Finite

In this post, I'll study the isoparametric formulation of the plane 4-node quadrilateral (Q4) element stiffness matrix. Additionally, Gauss integration will be used to calculate the stress at given points in a plane quadrilateral element. The isoparametric element equations are formulated using the natural coordinate system $s$, defined by the element geometry. The axial coordinate $s$ is affixe..

In this post, I'll solve an example of the LST element which was explained in previous post. The triangle has a six nodes, with a height of $h$ and a width of $b$ Displacement can be calculated as follows using the coordinates of the six nodes. Upon rearranging for ${a_i}$ By substituting the two equations into the displacement function, it can be expressed as follows. This can be represented in..

In this post, I'll explain the stiffness matrix and equations of the Linear-Strain Triangle(LST) element, which differs from the Constant-Strain Triangle(CST) element covered in the last post. Unlike CST, LST has six nodes and twelve degrees of freedom, and its displacement function is a quadratic function, unlike the linear function in CST. In LST, the node displacements are as follows. And the..

In this post, I intend to explain the process of deriving the stiffness matrix for a Q4 element, which is quadrilateral with four nodes, in a plane. The displacement function for each element should be linear as it is located at the edges. Starting from the lower left corner of the rectangle and numbering the nodes counterclockwise, with the geight from the center being $h$ and the width being $..

Based on the theory explained in previous posts, I intend to solve a simple example. In the element, $E$=210Gpa, $v$=0.25, $t$=20mm, ${u_1}$=0, ${v_1}$=0.05mm, ${u_2}$=0.025mm, ${v_v}$=0, ${u_3}$=0, and ${v_3}$=0.05mm. For an element as described above, $\beta $, $\gamma $ are as follows. Through this, the $B$ matrix can be obtained. Knowing the area, $D$ can be calculated as follows. Therefore,..

Continuing from the previous post, I intend to write about stress-strain relationship In the plane, the stress-strain relationship can be expressed as follows. Using the principle of minimum potential energy to express the equation of the CST element in its weak form yields the following. And the strain energy is Therefore The potential energy of the body forces is as follows. Where $\psi $ is t..

Continuing from the previous post, I intend to write about the strain in Constant-Strain Triangular(CST) elements. The strain of a two-demensional element is as follows. In the CST element, the strain within the element is always constant, and the edges remain straight even when the element deforms. Using the $u(x,y)$, $v(x,y)$ equations frome the previous post, it can be transformed as follows...

In this post, I'll explain the equations required for the Constant-Strain Triangular(CST) element. The basic triangle in the picture has nodes $i$, $j$, and $m$. Triangular elements are often used to approximate curved shapes. Each node has two degrees of freedom in $x$ and $y$, and the displacement matrix for the nodes is as follows. The linear displacement function for each element can be repr..

Continuing from the last post, today I intend to explain Castigliano's First Theorem. Castigliano's First Theorem can be summarized as 'the partial derivative of the strain energy of any elastic structure with respect to displacement is equal to the load' and I intend to explain this using the corresponding equation. In an elastic structure where $n$ external forces ${P_1},{P_2}, \cdots ,{P_n}$ ..

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..