Finite

In this post, I aim to solve an example that involves determining the temperature at each point along an insulated rod with a total lengh $L$. The left end is fixed at ${100^ \circ }{\text{C}}$, and a heat flux of $q = 5000{\text{W/}}{{\text{m}}^2}$ acts on the right end. Here, ${K_{xx}}$ is ${\text{6W/(m}}{{\text{ - }}^{\text{o}}}{\text{C)}}$, and the cross-section is a circle with an area of $..

In a previous post, I theoretically covered the method for calculating the stiffness matrix of a tetrahedron. In this post, based on that theory, I aim to derive the stiffness matrix of a tetrahedron through an example. The given tetrahedral element is as follows. In the given element, $E = 200{\text{GPa}},\,v = 0.3$. To calculate the element stiffness matrix first, it is necessary to determine ..

In the field of structural analysis and Finite Element Analysis (FEA), I intend to explain the process of deriving the stiffness matrix for a basic three-node bar element. Unlike previous postings(39_High Order Example), this time I plan to calculate the stiffness matrix of a three-node bar element using two-point Gaussian integration. A bar element with three nodes and two Gaussian points is as..

In this post, we are dealing with a linear plane rectangular element that has four nodes, and the issue at hand involves calculating the forces acting on each node when there is a uniform surface tension. The thickness of the element is 0.25cm, as follows. First, the formula for calculating the surface traction needed to solve the problem is as follows. To calculate the surface tension for edge ..

Today, I plan to solve a problem related to finite elements, something I haven't done in a while. The problem is as follows. Here, the goal is to fine the nodal displacements and element stresses. $E$ is $210GPa$, and Poission's ratio is 0.3. First, discretize the plate into two elements and simplify the tensil surface traction into nodal forces for calculation. Applying $\left[ F \right] = \lef..

We derive the fundamental differential equation for one-dimensional heat flow due to thermal conduction, convection, and mass transfer in a fluid. The Galerkin's residual method is used for problems where variational methods are not applicable. We start with the equation for thermal energy conservation discussed in the previous post(42_One-Dimensional Heat Conduction). The mass transfer term(${q..

Today, based on what we studied in the previous post, I'll solve an example problem. In the figure above, we calculate the temperature distribution along the insulated external boundary. The temperature at the left side is ${40^ \circ }{\text{C}}$, the convection coefficient $h = 100{\text{W/}}{{\text{m}}^{\text{2}}}{ \times ^{\text{o}}}{\text{C}}$, the free-steam temperature ${T_\infty } = {10^..

Having recently addressed a 1-D problem, today I aim to solve a 2-D problem. The goal of this post is to study the necessary formulas before solving an example. First, the basic element is a triangle with three nodes, and the temperature function is given as follows. Where, ${t_i}$, ${t_j}$ and ${t_m}$ are the nodal temperatures, and the shape functions are as follows. The same applies to ${N_j}..

Similar to the problem addressed in the previous post, in this post, I'll solve an example that calculates the temperature distribution and rate of heat in a rod fixed to a wall, without any insulated parts. In the figure above, the temperature at the left end is fixed at ${100^{\text{o}}}{\text{C}}$, ${K_{xx}} = 60{\text{W/(m}}{{\text{ - }}^{\text{o}}}{\text{C)}}$, radius is $50mm$, $h = 800{\t..

Based on what we studied in the previous post, I'll solve an example problem. In the figure above, we calculate the temperature distribution along the insulated external boundary. The temperature at the left end is fixed at ${40^ \circ }{\text{C}}$, and the free-stream temperature is ${-10^\circ}{\text{C}}$. The value of $h$ is ${\text{55W(}}{{\text{m}}^{\text{2}}}{{\text{ - }}^{\text{o}}}{\text..