Finite element-rod elements what is the pde

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The discussion focuses on the application of finite element analysis (FEA) using simple rod elements to derive partial differential equations (PDEs). Specifically, it highlights that the stiffness matrix and displacements can be determined from the governing equation derived from Hooke's Law, expressed as Stress = E * Strain. The conversation emphasizes that the PDE varies based on the structure and configuration, such as three rods with different cross-sectional areas connected in series. It also suggests using Galerkin's method and Gauss' theorem to relate finite element equations back to the PDE.

PREREQUISITES
  • Understanding of finite element analysis (FEA)
  • Familiarity with Hooke's Law and material properties
  • Knowledge of Galerkin's method
  • Basic principles of partial differential equations (PDEs)
NEXT STEPS
  • Study the derivation of finite element equations from Hooke's Law
  • Learn about Galerkin's method and its applications in FEA
  • Explore the equation of virtual work in structural analysis
  • Review potential energy methods in finite element analysis
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Engineers, researchers, and students in structural analysis, particularly those focused on finite element methods and the mathematical foundations of PDEs in engineering applications.

chandran
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case 1)in finite element analysis of structures using simple rod elements we do the stiffness matrix and then find the displacements from loads and constraints


case 2)finite element method is a technique for solving partial differential equations. In the case1 what is the partial differential equation and what it looks like?
 
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?? The partial differential equation depends on exactly what the problem is. There are many different "structures" that will give different differential equations.
 
i have an example. There are three rods of different cross sectional areas
A1,A2,A3. with lengths L1,L2,L3 connected in series. What will be the pde
which when tried to solve by fem will give the stiffness matrix and force,displacements
 
For your simple 1D rod example, the simple relationship for Hooke's Law will provide the governing equation you require.

Stress = E* strain (from d(stress)/dx = 0)

You can then build the finite element equations from that point. Reference to any reasonably simple FEM textbook will assist you to solve those fundamentals.
 
If you've your FE equations, you can interpret FE as an application of Galerkin's method and work your way backward to the PDE with a suitable application of Gauss' theorem, integration by parts etc. Kind of like deriving BEM.
 
In most cases of structural analysis FEM, the differential equation is the equation of virtual work. Or, you can apply an energetic approach and derive potential energy equations.
 

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