can
verify that the reaction force along the
x

and
z

direction is zero except for small
numerical errors.
34 
Parametric Study
Parametric Study
In the previous section, a bracket loaded by an actuator was analyzed. This section
extends this analysis to study the effect of the actuator’s position. This is equivalent
to changing the direction of the applied load. Use the load-angle parameter that the
model already contains to set up a parametric study.
COMSOL Multiphysics has two ways to perform parametric studies—using either a
Parametric Sweep
node or the continuation feature from the
Stationary Solver
node. In
this example, both methods are applicable and the continuation feature in the
Solver

node is used. This feature uses the solution from the previous parameter as an initial
guess to calculate the current parameter value, and is the preferred option for
nonlinear problems. Using the
Parametric Sweep
node is preferable for applications
requiring geometric parametrization.
LOADING THE MODEL
Either continue working on the existing model or open a saved version of the model
from the Model Library. See
“Opening the Model Library” on page 14
to browse to
the
Structural_Mechanics_ModuleTutorial_Models
folder. Double-click to open
bracket_static.mph
.
EXTENDING THE STUDY WITH A PARAMETRIC
CONTINUATION
Parametric studies can be set up from scratch or, as in this example, added to an
existing study.
1 In the
Model Builder
, expand the
Study 1
node and click
Step 1: Stationary
.
Parametric Study
 35
2 In the
Stationary
settings window, click to
expand the
Study

Extensions
section and
select the
Continuation
check box.
3 Under
Continuation parameter
click
Add
.
4 In the
Add
dialog box, in the
Continuation
parameter
list, select
theta0 (Direction of
Load)
and click
OK
.
5 In the
Stationary
settings window under
Study

Extensions
, click the
Range
button .
6 In the
Range
dialog box, set the
Start
,
Stop
,
and
Step
fields as in the figure to the right
and below:
-
Select
Step
as the
Entry method
.
-
In the
Start
field, enter
0
.
-
In the
Step
field enter
45[deg]
.
-
In the
Stop
field enter
180[deg]
.
Note: Or copy and paste
range(0,180[deg],45[deg])
in the
Parameter Values
field.
7 Click
Add
. The
Study Extensions
section
adds the parameter values.
8 In the
Model Builder
, right-click
Study 1

and select
Compute
.
RESULTS
The default plot shows the solution for the last parameter value (
180[deg]
), which
corresponds to the case study in the previous section (see
“Results” on page 28
).
You can easily change the parameter value to display the plot and then compare
solutions for different load cases.
36 
Parametric Study
Note: Click the
Zoom Extents
button
to view the new default plots.
1 Click the
Stress (solid)
node . Under
Data
, from the
Parameter value (theta0)
list,
select
0
. Click the
Plot
button .
Note: In the case where compression occurs, the deformation of the bracket arms are
in the opposite direction to that in the tensile case. Also, the maximum von Mises stress
value is lower.
2 Click the
Stress (solid)
node again. In the
3D Plot Group
settings window under
Data
,
select a different value from the
Parameter value (theta0)
list, for example
1.570796
.
3 Click the
Plot
button .
Parametric Study
 37
The plot displays the stress distribution for a bending load case. The maximum stress
values are no longer located around the holes but in the region where the arms
connect to the bolt supports.
Reaction Forces
To evaluate the total reaction forces for each parameter, each component of the
reaction force is integrated over the bolt domains. Because no load is acting in the
global
x

direction, the reaction force in this direction is zero.
1 In the
Model Builder
under
Results
, right-click
Derived Values
and select
IntegrationVolume Integration
.
Note: If working with the Model Library file, this adds a second
Volume Integration

node to the
Model Builder
.
2 In the
Volume Integration
settings window under
Selection
, select
Box 1
from the
Selection
list.
3 Under
Expression
, click
Replace Expression
. Select
Solid
MechanicsReactionsReaction force (Spatial)Reaction force, y component (solid.RFy)
from
the list (or enter
solid.RFy
in the
Expression
field and replace the default). Click the
Evaluate
button .
38 
Parametric Study
4 Click
Replace Expression
. Select
Solid MechanicsReactionsReaction force
(Spatial)Reaction force, z component (solid.RFz)
from the list. Click the
Evaluate

button (or enter
solid.RFz
in the
Expression
field and replace previous value).
The results are available in the table located in the lower right side of the desktop.
Or select
ViewResults

from the main menu to open the
Results
window.
The reaction force in the
y

direction is symmetric around 90
o
—negative from 0
o
to
90
o
and positive from 90
o
to 180
o
. The reaction force in the
z

direction has a
maximum at 90
o
.
Plot the Reaction Force
1 In the
Model Builder
under
Results
, click
1D Plot Group 4
.
2 In the
ID Plot Group
settings window, click to expand the
Legend
section. From the
Position
list, select
Lower right
.
3 Expand the
1D Plot Group 4
node and select
Table Graph 1
. Click to expand the
Legends
section. Select the
Show legends
check box. Click the
Plot
button .
Note: You may need to select
Table 2
from the
Table
list under
Data
to produce this plot.
Parametric Study
 39
Modeling Techniques for Structural Mechanics
The next tutorials build on the previous examples to demonstrate the following
structural mechanics modeling techniques: How can I model thermal stress? How can
I add an initial strain to the simulation? To get started go to
“Including Initial Strain”
on page 40
or
“Modeling Thermal Expansion” on page 45
.
40 
Including Initial Strain
Including Initial Strain
Initial stresses and strains can be specified as a subfeature to a material model. You
can define a stress/strain distribution with constant values or as an expression which
can, for example, be space- or time-dependent. The initial stresses and strains can
also be results from another study, or even from another physics interface in the
same study.
In this example, you add a pin geometry to the bracket assembly. Then you specify
an initial strain to simulate that the pin is prestrained in the axial direction, and
investigate how it affects the assembly.
LOAD MODEL AND CHOOSE THE STUDY
1 See “Opening the Model Library” on page 14 then browse to the
Structural_Mechanics_ModuleTutorial_Models
folder and double-click to open
bracket_basic.mph
.
2 In the
Model Builder
, right-click
Model 1 (mod1)
and select
Add Physics
.
3 Under
Structural Mechanics
, double-click
Solid Mechanics (solid)
to add it to the
Selected physics
list. Click
Next
.
4 In the
Studies
window under
Preset Studies
, click
Stationary
.
5 Click
Finish
.
DEFINITIONS - PARAMETERS
Parameters defining the original length of the pin,
L0
, and the current length,
L
, are
used to calculate the initial strain. The prestrain is the only load acting on the
structure, which is fixed by fully constrained mounting bolts.
In the
Parameters
table, define a strain value that corresponds to a reduction of the
pin length from 216 mm to 215 mm.
1 Expand the
Global Definitions
node and click
Parameters
.
Including Initial Strain
 41
2 In the
Parameters
settings window in the
Parameters
table, add the new parameters
L
,
L0
, and
InitStrain
as in the figure.
GEOMETRY
Next, the pin geometry is added to the bracket assembly. This is done by importing
it into the existing geometry.
Note: The location of the files used in this exercise varies based on your installation. For
example, if the installation is on your hard drive, the file path might be similar to
C:\Program Files\COMSOL43\models\
.
1 In the
Model Builder
under
Model 1 (mod1)
, right-click
Geometry 1
and select
Import
. An
Import 2
node is added to the
Model Builder
.
2 In the
Import
settings window under
Import
, click
Browse
.
3 Browse to the model folder
Structural_Mechanics_ModuleTutorial_Models
and
double-click the file
bracket_pin.mphbin
.
4 Click
Import
.
42 
Including Initial Strain
ADDING INITIAL STRAIN AND OTHER PHYSICS SETTINGS
The initial strain is specified under the
Linear Elastic Material
node.
1 In the
Model Builder
under
Solid Mechanics (solid)
, right-click
Linear Elastic Material 1

then select
Initial Stress and Strain
. The ‘D’ in the upper left corner of the node
means that it is a default node.
The prestrain direction is the axial direction of the bolt, which coincides with the
global
x

direction.
2 In the
Initial Stress and Strain
settings window to the right of the
Selection
list, click the
Clear Selection
button . Then select only Domain

3
.
Including Initial Strain
 43
3 Under the
Initial Stress and Strain
section, enter
InitStrain
in the first component of the

0

table.
Fixed Constraints
1 Right-click
Solid Mechanics (solid)
and at the domain level, select
MoreFixed
Constraint
.
2 In the
Fixed Constraint
settings window under
Domain Selection
, select
Box 1
from the
Selection
list.
COMPUTE AND DISPLAY RESULTS
1 In the
Model Builder
, right-click
Study 1
and select
Compute
.
The default plot shows the von Mises stress in the bracket.
44 
Including Initial Strain
The results show how the pin compresses the bracket arms, and that the greatest
stresses are found in the region where the bracket arms are joined to the bolt
supports.
Note: The structural analysis from the previous tutorial is not included here.
You can also plot the
x
-component of the strain tensor so as to visualize the total
strain in the structure. As the pin is stiff in comparison to the bracket, the total strain
in the pin is almost the same as the initial strain given in the example.
1 Right-click
Results
and select
3D Plot Group
.
2 Right-click
3D Plot Group 2
and select
Surface
to add a second 3D plot group with
a surface plot.
3 In the
Surface
settings window under
Expression
, enter
solid.eX
in the
Expression
field).
4 Click the
Plot
button .
Modeling Thermal Expansion
 45
Modeling Thermal Expansion
In this example, a thermal field is applied to the bracket and pin assembly and the
thermal stresses are calculated.
LOAD THE MODEL AND CHOOSE THE STUDY
1 As in the previous tutorial, see “Opening the Model Library” on page 14 then
browse to the
Structural_Mechanics_ModuleTutorial_Models
folder and double-click to
open
bracket_basic.mph
.
Note: You can also select
FileRevert to Saved
to open the original file or select
File
and
choose the file from the history list.
2 In the
Model Builder
, right-click
Model 1 (mod1)
and select
Add Physics
.
3 Under
Structural Mechanics,
double-click
Thermal Stress (ts)
to add it to the
Selected
physics
list. Click
Next
.
4 In the
Studies
window under
Preset Studies
, click
Stationary
.
5 Click
Finish
.
THERMAL STRESS (TS)
COMSOL Multiphysics contains physics interfaces for structural analysis as well as
thermal analysis. You can define the analyses separately and then simulate the
thermal-structure interaction by coupling them using the appropriate variables and
terms in the structural analysis equations. If you were using a Solid Mechanics
interface, adding thermal expansion to a material model is as easy as it was to add
the initial strain—you would add a Thermal Expansion node to the Linear Elastic
Material Model.
However, an easier option is to utilize a pre-defined physics interface, the Thermal
Stress interface. This interface contains both the structural and thermal equations
along with the coupling, which is included by default.
46 
Modeling Thermal Expansion
1 In the
Model Builder
, expand the
Thermal Stress
node. Click the
Thermal Linear Elastic
Material 1
node to see that both structural and heat conduction material
properties can be specified in the settings window.
Note: The thermal expansion requires both a thermal expansion coefficient and a strain
reference temperature, which is the temperature reference at which there are no
thermal strains. As a material (structural steel) has already been defined for the model,
you do not need to modify anything here.
2 In the
Model Builder
, right-click
Thermal Stress (ts)
and select
Solid MechanicsFixed
Constraint
.
3 In the
Fixed Constraint
settings window under
Domain Selection
, select
Box 1
from the
Selection
list.
As the Thermal Stress interface includes a heat balance, the thermal boundary
conditions also must be set. Assume that the mounting bolts are rigid and fully
constrained and that the temperature is maintained at 20
o
C. Also assume that the
arms of the bracket are holding the pin, which itself has a constant temperature of
100
o
C. Finally, assume that heat is lost by convection to the surroundings from the
surfaces of the arms and the bolt support.
4 Right-click
Thermal Stress (ts)
and at the boundary level, select
Heat TransferHeat
Flux
.
5 In the
Heat Flux
settings window under
Boundaries
, select
All boundaries
from the
Selection
list.
Modeling Thermal Expansion
 47
6 Under
Heat Flux
click the
Inward heat flux
button.
In the
h
field, enter
10
.
7 Right-click
Thermal Stress (ts)
and at the
boundary level, select
Heat
TransferTemperature
.
8 Select boundaries
22
,
25
,
30
,
33
,
46
,
49
,
54
, and
57
only (or use the
Paste Selection
button).
9 Add a second Temperature node. Right-click
Thermal Stress (ts)
and select
Heat TransferTemperature
.
10In the
Temperature
settings window under
Boundary Selection
, select
Box 2
from the
Selection
list.
11Under
Temperature
enter
100[degC]
in the
T
0
field.
COMPUTE AND DISPLAY RESULTS
1 In the
Model Builder
, right-click
Study 1
and select
Compute
.
48 
Modeling Thermal Expansion
Under the
Results
node, two plot groups are automatically added to show the default
results for a structural analysis and a thermal analysis. The first plot group,
Stress (ts)
,
shows the von Mises stresses on a scaled deformed geometry.
You can see how the bracket arms holding the pin are deformed through thermal
expansion and that thermal stresses are developed.
The second default plot group,
Temperature (ts)
, displays the temperature distribution
on a scaled, deformed geometry.
Modeling Thermal Expansion
 49
This concludes this introduction. For additional tutorials using the bracket geometry,
go to the Structural Mechanics Module Model Library and browse the Tutorial
Models folder. Click the
Model PDF
button to open instructions for each of these
models.
As a final step, pick one of the plots to use as a model thumbnail.
1 In the
Model Builder
under
Results
click
3D Plot Group 2
.
2 From the
File
menu, choose
Save Model Thumbnail
.
To view the thumbnail image, click the
Root
node and look under the
Model Thumbnail

section. Make adjustments to the image in the
Graphics
window using the toolbar
buttons until the image is one that is suitable to your purposes.
If you want to explore further, in the Structural Mechanics Module Model Library
there are several more detailed examples that use the bracket model to extend this
tutorial. See
“Model Examples in this Guide” on page 13
for more information.
50 
Modeling Thermal Expansion