Introduction to the Structural Mechanics Module · PDF fileIntroduction | 5 Introduction The...

VERSION 4.3

Introduction toStructural Mechanics Module

C o n t a c t I n f o r m a 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 M e c h a n i c s M o d u l e 1998–2012 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

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Version: May 2012 COMSOL 4.3

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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

| 3

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

Introduction | 5

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,

6 | Introduction

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.

Introduction | 7

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 InterfacesThe figure below shows the physics interfaces in the COMSOL Model Wizard and included with this license. In addition to the structural mechanics capabilities, the

8 | Introduction

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 O L I D M E C H A N I C S

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

Introduction | 9

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 MaterialsThere 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 H E L L S A N D P L A T E 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.

10 | Introduction

B E A M S

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.

TR U S 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 M B R A N E 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 T H E R S T R U C T U R A L M E C H A N I C S I N T E R F A C E 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

Introduction | 11

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 U I D F L O W

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 TypeThe 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

12 | Introduction

Model Examples in this GuideIn 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

Introduction | 13

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 LibraryTo 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.

14 | Introduction

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.

Introduction | 15

The Fundamentals: A Static Linear AnalysisThis 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 P0 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 P0 0– 0 0– sin=

0

16 | The Fundamentals: A Static Linear Analysis

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 .

The Fundamentals: A Static Linear Analysis | 17

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 .


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


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.


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.


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.

VariablesWhen 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


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 SystemsCreate 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 .


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 .


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 ConstraintAssume 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.


Boundary LoadApply 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.


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.


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.


DisplacementsTo 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 .


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


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, enter4e-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.


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


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 ForcesA 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 .


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.


Parametric StudyIn 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 .

34 | Parametric Study

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.


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.

Parametric Study | 35

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.



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 ForcesTo 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 .


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 90o—negative from 0o to 90o and positive from 90o to 180o. The reaction force in the z direction has a maximum at 90o.

Plot the Reaction Force1 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.


Modeling Techniques for Structural MechanicsThe 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 star ted go to “Including Initial Strain” on page 40 or “Modeling Thermal Expansion” on page 45.


Including Initial StrainInitial 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 .

40 | Including Initial Strain

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.

Including Initial Strain | 41

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.


3 Under the Initial Stress and Strain section, enter InitStrain in the first component of the 0 table.

Fixed Constraints1 Right-click Solid Mechanics (solid) and at the domain level, select More>Fixed

Constraint .


COMPUTE AND DISPLAY RESULTS


The default plot shows the von Mises stress in the bracket.

Including Initial Strain | 43

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).



Modeling Thermal ExpansionIn 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.

Modeling Thermal Expansion | 45

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 .


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 oC. Also assume that the arms of the bracket are holding the pin, which itself has a constant temperature of 100 oC. 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.

46 | Modeling Thermal Expansion

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 .

10 In the Temperature settings window under Boundary Selection, select Box 2 from the Selection list.

11Under Temperature enter 100[degC] in the T0 field.

COMPUTE AND DISPLAY RESULTS



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.


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 fur ther, 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.


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