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AEE 3150
Aerospace Computational
Techniques (ACT)
Finite Element Method
– Homework 6
Dr Eric Swenson
Last daily quiz is 18 April
1
Overview
- Complete HW #6 in today’s class
- Create a FE model of a cantilever beam under load
- Discuss answers to HW6 questions and results
- May need to download MATLAB’s Partial Differential Equation Toolbox
- FIT AEE Finite Element courses
HW 6 Directions
- This slide deck is HW #6
- Where you see “(5 pts.)”, “(10 pts.)”, or “(15 pts.)” - these are the HW
- Copy code from these slides into file to create FE model
“ACT_Lec_19_FEM_cantilever_beam_STUDENT_VERSION.mlx”
- Slide #2 provides all necessary measurements
- Hyperlinks are provided for most steps to link you to MATLAB help for each
command
Directions:
- Turn in code (.m or .mlx) into CANVAS
- Turn in results (PDF preferred) into CANVAS
- All general rules of plotting, such as having labels, titles, etc., apply
Cantilever Beam
Beam length 768 mm
X-direction
displacement
64 mm
Fixed
end
rigid
support
Tip
deflection
in zdirection
267 mm
(~10.5”)
Z
Coordinate
system
X
Material
Label
Tip
weight
0.54 N
4
Geometry & Materials
E = 69E9; % Aluminum Young's Modulus (GPa)
nu = 0.3; % Aluminum Poissons Ratio (unitless)
rho = 2700; % Aluminum Density (kg/m^3)
Z
X
Rigid constraint on face 5
Tip load on vertex 6 in
negative z-direction
5
Is this a good mesh?
generateMesh(sModel,"Hmax",0.0025);
6
Stress in x-direction & Displacement
(1) Stress through the thickness depicted in xdirection colorized what is the legend showing?
Legend color depicts
stress in x-direction
(2) What are the units?
Pa
Tension (top)
Compression (bottom)
(3) Why is the top red and bottom blue?
(4) Has the 400 MPa elastic limit been
reached?
No
(4) What is the max tip deflection in zdirection? (example shown – using
“data tips” in figure menu)
(5) What are the units?
meters
7
Is displacement in x-direction
correct?
How do you calculate tip deflection in
x-direction? (example shown)
- Predicted tip deflection in x-direction is 0.000369 m
- Because loads are only in z-direction, this linear FE model will not capture xdisplacements
- Only a non-linear model to capture x-deflections because the load direction
with respect to structure will change as the structure deforms
8
Create Geometry, apply material properties,
and set BCs
Step 1: Construct the structural model using createpde
sModel = createpde('structural','static-solid’);
Step 2: Create beam geometry using multicuboid
gm = multicuboid(0.711,0.019,0.0015); %length (m), width (m), height (m)
sModel.Geometry = gm;
Step 3: Set material properties using structuralProperties
E = 69E9; % Aluminum Young's Modulus (GPa)
nu = 0.3; % Aluminum Poissons Ratio (unitless)
rho = 2700; % Aluminum Density (kg/m^3)
structuralProperties(sModel,'YoungsModulus',E,'PoissonsRatio',nu,'MassDensity',rho);
Step 4: Set boundary conditions using structuralBC
structuralBC(sModel,'Face',5,'Constraint','fixed');
Step 5: Visualize the beam geometry using pdegplot
figure
pdegplot(sModel,'VertexLabels','on','FaceAlpha',0.5,'FaceLabels','on’)
title('Faces and Vertices')
9
Apply loads, generate mesh, and solve
Step 6: Apply loads using structuralBoundaryLoad and structuralBodyLoad
structuralBoundaryLoad(sModel,'Vertex',6,'Force',[0;0;-0.5396]); % 0.54 N (2 oz mass)
structuralBodyLoad(sModel, ...
"GravitationalAcceleration",[0;0;-9.8]); % 65 mm tip deflection
Step 7: Generate mesh using generateMesh
generateMesh(sModel);
figure % plot mesh
pdemesh(sModel); title('FE Mesh')
a. (10 pts) How many elements, nodes, and DOF
(nodes times 3 for solid elements) are there?
Step 8: Solve the PDE using solve
R = solve(sModel)
10
Plot stress and displacements
Step 9: Plot x-stress contours using pdeplot3D
pdeplot3D(sModel, ...
"ColorMapData",R.Stress.sxx, ...
"Deformation",R.Displacement)
title('X Stress')
figure
pdeplot3D(R.Mesh, ...
"ColorMapData",R.VonMisesStress, ...
"Deformation",R.Displacement, ...
"DeformationScaleFactor",2)
a. (10 pts) Why is x-dir stress through thickness colorized red on top and blue on bottom?
b. (10 pts) What does the legend show
c. (10 pts) What are the units?
11
HW6 Directions and Deliverables
Step 10: Plot x- and z- displacements using pdeplot3D
figure
pdeplot3D(sModel, ...
"ColorMapData",R.Displacement.uz, ...
"Deformation",R.Displacement, ...
"DeformationScaleFactor",2)
title('Z Displacement')
12
HW6 Directions and Deliverables
Step 11: Calculate maximum displacements in x- & z-direction from R
disp(sprintf('Max absolute X-displacement %8.6f m',max(abs(R.Displacement.ux))))
disp(sprintf('Max absolute Z-displacement %8.6f m',max(abs(R.Displacement.uz))))
a. (10 pts) Compare max FE vs measured (see slide 2) tip deflection in x-direction?
b. (10 pts) Compare max FE vs measured (see slide 2) tip deflection in z-direction?
c. (10 pts) Discuss agreement/disagreement between measured and predicted
Step 12: Calculate maximum x- and Von Mises stresses from R
disp(sprintf('Max x-direction stress = %8.3f MPa',max(abs(R.Stress.sxx))/1E6))
disp(sprintf('Von Mises stress = %8.3f MPa',max(abs(R.VonMisesStress))/1E6))
a. (10 pts) What is the max x-direction stress?
b. (10 pts) What is the max Von Mises stress?
c. (10 pts) Has the 400 MPa elastic limit been reached – either x- or Von Mises?
13
Conclusions
- MATLAB’s Partial Differential Equation Toolbox can be used to solve simple
finite element models
- Far more robust FE model packages are available
- For static analysis of the cantilever beam, we
found that our predicted z-displacements were in
somewhat good agreement with measured
- We learned there are limitations on linear
analyses, especially as structure deforms and
loads change with respect to structure
Top 10 FEA Software Used By
Engineers (mechanicalland.com)

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