Gregory Hew
The methodology required to use COMSOL as a tool to examine the deflection of cantilever beams of different shaped cross sections subjected to an end load is detailed. The specific cross sections utilized were square, circular, triangular, and I-
Beam. The steps required to model, solve, and validate the solution of the systems in question are shown. The impact of geometry on the computational power required for solution was confirmed as the I-beam and circular were more calculation intensive. The modeled solutions themselves were in good correlation with the classical solution obtained though stress-strain analysis, and were verified to adequately converge via mesh extensions, however the affect of mesh shape on the convergence for circular geometries was noted.
Detail the methodology needed to utilize
COMSOL to examine the deflection of end loaded cantilever beams of different shaped cross sections. This report is focused on:

How to use COMSOL to accurately model the respective systems.

The utilization of COMSOL to solve the respective systems and view results of interest.

How to verify the results and the validity of the solution.
1
Engineering by it's very nature, requires examination of complex systems. In many cases, these systems are unable to be solved through analytical methods, which means solutions to these systems must be obtained by numerical methods. One tool that can be used is finite element analysis (FEA) software to obtain an approximate solution to the system under study.
However, since each individual FEA software is different, specific methodology must be used to ensure the software gives the appropriate solution as, without confidence in the solution obtained by the software, the results are of little consequence. Even worse, if the solution is incorrect but utilized, there could be serious real world repercussions.
This project details how to use a specific FEA software, COMSOL, to determine the deflections of end loaded cantilever beams of various cross sections. This analysis is useful for this purpose, as an analytical solution to this problem is available. This allows for a comparison of the model solution to the actual analytical solution, which can be used to validate the solution obtained by COMSOL and the methodology detailed in this report.
The basic system under review is a steel cantilever beam fixed at one end and subjected to an end load on the other. This basic arrangement is shown in Figure 1 below.
Figure 1.
Cantilever Beam Subjected to End Load .
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2
This basic arrangement will be analyzed for four different cross sections. These cross sections are square, circular, triangular, and that of an I-
Beam. The shapes and dimensions are detailed in Figure 2.
It should be noted that out of these three steps, the majority of the users time is spent on pre and post processing.
Pre Processing
During pre processing, the user creates the model of the system that is to be studied. This includes creating geometries, meshes, boundary conditions, loads, and defining other important material properties. This step is crucial as if the problem to be analyzed is not modeled properly, the solution obtained will not be correct for the system it was intended for.
Figure 2
Cross Sections and Pertinent Dimensions
Other important decisions are also made to
In addition to the forces applied and the improve the efficiency of the solving process, dimensions describing the system, the material such as the type of mesh that will be used and properties of the steel beam must be stated in potential simplifications to the actual situation order to determine the deflection. The properties that can be used for the model. Proper decisions used for this analysis will be the constant values listed in Table 1. at this juncture result in an efficient solving process maximizing the computational resources utilized and positively impacting costs.
Table 1
Beam Material Properties
The preprocessing done for this COMSOL analysis is broken down as follows:
Steel Properties
Young's Modulus 2*10 11 Pa
Poisson's Ratio 0.3
Density 7800 kg/m3
With the pertinent dimensions, forces, and material properties defined, as well as a plan to validate the numerical solution; an analysis of

Define study

Define geometries

Define material properties

Add loads and boundary conditions

Specify meshing
The define study options are contained in the the system using COMSOL can be performed. model wizard, while all the other steps are included under the model option found under the
model builder.
Define Study
Any finite element analysis can be broken down into three distinct steps:
1) Pre processing
COMSOL begins any analysis with the model wizard, which sets the framework with which the solution will be based off of.
2) Solving
3) Post Processing
First, the number of dimensions that are required for analysis are specified. This impacts the geometric model options that can be used later
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3 in the analysis. For this case, a three dimensional model was used so that the height, length and the different cross sectional areas could be represented in the model.
Once the number of dimensions is specified, the model wizard requests for the physics of the system to be specified. The proper selection at this juncture ensures that COMSOL will be request appropriate properties and boundary conditions, as well as the appropriate post processing options. For this case, it is appropriate to use the Solid Mechanics option, located under the Structural Mechanics option.
Next the study type must be defined, this is similar to the previous physics step, except tailored specifically to the type of physics previously chosen. Special attention should be paid to how this section allows for the analysis to be specified as transient or steady state. As this analysis is steady state deflection of cantilever beams subjected to end loads; the selection of stationary is appropriate. Once this step is compete, the model wizard is finished and the user is placed into the model builder, where the geometry of the system can now be defined,
Define Geometries
In this section, the physical dimensions of the system are defined. After the units of the system are confirmed to be the appropriate type, the user can right click the geometry tab in the model builder and left click on a shape that is appropriate for the system. After the shape is created, the placement of the shape as well as the pertinent dimensions are defined and the shape can be created and displayed in the graphics window. Due to multiple cross sections utilized in this analysis, different methods to create the 3D models were required.
For the square and circular cross section beams; these shapes are readily built into COMSOL and can be defined accordingly.
The triangular cross section beam required a triangle to be drawn using the polygon option under the more primitives subtab. Once this was created the polygon was made into a face, using the cap faces option, and then extruded to the appropriate length. Both the cap faces option and the extrude option are located under the geometry tab.
The I-Beam cross section beam was created by making three distinct beams: the two outer rectangular cross section beams and the center square cross section beam. These were then unioned together, using the union suboption, to make one solid piece to be analyzed. This model is shown in Figure 3 below.
Figure 3
I-Beam Geometry
Define Material Properties
Now that the system geometries have been set, the material properties must be specified, which is done under the materials subtab.
Right clicking the materials subtab allows for options of material or material browser. The material browser allows access to a material properties library that comes preloaded with
COMSOL, while the material option allows the
MANE 4200 – Introduction to Finite Elements, Professor E. Gutierrez-Miravete

Using the physics controlled mesh, all mesh size can be specified from extremely course to extremely fine. Extremely course meshes are very rough solutions do not require much computational power, but at the cost of accuracy. However, the meshes in the course range may be good to obtain an initial idea regarding the accuracy of the model prior to committing to more intensive calculations. The extremely fine mesh, is the opposite of the extremely course, used primarily to obtain high accuracy results or to validate results in post processing. A representation of the amount of elements corresponding to a course and an extremely fine mesh can be seen in Figure 4 for the sake of comparison.
4 user to specify material properties relevant to the system. The relevant properties are chosen based on the physics of the system under examination.
In this case, the material option was used and the material properties listed in Table 1 were inputted. These were then selected to the geometries created previously, so that the material properties can be defined to the beams.
Apply Loads and Boundary Conditions
The next step is to go to the solid mechanics subtab. Here the loads, boundary conditions, and other initial values are defined for the system.
Since it is a cantilever beam, one end of the beam must be fixed. To apply this to the system, the solid mechanics tab was right clicked and the fixed constraint condition was chosen. This was then applied to the surface at the front of the beam, to prevent any deflection due to the end load.
The end load was applied by similarly right clicking the solid mechanics tab and using the edge load option. The edge selected was the top edge at the far end of the beam, or the top edges at the far end of the beam for the case of the triangle and circular cross sections. Once the edges are selected, the end load must also be specified. COMSOL gives options for total force or force per unit area. In this case a negative 100
N total force was applied.
Specify Meshing
One of the most important aspects of any FEA analysis is meshing, which has direct results on the accuracy of the solution as well as the efficiency in which it is obtained. In COMSOL, the meshes can be user controlled or physics controlled, the latter being the COMSOL default which uses tetrahedral shapes. The user controlled mesh gives the user a greater range of control regarding the type and methodology in which the meshes are applied to the system.
MANE 4200 – Introduction to Finite Elements, Professor E. Gutierrez-Miravete
Figure 4
Comparison of Meshes for Square Cross section
(Course left) (Extremely Fine Right)
For this analysis, the physics controlled mesh was used for courser, normal, finer, and extremely fine meshes. With the mesh set, the solution can now be obtained.
Solving
The solution process is not very involved from the users perspective, however it is where the majority of the computational power is used.
The efficiency and time required to obtain a solution is very dependant upon the complexity of the model as well as the mesh chosen. This being the case, proper attention must be paid to the pre processing process to ensure the success of the analysis.

In terms of solving in COMSOL, all the user needs to do is go to the study subtab in the model builder. The user can then right click the subtab to define parameters of the solution, such as step size, and simply press compute to find the solution, In the case of this study, no time steps are required since it is steady state.
Therefore, the study was merely computed using
COMSOL defaults.
Post Processing
Post processing in COMSOL can be divided into three distinct parts:

Analyzing and refining the solution

Verifying the solution

Confirming stability of solution
Analyzing and Refining Solution
Once COMSOL has processed the solution, the user has a variety of options in regards to analyzing and refining the data. Some of these include exporting tabulated data, having
COMSOL produce simple reports, or viewing results based on the particular physics chosen directly in COMSOL. In this particular case, both exportation of results and using the internal COMSOL graphics were used.
All options to analyze given results in COMSOL are located under the results subtab. To see results for this particular physics, the stress subtab was expanded and surface one selected.
Originally, COMSOL displays the stress field for the system, In order to display deflection results, the user simply needs to click the replace expression button, go to solid mechanics, go to displacement, then total displacement, and plot the results. This gives the total deflection of the beam along it's entire length, with the maximum value shown at the top of the color legend.
Additionally, additional focus can be obtained by using different plot groups. If a 1D plot group
5 is created by right clicking the results section and then right clicking the newly created plot group and selecting line graph; the user can create a 2D graph displaying displacement as a function of beam length. All that is now needed is for the user to select an edge along the beam length and making the y axis data pertain to total displacement, using the same replace expression button as used on the 3D plot.
A similar process is used to export data from
COMSOL. Under the export tab, the user can choose the correct expressions and specify the form and destination for the data that is to be exported. This exported data can then be utilized in other programs to do additional analysis.
Verifying Solution
In terms of using the FEA method, this step may be the most important, as the correctness of the
COMSOL solution must be verified for it to have potential use. The best way to verify the
COMSOL solution is to verify it against an analytically obtained solution.
If this is not possible due to the complexity of the system, the integrity of the COMSOL model can be verified by using the COMSOL model to solve a simpler but equivalent system that possesses an analytical solution. This process at least proves the COMSOL model created has some merit.
In this particular instance, an analytical solution does exist and can be used to verify the accuracy of the COMSOL solution. The deflection of an end loaded cantilever beam with respect to position along the beam has been studied extensively, and an analytical solution has been obtained as shown in Equation 1, obtained from
Reference (1).
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6
Using the methodology detailed in the previous
P = Load The physical representation of the deflection for x = Position along the beam from fixed point each respective cross section as obtained by
L = Beam length
I = Area moment of Inertia for z (thickness of beam)
All of the above quantities in the equation are known except for the area moment of inertia.
Again, the method to determine the area moment of inertia for a variety of geometries is well known. These were calculated using the methodology detailed in Reference (2) and the equations derived are contained in Appendix I.
Confirming Stability of Solution
The deflections were then recorded for each mesh and compared to the analytical solution.
The analytical solution was computed using the
Matlab code contained in Appendix III. The results of this process for the maximum magnitude of deflection are detailed in Table 2.
This code was also used to compare the analytical solution graphically versus a data export of COMSOL data, which are shown in
Figure 5.
Even if a COMSOL solution obtained is reasonably close to the actual solution, the
COMSOL solution may only be correct for that particular mesh size. If the model is not quite fit to obtain an appropriate solution, results obtained at finer meshes may prove to be inaccurate to the actual solution.
To ensure this does not occur, mesh extensions are utilized. Mesh extensions verify that the
COMSOL solution eventually converges towards a particular value with finer meshes. To do this in COMSOL, the user simply goes back to the preprocessing section and refines the mesh. The model is then solved again and the new solution is obtained and compared to the previous mesh. This process should be repeated a number of times until confidence that the solutions obtained converge toward one value.
For this particular analysis, the solution was analyzed at four different meshes. These were courser, normal, finer, and extra fine.
Mesh Size
Courser
Normal
Finer
Extremely
Fine
Analytical
Table 2
Magnitude of Deflection in y direction
Maximum Deflection Times 10 5 (meters)
Square Circular Triangular I-Beam
1.9662 3.4071
2.0136 3.4125
5.8880
5.9586
2.1496
2.1567
2.0266 3.4138
2.0336 3.4159
6.0157
6.0348
2.1615
2.1644
2.000 3.395 6.000 2.133
Analyzing the data, there is a good correlation between the COMSOL solution at an extremely fine mesh and the analytical solution, with percent errors ranging from 0.58% to 1.68%. A point of note is that the percent error is largest for the square and I-Beam cross sections. The reason for the larger percent error values is a product of these geometries having less deflection than the triangular or circular cross sections.
Figure 5 can be used to visually see the correlation between the analytical solution
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7 versus the COMSOL solution. This figure depicts the analytical and COMSOL normal mesh deflections versus distance from the fixed end of the cantilever.
2.5
x 10
-5
2
Displacement of Square Cross Section
Analytical
COMSOL
1.5
One feature that should be apparent is the larger amount of COMSOL solution data points contained on the plots for the circular and I-
Beam cross sections. This is due to these geometries being more difficult to analyze than the triangular or square geometries.
1
0.5
0
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
6 x 10
-5
Displacement of Triangular Cross Section
Analytical
COMSOL
The I-Beam is a non-standard geometry which requires a larger amount of complexity than a triangle or square. The circular cross section is difficult to mesh, as the meshes used by
COMSOL are tetrahedral. These are readily used for triangular or square cross sections, but the round features of the circular cross section may have caused some difficulties. This behavior shows the relationship between geometry and mesh choice on the computational power required in solution.
5
4
3
2
1
0
0 0.1
0.2
0.3
0.4
0.5
0.6
Distance from Fixed End (m)
0.7
0.8
0.9
3.5
x 10
-5
Displacement of Circular Cross Section
Analytical
COMSOL
3
Based on the error values obtained and as evident by Figures 5, the COMSOL model gives a solution that accurately solves the deflection of the system.
2.5
2
1.5
1
Finally, to validate the convergence of the system, the solutions obtained for the different meshes were compared to one another. The results of this mesh extension are shown in
Table 3.
0.5
0
0
2.5
x 10
-5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Displacement of I-Beam Cross Section
2
Table 3
Mesh Extension
1.5
1
Analytical
COMSOL
1
Difference Between Respective Mesh/Actual
Solutions X 10 5
Solutions Compared Square Circular Triangular I-Beam
Courser - Normal 0.0474 0.0054 0.0706 0.0071
Normal - Finer 0.0130 0.0013 0.0571 0.0048
0.0070 0.0021 0.0191 0.0029 Finer - Extremely
Fine
Actual - Extremely
Fine
0.0336 0.0209 0.0348 0.0314
1
0.5
0
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6
Comparison Analytical and COMSOL Solution for Different Cross Sections "Listed Top to Bottom"
(Square, Triangular, Circular, I-Beam)
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The values of Table 3 show that the solutions generally converge for all the cross sections analyzed. The only exception would be for the circular cross section, which shows a slight divergence in going from the finer mesh to extremely fine mesh when compared to the going from the normal to the finer mesh. This is again likely due to difficulties with using tetrahedral meshes to model circular geometries.
The use of a different mesh may resolve this issue, however, since the divergence is very small and the result still in good correlation to the analytical solution; the effort involved in using a more computationally intensive solution and the necessary changes to the model are not warranted for the anticipated improvements in accuracy.
Based on the results obtained, the methodology used to model the system in question was validated. Of special emphasis was the preprocessing and post processing portions of the analysis.
The pre processing phase was demonstrated in detail, and the importance of this portion on the computational requirements and the solutions obtained was emphasized. Based on post processing results, the affect of mesh shape on particular geometries was noted, as the tetrahedral meshing had some difficulties with cylindrical geometries.
The solving process was shown to be completely dependant upon the pre processing phase. Based on the resources required for solution, the user must ensure the pre processing phase has modeled the system appropriately for their needs. Additionally, the impact of geometry on required calculations was found, as more calculations were required for the I-beam or cylindrical geometries. This is because the Ibeam required multiple shapes to be utilized and
8 the cylindrical geometry is less compatible with tetrahedral meshes.
Finally, the post processing phase was detailed and utilized to verify the integrity of the solution and the accuracy of the solution compared to analytical results. Slight divergence was noted for the circular cross section, however based on the magnitude of the divergence and the accuracy of the solution compared to the analytical results, the model was still deemed acceptable for this application.
[1] Riley, William F., Leroy D. Sturges, and
Don H. Morris. Mechanics of Materials . 6th ed.
Hoboken, NJ: John Wiley, 2007. Print.
[2] Meriam, James L., and L. G. Kraige.
Engineering Mechanics: Statics . 6th ed.
Hoboken, NJ: Wiley, 2007. Print.
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