Finite Element Modeling for Stress-Strain Analysis in 2D Structures
Explore the implementation of 2D stress-strain Finite Element Modeling using MATLAB. Understand the concepts of stress, strain, plane stress conditions, stress-strain relation, Turner Triangle in FEM, linear interpolation, stress and strain vectors, static equilibrium, and stiffness matrix. Learn how to differentiate displacement to obtain strain, use stress-strain relation to calculate stress, and achieve static equilibrium by minimizing potential energy, along with analyzing stiffness matrix.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
Implementation of 2D stress-strain Finite Element Modeling on MATLAB Xingzhou Tu
What is stress? Stress is a tensor ??? ??? ??? ??? ??? ??? ??? ??? ??? ?? ?? ?? ?? ?? ?? = 2
What is strain? Strain is also a tensor ? + ?. ?. Deformation +??? ???) ???=1 2(??? ??? 3
2D case Plane stress no stress in z-direction Things need to consider: ??? ??? ??? ??? ??? ??? 4
Stress-Strain Relation in 2D case E: Young s Modulus V: Poisson Ratio ? ??? ??? 2??? ??? ??? ??? 1 ? 0 ? 1 0 0 0 ? = 1 ?2 1 ? 2 ? 5
FEM: Turner Triangle Three nodes ?1,? ?1,? ?2,? ?2,? ?3,? ?3,? ?1,? ?1,? ?2,? ?2,? ?3,? ?3,? ? = ? = 6
FEM: Linear Interpolation Use linear interpolation to evaluate displacement at other points ??(?,?) ??(?,?)= ?1(?,?) 0 ?1?,? =2 ?2?3 ?3?2 + ?(?2 ?3) + ?(?3 ?2) ?2?,? =2 ?3?1 ?1?3 + ?(?3 ?1) + ?(?1 ?3) ?1?,? =2 ?1?2 ?2?1 + ?(?1 ?2) + ?(?2 ?1) 0 ?2(?,?) 0 0 ?2(?,?) 0 0 ?2(?,?)? ?1(?,?) ?2(?,?) 2? 2? 2? A is the area of the triangle. 7
FEM: stress and strain vectors Differentiate the displacement to get the strain vector Use the stress-strain relation to get the stress vector ? = ??, ?3 ?1 0 ?1 ?3 ?2 ?3 0 ?3 ?2 0 0 ?1 ?2 0 ?2 ?1 ? 1 0 0 1 ?3 ?2 ?2 ?3 ?1 ?3 ?3 ?1 ?2 ?1 ?1 ?2 ? = 2? 1 ? 0 0 0 ? ? = ? ? = ???, ? = 1 ?2 1 ? 2 8
FEM: static equilibrium Static equilibrium <-> Minimize Potential Energy ? =1 2 ????? ??? h is the thickness of the 2D domain in the z- direction. =1 2 ????????? ??? =1 2??(? ????)? ??? 9
FEM: Stiffness Matrix Minimize Potential Energy -> ?? = 0 ? = ? ????? = ??, ? = ? ????, K is the stiffness matrix of the element 10
Implementation: Problem Cantilever beam Beam Dimension: 100mm*10mm*10mm 10N load at the end of the beam Aluminum: Young s modulus = 70000N/mm^2 Poisson ratio = 0.33 11
Implementation: Meshing 4000 elements 2111 nodes 1mm 12
Implementation: System Stiffness Matrix 2111 nodes: Each node has two degrees of freedom 4222*4222 system stiffness matrix Build a 4222-by-4222 zero matrix Traversing each element: Calculate the stiffness matrix of this element Adding the matrix into the corresponding row and column of the system matrix. 13
Implementation: Boundary Condition Two kinds of Boundary Condition: Fixed Support: zero displacement at the fixed node External load: 10N load at the end of the beam 0N load inside the beam Unknown load (reaction force) at the fixed node 14
Implementation: Solving 4222 unknowns, 4222 equations. ? :: ? 0: : 0 0 Unknown reaction force at fixed nodes 0 :: 0 ? :: ? Zero displacement at fixed nodes Zero load inside the beam = ? Unknown displacement at other nodes 10N load on the y direction at the end of the beam 10 15
Implementation: Result (Displacement plot) Maximum ?? = 57.124 ?? 16
Implementation: Theory Prediction FEM simulation result: ? = 57.124 ?? Theory Prediction: ? =??3 10? 100??3 3??= 3 70000??? 2 1 12 10?? 10??3 = 57.143?? Matches perfectly! 17
Questions? Thanks for your listening! 18
