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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