COMSOL 4.3b - Introduction to Structural Mechanics Module

8/16/2019 COMSOL 4.3b - Introduction to Structural Mechanics Module

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VERSION 4.4

Introduction to

Structural Mechanics Module


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C o n t a c t I n f o r m a t i o n

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Part number. 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 M e c h a n i c s M o d u l e

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Structural Mechanics Module Physics Interfaces. . . . . . . . . . 5

The Physics List by Space Dimension and Study Type . . . . . . . . . . 8

The Model Libraries Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Model Examples in this Guide and the Model Library . . . . . . . . . . 12

The Fundamentals: A Static Linear Analysis . . . . . . . . . . . . . . . . 13

Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Modeling Techniques for Structural Mechanics. . . . . . . . . . . . . . . . 37

Including Initial Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Modeling Thermal Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


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Introduction | 1

Introduction

The Structural Mechanics Module is tailor-made for modeling and simulatingapplications and designs in the fields of structural and solid mechanics. Engineers

and scientists use it to design new structures and devices, and to study theperformance of existing structures.

This module can model static and dynamic analyses in 2D, 2D axisymmetric, and3D coordinate systems for solids, shells (3D), plates (2D), trusses (2D, 3D),membranes (2D axisymmetry, 3D), and beams (2D, 3D). The material modelsinclude linear descriptions, such as linear elastic and viscoelastic material models.Other capabilities are for thermal stress, geometric nonlinearities (largedeformations), and structural contact.

Figure 1: Von Mises stresses caused by thermal expansion in a turbine stator model. From the HeatTransfer Module Model Library: Thermal Stress Analysis of a Turbine Stator Blade (turbine_stator). Thismodel 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 described above. Thisguide gives an overview of these interfaces as well as examples of the modelingprocedures used in them.

The ApplicationsSimulations in structural mechanics are used in a wide range of applications—fromthe microscale of MEMS components to the geomechanics scale of civil


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2 | Introduction

engineering. These types of simulations are also frequently used to study thebehavior of existing structures, from microscopic biostructures to glaciers.

Structural mechanics was the first engineering field to use the concept of finiteelements as a standard tool. Over time, these verifiable and validated formulationshave been developed and are applicable to a wide range of materials. Simulations

can often replace experimental measurements. For example, finite elementsimulations are used extensively in safety-critical applications within the aerospaceand nuclear industries.

A traditional use of structural analysis tools is depicted below. The device beingstudied is a pipe with a bolted flange, and the purpose of the study is two-fold: toestimate 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 thepipe.

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 modelalso requires the CAD Import Module.

A less traditional application of mechanical design for a MEMS device is shown inFigure 3. The microactuator is subjected to controlled thermal expansion by theapplication of a current through parts of the structure, which induces Jouleheating. The thermal expansion then causes the actuator to deflect and thesimulation predicts the deflection as a function of the operating conditions. The


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Introduction | 3

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 outside of traditional structural engineering,for example in the field of bioscience. Figure 4 shows the results of a simulation of

part of the vascular system in a young child. The purpose of the simulation is tostudy 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.Pressure from the moving fluid is applied as a face load in the structural analysis.The Structural Mechanics Module includes the Fluid-Structure Interactioninterface, which is a predefined multiphysics interface dedicated to these types ofstudies.


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4 | Introduction

Figure 4: Displacements in the blood vessel. From the Structural Mechanics Module Model Library:Fluid-Structure Interaction in a Network of Blood Vessels (blood_vessel).


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The Structural Mechanics Module Physics Interfaces | 5

The Structural Mechanics Module Physics Interfaces

The figure below shows the physics interfaces that are in the COMSOL Model Wizard and included with the Structural Mechanics Module. In addition to the

structural mechanics capabilities, the module also has substantial multiphysicscapabilities, including AC/DC, acoustics, CFD, heat transfer, fluid-structuralinteractions, Joule heating, and piezoelectricity.

Figure 5: The 3D model physics list available with the Structural Mechanics Module. The Beam CrossSection and Plate interfaces are not shown, but are available in 2D. In addition, the Multibody Dynamicsand Fatigue interfaces are also included in the list but require additional licenses.

A short description of the Structural Mechanics physics interfaces follow.

SOLID MECHANICS

The Solid Mechanics interface ( ) defines the quantities and features for stressanalysis and general linear and nonlinear solid mechanics. The Linear ElasticMaterial node is the default material model. The elastic material model can beexpanded with features for thermal expansion, viscoelasticity, damping, and initialstress and strain. The description of elastic materials in the module includesisotropic, orthotropic, and fully anisotropic materials. A number of preset study

types are available for solid mechanics (shown below). Also see “The Physics List


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6 | The Structural Mechanics Module Physics Interfaces

by Space Dimension and Study Type” on page 8.

Geomechanics, Nonlinear Structural Materials, Multibody Dynamics and

Fatigue

There are four additional modules available to enhance the Solid Mechanics

interface—the Geomechanics Module, the Nonlinear Structural MaterialsModule, the Multibody Dynamics Module, and the Fatigue Module.

With the addition of the Geomechanics Module, you can add Soil Plasticity,Concrete, and Rocks features to the physics interface.

• With the Nonlinear Structural Materials Module, hyperelasticity, plasticity,creep, and viscoplasticity can be modeled.

• The Multibody Dynamics Module makes it possible to study rigid andflexible bodies connected by different types of mechanical joints.

• Using the Fatigue Module, you can augment your stress analysis with low orhigh cycle fatigue evaluation.

The exact contents of the menus and windows shown in this document may varydepending on the license or licenses you have. All illustrations assume the presenceof these additional modules, even though the examples given only require theStructural Mechanics Module.

SHELLS AND PLATES

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 cantherefore be used for rather thick shells as well as thin ones. It is possible toprescribe an offset in a direction normal to a selected surface. The Shell interfacealso includes other features such as damping, thermal expansion, and initialstresses and strains. The preset studies available are the same as for the SolidMechanics interface.


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The Plate interface ( ) is a 2D analogy to the 3D Shell interface. Plates aresimilar to shells, but act in a single plane and usually act only with out-of-planeloads. The formulation and features for this physics interface are similar to thosefor the Shell interface.

BEAMS

The Beam interface ( ) is intended for the modeling of slender structures(beams) that can be fully described by their cross-section properties, such as areaand moment of inertia. The Beam interface defines stresses and strains usingHermitian elements and Euler-Bernoulli or Timoshenko theory. Beam elementsare used to model both planar and three-dimensional frame structures. Thisinterface is also suitable for modeling reinforcements of solid and shell structures.It includes a library for rectangular, box, circular, pipe, H-profile, U-profile, andT-profile beam sections. Additional features include damping, thermal expansion,

and initial stresses and strains. The preset studies for this physics interface arealmost identical to those for the Solid Mechanics interface, with two exceptions—the Linear Buckling and Prestressed study types are not included.

A separate 2D physics interface called the Beam Cross Sections interface ( ) canbe used to evaluate properties for general cross sections, to use as inputs in beamanalysis.

TRUSSES

The Truss interface ( ) may be used to model slender structures that can only

sustain axial forces. Trusses are modeled using Lagrange shape functions, whichallow the specification of small strains as well as Green-Lagrange strains for largedeformations. Examples of truss structures are truss works with straight edges andcables exposed to gravity forces (sagging cables). Additional features includedamping, thermal expansion, and initial stresses and strains. The preset studies forthis physics interface are the same as for the Solid Mechanics interface.

MEMBRANES

The Membrane interface ( ) is used for membranes, which can be modeled as

plane stress elements in 3D and can deform in both in-plane and out-of-planedirections. The difference between a shell and a membrane is that a membranedoes not have any bending stiffness. Thin films and fabric are structures suited formodeling with the Membrane interface. The preset studies for this physicsinterface are almost the same as for the Solid Mechanics interface, with oneexception—they do not include the Linear Buckling study type.


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OTHER STRUCTURAL MECHANICS INTERFACES

The Thermal Stress multiphysics interface ( ) combines the Solid Mechanicsinterface with the Heat Transfer in Solids interface. The temperature field isautomatically coupled to a structure’s (material) expansion.

The Joule Heating and Thermal Expansion multiphysics interface ( ) combinesthree physics interfaces: Joule Heating, Heat Transfer in Solids, and SolidMechanics. It describes the conduction of electric current in a structure, thesubsequent electric heating caused by the ohmic losses, and the thermal stressesinduced by the temperature field.

The Piezoelectric Devices interface ( ) combines the Solid Mechanics andElectrostatics interfaces to model piezoelectric materials. The piezoelectriccoupling can be in stress-charge or strain-charge form. All solid mechanics andelectrostatics functionalities are also accessible through this physics interface, forexample, for modeling the surrounding linear elastic solids or air domains.

FLUID FLOW

The Fluid-Structure Interaction (FSI) multiphysics interface ( ), found underthe Fluid Flow branch in the Model Wizard, combines fluid flow with solidmechanics to capture the interactions between fluids and solid structures. A SolidMechanics interface and a Single Phase Flow interface model the solid and thefluid, respectively. The flow can be either laminar or turbulent (turbulent flowrequires the CFD Module). The FSI couplings appear on the boundaries betweenthe fluid and the solid. The FSI interface uses an arbitrary Lagrangian-Eulerian

(ALE) method; it combines the fluid flow formulated by an Eulerian descriptionand a spatial frame with solid mechanics formulated by a Lagrangian descriptionand a material (reference) frame.

The Physics List by Space Dimension and Study TypeThe table below lists the physics interfaces available specifically with this modulein addition to those included in the basic COMSOL Multiphysics license.

INTERFACE ICON TAG SPACE DIMENSION PRESET STUDY TYPE

Fluid Flow

Fluid-Structure

Interaction

fsi 3D, 2D, 2D

axisymmetric

stationary; time dependent


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Structural Mechanics

Solid Mechanics* solid 3D, 2D, 2D

axisymmetric

stationary; eigenfrequency;

prestressed analysis,

eigenfrequency; time dependent; time dependent modal; frequency

domain; frequency-domain modal;

modal reduced order model;

prestressed analysis, frequency

domain; linear buckling

Thermal Stress** — 3D, 2D, 2D

axisymmetric


Shell shell 3D stationary; eigenfrequency;


eigenfrequency; time dependent;

time dependent modal; frequency





Plate plate 2D stationary; eigenfrequency;








Beam beam 3D, 2D stationary; eigenfrequency;

frequency domain;

frequency-domain modal; modal

reduced order model; time

dependent; time dependent

modal

Beam Cross Section bcs 2D stationary



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Truss truss 3D, 2D stationary; eigenfrequency;




domain; frequency-domain modal;modal reduced order model;



Membrane mem 3D, 2D, 2D

axisymmetric

stationary; eigenfrequency;






domain

Joule Heating and

Thermal Expansion**

— 3D, 2D, 2D

axisymmetric


Piezoelectric Devices pzd 3D, 2D, 2D

axisymmetric

stationary; eigenfrequency; time

dependent; time-dependent

modal; frequency domain;

frequency domain modal; modal

reduced order model

* This is an enhanced interface, which is included with the base COMSOL package but has

added functionality for this module.** This interface is a predefined multiphysics coupling that automatically adds all the physics

interfaces and coupling features required.



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The Model Libraries Window | 11

The Model Libraries Window

To open a Structural Mechanics Module model library model, click Blank Modelin the New screen. Then on the Home or Main toolbar click Model Libraries .

In the Model Libraries window that opens, expand the Structural MechanicsModule folder and browse or search the contents.

Click Open Model to open the model in COMSOL Multiphysics or clickOpen PDF Document to read background about the model including thestep-by-step instructions to build it. The MPH-files in the COMSOL modellibrary can have two formats—Full MPH-files or Compact MPH-files.

• Full MPH-files, including all meshes and solutions. In the Model Libraries window these models appear with the icon. If the MPH-file’s sizeexceeds 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 Libraries 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 nosolutions for other reasons). You can open these models to study the settingsand 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 when youupdate your model library. These models appear in the Model Libraries window with the icon. If you position the cursor at a compact model inthe Model Libraries window, a No solutions stored message appears. If a full

MPH-file is available for download, the corresponding node’s context menuincludes a Download Full Model item ( ).

To check all available Model Libraries updates, select Update COMSOL ModelLibrary ( ) from the File>Help menu (Windows users) or from the Help menu(Mac and Linux users).


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The Fundamentals: A Static Linear Analysis | 13

The Fundamentals: A Static Linear Analysis

This section summarizes the fundamentals for modeling structural mechanicsproblems and applying them in COMSOL Multiphysics and the Structural

Mechanics Module. It includes instructions for creating a geometry, as well asdefining material properties and boundary constraints and conditions. After thesolution is computed, you will learn how to display and analyze the results.

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 actuatorthat is mounted on a pin placed between the two holes in the bracket arms. Thegeometry 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 bondedto 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 P0 is the peak load intensity. The main direction of the load is defined by, the angle from the negative y-axis. The direction of the load is given in a local

coordinate system controlled by a parameter theta0. The type of loaddistribution is shown in Figure 7:

p P0

0

– 0 0

– sin=

0


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14 | The Fundamentals: A Static Linear Analysis

Figure 7: The load distribution on the bracket.

In the first study, the load will be pointing along the negative y-axis, but the moregeneral definition will be useful later.

Model Wizard

The first step to build a model is to open COMSOL and then specify the type ofanalysis you want to do—in this case, a stationary, solid mechanics analysis.

Note: These instructions are for Windows users, but apply with minor differences

to Linux and Mac.

1 To start the software, double-click the COMSOL icon on the desktop. Whenthe software opens, you can choose to use the Model Wizard to create a new

COMSOL model or Blank Model to create one manually. For this tutorial, clickthe Model Wizard button.

If COMSOL is already open, you can start the Model Wizard by selectingNew from the File menu and then click Model Wizard .

The Model Wizard guides you through the first steps of setting up a model. Thenext window lets you select the dimension of the modeling space.

2 In the Space Dimension window click 3D .


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3 In the Select Physics tree under Structural Mechanics, select Solid Mechanics(solid) .

4 Click Add, then click Study .

5 In the Select Study window under Preset Studies, select PresetStudies>Stationary .

6 Click Done .

Global Definit ions - Parameters

It is good modeling practice to gather the constants and parameters in one placeso that you can change and vary them easily. In this model, the parameter setincludes theta0, the angle at which the maximum pressure from the pin is appliedto the holes—this is a parameter you may want to vary later on. In general, theseconstants and parameters are valid throughout the model, and are thereforedenoted as global parameters.

The following parameters will be defined: the load orientation angle theta0, themaximum load value P0, and the y- and z-coordinate of the center of the bracketholes, y0 and z0.

1 On the Home toolbar click Parameters .

Note:

On Linux and Mac, the Home toolbar refers to the specific set of controlsnear the top of the Desktop.

2 In the Parameters settings window under the Parameters section, enter theta0 in the Name column and 180[deg] in the Expression column.

3 Fill in the Parameters table with the other parameters and expressions asdisplayed in the figure below.


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Importing the Geometry

The next step is to create your geometry, which also can be imported from anexternal program. COMSOL Multiphysics supports a multitude of CADprograms and file formats. In this example, import a file in the COMSOLMultiphysics geometry file format (.mphbin). The file contains the assembly ofboth 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\COMSOL44\models\.

1 On the Geometry toolbar click Import .

2 In the Import settings window under Import, click Browse. Then browse to thefolder 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 choose BuildSelected .

The Finalize feature node determines how the parts of the assembly will beconsidered in the analysis. By using the default setting, Form a union, themounting bolts are automatically bonded to the bracket and the internal boundaryassumes continuity. If Form an assembly were selected instead, the mounting bolts


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would not be connected to the bracket, and the structural contact between thebolts and the bracket could be modeled. This first example assumes that themounting bolt is bonded to the bracket.

Definit ions - Selections

Regardless of whether the model is treated as a single entity or as an assembly, youstill may need to access different parts of it for definitions, such as multiplespecifications of similar boundary conditions or solid domain specifications. Theuse of arbitrary box selections is one way to do this. In this example, one selectioncontains the bolt domains, while a second selection contains the area around the

load-bearing boundaries of the holes. The box selection is independent of thegeometry, and allows you to change the geometry topology while still keeping thedesired selection.

1 On the Definitions toolbar click Box .

2 In the Box settings window under Box Limits,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 iflist, select Entity inside box.


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4 Click the Wireframe Rendering button on the Graphics toolbar to displaythe selection as in the figure:

5 Add a second Box selection, as describedin step 1.

6 In the Box settings window, select Boundaryfrom the Geometric entity level list.

7 Under Box Limits, set the values as in thefigure 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 entityif list, select Entity inside box.


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View the selection in the Graphics window. It should match this figure.

Definit ions - Variables and Coordinate Systems

In this section you define variables and add a local coordinate system.

Variables

Here, you want to define expressions for the load applied to the load-carryingholes. This is done by first evaluating the radial angle based on the position alongthe global z-coordinate. Using a sinusoidal function, the second expression is thendefined by the pressure distribution.

1 On the Definitions toolbar click Local Variables .

2 In the Variables settings window, enter angle in the Name column andatan2((y-y0),(z-z0)) in the Expression column.


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3 Add a second variable loadIntensity with the expressionP0*sin(angle-theta0)*(sin(angle-theta0)>0).

The last part of the expression is a condition that ensures that the load is appliedto only one half of the hole.

Coordinate Systems

Create a rotated coordinate system, which defines the orientation of the loadapplied to the bracket holes.

1 On the Definitions toolbar click Coordinate systems and choose RotatedSystem .


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2 In the Rotated System settings window under Settings, enter -theta0 as the value for beta.

Materials

COMSOL Multiphysics is equipped with built-in material properties for a numberof common materials. Here, choose structural steel as the material for both thebracket and the bolts. The material is automatically assigned to all domains.

1 On the Home toolbar click Add Material .


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2 In the Material Browser under Built-In,right-click Structural steel and select Addto Component 1.

3 On the Home toolbar click AddMaterial again to close the window.

Solid Mechanics

Now define the physics for the bracket assembly. Initially, the analysis wasspecified to be stationary using the classical equations associated with solidmechanics. In this step, you are more specific concerning the different modeling

domains.By default, the Solid Mechanics interface assumes that the participating materialmodels are linear elastic, which is appropriate for this example. All that is left to dois set the constraints and loads acting on the structure, making use of the boxselections defined previously (see “Definitions - Selections” on page 17).

Fixed Constraint

Assume that the bolts are rigid and that the displacements are perfectlyconstrained.


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1 On the Physics toolbar click Domains and choose Fixed Constraint.

2 In the Fixed Constraint settings window under Domain Selection, select Box 1from the Selection list.


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Boundary Load

Apply a boundary load to the bracketholes.

1 On the Physics toolbar, clickBoundaries and choose BoundaryLoad .

2 In the Boundary Load settings windowunder Boundary selection, select Box 2from the Selection list.

Use the rotated coordinate system tochange the load orientation by a value oftheta0.

3 Under Coordinate System Selection,

select Rotated System 2.4 Under Force, enter loadIntensity

for the x2 component.

DISPLAYING PHYSICS SYMBOLS ON THE GRAPHICS WINDOW

The Graphics window shows the domain selection, including symbols to describethe type of load applied to the selection.

The symbols indicate only the type of settings applied to the model and not themagnitude. If you are interested in visualizing the actual applied pressuredistribution, a solution must be computed first.

You turn the physics symbols on from the Preferences menu. For Windows users,select File>Preferences. You can also customize the Quick Access Toolbar andthen click a button. For Mac and Linux users, select Options>Preferences.


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In the Preferences dialog box, click Graphics and click to select the Show physicssymbols check box. Click Apply and OK. Click anywhere in the Model Builder,then click the node again. The symbols are now displayed in the geometry.

Study

You are now ready to compute the solution.

1 On the Study toolbar click Compute .

The Study node automatically defines a solversequence for the simulation based on theselected physics (Solid Mechanics) and studytype (Stationary). Since a mesh is required, andit has not been created yet, the Study nodeautomatically generates this based on defaultsettings at the same time as the solver sequence.

Note: In general, do not rely only on the

default mesh settings. For most real problems, suitable meshing parameters

should be set up from the Mesh toolbar.


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1 In the Model Builder under Results, right-click Stress (solid) and select Arrow Surface . Alternatively, you can select Stress (solid) and click ArrowSurface on the Stress (solid) toolbar.

2 In the Arrow Surface settings window under Expression, click ReplaceExpression . Choose Solid Mechanics>Load>Load(Spatial)(solid.FperAreax,...,solid.FperAreaz) from the list. Double click toselect it.

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 asintended. Note that the arrows by default are plotted on the undeformedstructure.

Displacements

To study the displacement, add a second 3D plot group to the Results node.


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1 On the Results toolbar click 3D Plot Group .

2 On the 3D Plot Group 2 toolbar, click Surface . The plot you just addedshows the displacement by default, so you do not need to change the resultexpression.

In the figure below you can see that the bracket base remains fixed while only thearms are deformed. The maximum total displacement is about 27 m, which is inagreement with the assumption of small deformations.

Principal Stresses

Create another plot to display the principal stresses in the bracket.

1 On the Results toolbar click 3D Plot Group .

2 On the 3D Plot Group 3 toolbar, click More Plots and choose Principal Stress

Volume .


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3 In the Principal Stress Volume settings window under Positioning, find the Xgrid points subsection. From the Entrymethod list, select Coordinates.

4 In the Coordinates field, enter

4e-3 15e-3 30e-3 1e-2. A spacebetween each of the values is required.

5 For both Y grid points and Z grid points,enter 15 in the Points field.

6 In the Coloring and Style section, selectScale factor. In the field enter 1.5e-9.

7 Click the Plot button .

As the load is oriented along the negative y direction, the principal stress plotshows tensile stress in the arm of the bracket.


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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. If the stress is compressive, the two arrow heads will aim

inward. If the stress is tensile, the two arrow heads will aim outward, as shown

below.

Reaction Forces

A final check is done to compute the total reaction force along the x, y, and z directions. Because the mounting bolts are fully constrained, use a volumeintegration over the constrained domains.

1 On the Results toolbar click More Derived Values and choose VolumeIntegration .


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2 In the Volume Integration settings window under Selection, select Box 1 fromthe Selection list.

3 Under Expression click ReplaceExpression . Select Solid

Mechanics>Reactions>Reaction force(Spatial) and double click Reaction force, xcomponent (solid.RFx) (or entersolid.RFx in the Expression field). Onthe Results toolbar, click Evaluate .

4 Change the expression in the Expressionfield to solid.RFy. Click the Evaluatebutton .

5 Change the expression in the Expression

field to solid.RFz. Click the Evaluatebutton .

The results are visible in the table located in the lower-right side of the desktop.To open the Table window you can also click More Windows on the Hometoolbar and choose Table . You can verify that the total reaction force in the

load direction ( y) is close to the 8 kN applied load. Along the x and z directionsthe values are zero except for small numerical errors.


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32 | Parametric Study

Parametric Study

In the previous section, a bracket loaded by an actuator was analyzed. This sectionextends this analysis to study the effect of the actuator’s position. This is equivalent

to changing the direction of the applied load. You will use the load-angleparameter that the model already contains to set up a parametric study.

COMSOL Multiphysics has two ways to perform parametric studies—aParametric Sweep node or an Auxiliary sweep from the Stationary Solver node. Inthis example, both methods are applicable. The continuation solver in the Solvernode is not used. This is a feature which uses the solution from the previousparameter as an initial guess to calculate the current parameter value, and is thepreferred option for nonlinear problems. Using the Parametric Sweep node ispreferable for applications requiring, for example, geometric parametrization.

Loading the Model

Either continue working on the existing model or open a saved version of themodel from the Model Library. (See “The Model Libraries Window” on page 11.)Browse to the Structural_Mechanics_Module\Tutorial_Models folder.Double-click to open bracket_static.mph.

Extending the Study with an Auxil iary Sweep

Parametric studies can be set up from scratch or, as in this example, added to anexisting study.

1 In the Model Builder expand the Study 1 node and click Step 1: Stationary .


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34 | Parametric Study

Results

The default plot shows the solution for the last parameter value (180[deg]), which is the same as the case studied in the previous section (see “Results” on page26). You can easily change the parameter value to display the plot and thencompare solutions for different load cases.

Note: Click the Zoom Extents button to view the new default plots.

1 In the Model Builder click the Stress (solid) node .

2 Under Data in the 3D Plot Group settings window, select 0 from the Parameter value (theta0) list.


Note: Under the compressive load, the maximum von Mises stress is significantly

lower than under the tensile load. Also, the deformation of the bracket armsoccurs in the opposite direction.

4 Click the Stress (solid) node again. In the 3D Plot Group settings windowunder Data, select a different value from the Parameter value (theta0) list, forexample 1.570796.


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Parametric Study | 35


The plot displays the stress distribution for a bending load case. The maximumstress values are no longer located around the holes but in the region where thearms connect to the bolt supports.

Reaction Forces

To evaluate the total reaction forces for all parameter values, each component ofthe reaction force is integrated over the bolt domains. Because no load is acting inthe global x direction, the reaction force in this direction is zero.


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Parametric Study | 37

Note: You may need to select Table 2 from the Table list in the information

window to produce this plot.

Modeling Techniques for Structural MechanicsThe next set of tutorials build on the previous examples to demonstrate thefollowing structural mechanics modeling techniques: How can I model thermalstress? How can I add an initial strain to the simulation? To get started, go to“Including Initial Strain” on page 38 or “Modeling Thermal Expansion” on page43.


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38 | Including Initial Strain

Including Initial Strain

Initial stresses and strains can be specified in a subnode to a material model. Youcan define a stress/strain distribution with constant values or an expression which

can, for example, be space- or time-dependent. The initial stresses and strains canalso come from another physics interface in the same study, or even results fromanother study.

In this example, you add a pin geometry to the bracket assembly. Then you specifyan initial strain to simulate that the pin is prestrained in the axial direction, andthen you investigate its effect on the assembly.

Load Model and Choose the Study

1 See “The Model Libraries Window” on page 11 and then browse to theStructural_Mechanics_Module\Tutorial_Models folder and double-click toopen bracket_basic.mph.

2 On the Home toolbar click Add Physics . Go to the Add Physics window.

3 In the Add Physics tree, under Structural Mechanics select Solid Mechanics(solid) .

4 In the Add Physics window click Add to Component .5 On the Home toolbar click Add Study . Go to the Add Study window.

6 In the Studies tree under Preset Studies, select Stationary .

7 In the Add Study window click Add Study .

8 On the Home Toolbar click the Add Study and Add Physics buttons again toclose the windows.

Definit ions - 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 thestructure, which is fixed by fully constraining the mounting bolts.


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Including Initial Strain | 39

In the Parameters table, define a strain value that corresponds to a reduction ofthe pin length from 216 mm to 215 mm.

1 In the Model Builder expand the Global Definitions node and clickParameters .

2 In the Parameters settings window in the Parameters table, add the newparameters L, L0, and InitStrain as in the figure.

Geometry

Next, the pin geometry is added to the bracket assembly. This is done byimporting 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\COMSOL44\models\.

1 On the Home toolbar click Import . An Import 2 node is added to theModel Builder.

2 In the Import settings window under Import, click Browse.

3 Browse to the model folder \Structural_Mechanics_Module\Tutorial_Modelsand double-click the file bracket_pin.mphbin.

4 Click Import.


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Adding Init ial Strain and Other Physics Settings

The initial strain is specified under the Linear Elastic Material node.

1 In the Model Builder under Solid Mechanics, right-click Linear ElasticMaterial 1 and select Initial Stress and Strain . The ‘D’ in the upper leftcorner of the Linear Elastic Material node means that it is a default node.

The prestrain direction is the axial direction of the bolt, which coincides with theglobal 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 (the pin).


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The results show how the pin compresses the bracket arms, and that the largeststresses are found in the region where the bracket arms are joined to the boltsupports.

Note: The structural analysis from the previous tutorial is not included here.

You can also plot the x-component of the strain tensor to visualize the total strainin the structure. As the pin is stiff in comparison to the bracket, the total strain inthe pin is almost the same as the initial strain given in the example.

1 On the Home toolbar click Add Plot Group and select 3D Plot Group .

2 To add a a surface plot, on the 3D Plot Group 2 toolbar click Surface .

3 In the Surface settings window under Expression, enter solid.eXX in theExpression field.



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Modeling Thermal Expansion | 43

Modeling Thermal Expansion

In this example, a thermal field is applied to the bracket and pin assembly and thethermal stresses are calculated.

COMSOL Multiphysics contains physics interfaces for structural analysis as well asthermal analysis. You can define the analyses separately and then simulate thethermal-structure interaction by coupling them using the appropriate variablesand terms in the structural analysis equations, or you can add a predefinedThermal Stress physics interface at the beginning of the setup.

If you were using a Solid Mechanics interface, adding thermal expansion to amaterial model is as easy as it was to add the initial strain—you would add aThermal Expansion node to the Linear Elastic Material Model.

However, an even easier option is to use a predefined physics interface, the

Thermal Stress interface. This interface contains both the structural and thermalequations along with the coupling, which is included by default.

Load the Model, select Physics, and Choose the Study

1 As in the previous tutorial, see “The Model Libraries Window” on page 11

and then browse to the Structural_Mechanics_Module\Tutorial_Models folderand 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 Recent list.

2 On the Home toolbar click Add Physics . Go to the Add Physics window.

3 In the Add Physics tree under Structural Mechanics select Thermal Stress

4 In the Add Physics window click Add to Component .

5 On the Home toolbar click Add Study . Go to the Add Study window.

6 In the Studies tree under Preset Studies select Stationary .7 In the Add Study window click Add Study .

8 On the Home toolbar click the Add Study and Add Physics buttons again toclose the windows.


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44 | Modeling Thermal Expansion

Thermal Stress

When you add a Thermal Stress interface, whatactually happens is that one Solid Mechanicsinterface and one Heat Transfer in Solids interfaceare added. In addition, a Multiphysics node is alsoadded to the model tree. Under the Multiphysicsnode there are two subnodes: TemperatureCoupling and Thermal Expansion. If you click eachof these nodes, you can see that the coupling is setup automatically.

Note: The thermal expansion requires both a coefficient of thermal expansion

and a strain reference temperature, which is the temperature reference at which

there are no thermal strains. Because a material (structural steel) has already been

defined for the model you do not need to modify anything here.


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You can now set the boundary conditions.

1 Make sure that Solid Mechanics is the currentlyactive physics interface. Do this selection either onthe Physics toolbar or by selecting a node under SolidMechanics in the model tree.

2 On the Physics toolbar, click Domains andchoose Fixed Constraint .

3 In the Fixed Constraint settings window under Domain Selection, select Box 1from the Selection list.

As the Thermal Stress interface includes a heat balance, the thermal boundaryconditions must also be set. Assume that the mounting bolts are kept at atemperature of 20 oC. Also assume that the arms of the bracket are holding thepin, which itself has a constant temperature of 100 oC. Finally, assume that heatis lost by convection to the surroundings from the surfaces of the arms and thebolt support.

4 On the Physics toolbar make sure that Heat Transfer in Solids is the currentlyactive physics interface.

5 On the Physics toolbar click Boundariesand choose Temperature .

6 Select boundaries 22, 25, 30, 33, 46, 49, 54,and 57 (or use the Paste Selection button).

7 Add a second Temperature node. On the

Physics toolbar click Boundaries andchoose Temperature .

8 In the Temperature settings window underBoundary Selection, select Box 2 from theSelection list.

9 Under Temperature enter 100[degC] in the T 0 field.


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46 | Modeling Thermal Expansion

10On the Physics toolbar click Boundariesand choose Heat Flux .

11In the Heat Flux settings window underBoundaries, select All boundaries from theSelection list.

12Under Heat Flux click the Inward heat fluxbutton. In the h field, enter 10.

Compute and Display Results

1 On the Study toolbar click Compute .

Under the Results node, three plot groups are automatically added to show thedefault results for a structural analysis and a thermal analysis. The first plot group,Stress (solid), shows the von Mises stresses on a scaled deformed geometry.


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You can see how the bracket arms holding the pin are deformed through thermalexpansion and that thermal stresses have developed.

The second default plot group, Temperature (ht), displays the temperaturedistribution on a scaled deformed geometry.

This concludes this introduction. For additional tutorials using the bracketgeometry, go to the Structural Mechanics Module model library and browse theTutorial Models folder. Click the Model PDF button to open instructions for eachof 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 any of the plot groups.

2 Click the Root node (the first node in the model tree). On the Root settings window under Model Thumbnail, click Set Model Thumbnail.

Make adjustments to the image in the Graphics window using the toolbar buttonsuntil the image is one that is suitable to your purposes.

If you want to explore further, in the Structural Mechanics Module model librarythere are several more detailed examples that use the bracket model to extend thistutorial. See “Model Examples in this Guide and the Model Library” on page 12 for more information.


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COMSOL 4.3b - Introduction to Structural Mechanics Module

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