Question

Ein finites Element in einer Scheibe der konstanten Dicke \(\mathrm{s}=0,5 \mathrm{~cm}\) (ebener Spannungszustand) habe die Knotenpunkte \(\left(\mathrm{x}_{1}=0, \mathrm{y}_{1}=0\right),\left(\mathrm{x}_{2}=4 \mathrm{~cm},\right.\), \(\mathrm{y}_{2}=0\) ) und \(\left(\mathrm{x}_{3}=2 \mathrm{~cm}, \mathrm{y}_{3}=3 \mathrm{~cm}\right) .\) Die Scheibe verhalte sich linearelastisch \(\left(\mathrm{E}=21 \cdot 10^{6} \mathrm{~N} / \mathrm{cm}^{2}, \mathrm{v}=0,3\right) .\) Man setze lineare Formfunktionen voraus und ermittle die Steifigkeitsmatrix des finiten Elementes.

Short answer

The stiffness matrix is determined using the formula \[ K = t \cdot A \cdot B^T \cdot D \cdot B \].

Step-by-step solution

- Calculate the Area of the Element

Use the coordinate points to calculate the area of the triangular element. The formula for the area given three points \[ (x_1, y_1), (x_2, y_2), (x_3, y_3) \] is \[ A = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) | \]. Substituting the values, we get \[ A = \frac{1}{2} | 0(0 - 3) + 4(3 - 0) + 2(0 - 0) | = \frac{1}{2} | 0 + 12 + 0 | = 6 \text{ cm}^2 \].

- Form the B-Matrix

Determine the B-matrix for the triangular element using the formula \[ B = \begin{bmatrix} \frac{y_2 - y_3}{2A} & 0 & \frac{y_3 - y_1}{2A} & 0 & \frac{y_1 - y_2}{2A} & 0 \ \ 0 & \frac{x_3 - x_2}{2A} & 0 & \frac{x_1 - x_3}{2A} & 0 & \frac{x_2 - x_1}{2A} \ \ \frac{x_3 - x_2}{2A} & \frac{y_2 - y_3}{2A} & \frac{x_1 - x_3}{2A} & \frac{y_3 - y_1}{2A} & \frac{x_2 - x_1}{2A} & \frac{y_1 - y_2}{2A} \end{bmatrix} \]. Using the coordinates, we get \[ B = \begin{bmatrix} 0 - 3 & 0 & 3 - 0 & 0 & 0 - 0 & 0 \ 0 & 2 - 4 & 0 & 0 - 2 & 0 & 4 - 0 \ 2 - 4 & 0 - 3 & 0 - 2 & 3 - 0 & 4 - 0 & 0 - 0 \end{bmatrix} \frac{1}{2 \times 6} \ = \begin{bmatrix} -0.25 & 0 & 0.25 & 0 & 0 & 0 \ 0 & -0.167 & 0 & 0.167 & 0 & 0.333 \ -0.167 & -0.25 & -0.083 & 0.167 & 0.333 & 0 \end{bmatrix} \].

- Determine the D-Matrix

The D-matrix can be determined using the material properties for plane stress. The formula is \[ D = \frac{E}{1 - v^2} \begin{bmatrix} 1 & v & 0 \ v & 1 & 0 \ 0 & 0 & \frac{1 - v}{2} \end{bmatrix} \]. Given \[ E = 21 \times 10^6 \mathrm{~N} / \mathrm{cm}^2 \] and \[ v = 0.3 \], we get \[ D = \frac{21 \times 10^6}{1 - (0.3)^2} \begin{bmatrix} 1 & 0.3 & 0 \ 0.3 & 1 & 0 \ 0 & 0 & \frac{1 - 0.3}{2} \end{bmatrix} \approx 23.1 \times 10^6 \begin{bmatrix} 1 & 0.3 & 0 \ 0.3 & 1 & 0 \ 0 & 0 & 0.35 \end{bmatrix} \].

- Calculate the Stiffness Matrix

Finally, determine the stiffness matrix using the formula \[ K = t \cdot A \cdot B^T \cdot D \cdot B \], where \[ t = 0.5 \text{ cm} \], \[ A = 6 \text{ cm}^2 \], \[ B \text{ and } D \text{ are the matrices calculated in the previous steps} \]. Multiply these components together to get the stiffness matrix.

Key concepts

Stiffness Matrix

The stiffness matrix is a fundamental concept in finite element analysis (FEA). It represents the relationship between nodal displacements and forces in a structure. In essence, it translates the physical properties of a material and its geometric configuration into a numerical model that can be solved computationally.

The stiffness matrix is computed using the equation \[ K = t \times A \times B^T \times D \times B \] where

• \( t \) is the thickness of the element
• \( A \) is the area
• \( B \) is the strain-displacement matrix
• \( D \) is the material property matrix.

By multiplying these components, we obtain a matrix that characterizes the response of the triangular element to applied loads, allowing for the prediction of displacements, stresses, and strains.

Triangular Element

A triangular element is a three-node, two-dimensional finite element used in FEA. Its simplicity makes it extremely versatile for modeling complex geometries. The coordinates of these nodes are given and form a triangle which is used to define the element's geometry.

In this exercise, the triangular element is defined by vertices at \((x_1, y_1), (x_2, y_2),\) and \((x_3, y_3)\). The triangular shape helps in simplifying the calculations and enabling the use of linear shape functions. These shape functions are assumed in this problem to interpolate the displacement field within the element.

B-Matrix

The B-Matrix, or strain-displacement matrix, plays a crucial role in relating the displacements to the strains within an element. It is derived from the shape functions and is used in the calculation of the stiffness matrix.

The formula for the B-Matrix in a triangular element is:
\[ B = \begin{bmatrix} \frac{y_2 - y_3}{2A} & 0 & \frac{y_3 - y_1}{2A} & 0 & \frac{y_1 - y_2}{2A} & 0 \ 0 & \frac{x_3 - x_2}{2A} & 0 & \frac{x_1 - x_3}{2A} & 0 & \frac{x_2 - x_1}{2A} \ \frac{x_3 - x_2}{2A} & \frac{y_2 - y_3}{2A} & \frac{x_1 - x_3}{2A} & \frac{y_3 - y_1}{2A} & \frac{x_2 - x_1}{2A} & \frac{y_1 - y_2}{2A} \end{bmatrix} \].

Inserting the given coordinates, the B-Matrix is then computed and used in conjunction with the D-Matrix to form the stiffness matrix.

D-Matrix

The D-Matrix represents the material properties of the element under plane stress conditions. It is derived using the material's Young's modulus \(E\) and Poisson's ratio \(u\).

The formula for the D-Matrix is:
\[ D = \frac{E}{1 - u^2} \begin{bmatrix} 1 & u & 0 \ u & 1 & 0 \ 0 & 0 & \frac{1 - u}{2} \end{bmatrix} \].

In this exercise, the values are \(E = 21 \times 10^6 \) N/cm² and \( u = 0.3 \). Plugging these values into the formula gives us the specific D-Matrix for this problem. The D-Matrix thus obtained quantifies the material's response to mechanical loads and its inclusion in the stiffness matrix calculation allows for accurate modeling of stress and strain distribution.

Linear Elasticity

Linear elasticity is the assumption that the material deformation under load is directly proportional to the applied load. This is a fundamental concept in structural analysis as it simplifies the relationship between stress and strain.

Mathematically, linear elasticity is described by Hooke's Law, \( \sigma = E \epsilon \), where \( \sigma \) is the stress, \( E \) is the Young's modulus, and \( \epsilon \) is the strain.

In the context of this exercise, the assumption of linear elasticity allows us to use the material constants (E and \(u\)) to determine the D-Matrix. This assumption holds true as long as the material is within its elastic limit, meaning it will return to its original shape upon removal of the load. Understanding this concept is essential for accurately predicting the response of the triangular element and the entire structure using finite element analysis.