Finite

Today, I'll explain Castigliano's Second Theorem, which is used to find the deflection of elastic structures based on the strain energy of the structure. Castigliano's Second Theorem can be summarized as 'the partial derivative of the strain energy of an elastic structre with respect to the load is equal to the displacement' and I intend to explain this using the corresponding equation. Assuming..

${\sigma _x}$ and ${\sigma _y}$ is the normal stress, ${\tau _{xy}}$ is the shear stress acts on the x edge in the $y$ direction, ${\tau _{yx}}$ is the shear stress acts on the $y$ edge in the $x$ direction. Represented by the vector column matrix as follows. And general definitions of normal and shear strins as follows. Represented by the vector column matrix as follows. For plane stress, the a..

Today, I intend to explain the last of the finite element equation methods : the weighted residual methods Among them, the most widely used Galerkin's method will be explained. Assuming there is a differential equation $Du(x)=f(x)$ for $u(x)$, and the approximate solution for $u$ is denoted as $\hat u$, then since $u \ne \hat u$, $Du(x)=f(x)$ is satisfied, but $D\hat u - f \ne 0$. This is referr..

The book(Darly L.Logan - A First Course in the Finite Element Method) explains the principle of virtual work as follows. If a deformable body in equilibrium is subjected to arbitrary virtual (imaginary) displacements associated with a compatible deformation of the body, the virtual work of external forces on the body is equal to the virtual strain energy of the internal stresses. In other words,..

I solve an example problem using the direct method explained in previous post. The problem as follows. Each element stiffness matrix as follows To expand each element stiffness matrix to the size of the global stiffness matrix, simply add rows and columns of zeros for displacements not related to that element. Assembled in to the global stiffness matrix, as follows. Organized in the form K=KU Gi..

Three main ways to derive finite element equations. 1. Direct method or Direct equilibrium method for structural analysis problems 2. Energy Methods and principle of virtual work 3. Weighted Residual Methods Of the three methods, the direct method is the simplest and can clarify the physical understanding of the FEM. However, it is limited to finding the stiffness matrix of a one-dimensional ele..

I remembered that in my previous post(13_Pendulums) I said that I would explain the Finite Element Method(FEM) later. 13_Pendulums There are a lot of interesting examples in the RADIOSS example, so I'll be working through them for a while. In this post, I'll going to perform a Pendulums example. There was also an example file in HyperMesh's installation path. However, due to the diffe finite-ele..