## Finite Element Analysis: FEA in Two and Three Dimensions

two-dimensional elements is presented.

### Stiffness Matrix and Nodal Forces Vector for a General 3D Linear Elastic Element

In three dimensions, the displacement vector of an element has three components designated , , and such that: Assuming that the element has nodes, then, each node has 3 nodal degrees of freedom designated , , and . So, the nodal degrees of freedom vector can have the form: If is the shape function associated with node , then the displacement vector function has the form: For simplicity, the vector representation of the small strain matrix will be used. In this case, the small strain matrix can be written in vector form as a function of the nodal degrees of freedom vector as follows: For a general linear elastic material, the vector representation of the stress matrix is related to the vector representation of the small strain matrix by a symmetric material coefficients matrix : Utilizing the principle of virtual work, the virtual displacement field and the associated virtual strain field can have the following forms with being an arbitrary “virtual” vector of degrees of freedom of the system: where The internal virtual work integral term for this particular element will have the form: where the integration is done on each term in the matrix and is performed over the volume (in 3D problems) of the element. The local stiffness matrix has dimensions and has the form: The external virtual work integral term for this particular element will have the form: Where is the traction vectors on the surface of the element, is the mass density, and is the body forces vector on the element. The integration in the first term is done over the element surface while in the second term is done over the element volume. The local nodal forces vector will components and will have the form: ### Stiffness Matrix and Nodal Forces Vector for a General 2D Linear Elastic Element

The equations in the previous section are repeated after reducing them to two dimensions. In this case, the displacement vector of an element has two components designated and such that: Assuming that the element has nodes, then, each node has 2 nodal degrees of freedom designated and . So, the nodal degrees of freedom vector can have the form: If is the shape function associated with node , then the displacement vector function has the form: For simplicity, the vector representation of the small strain matrix will be used. In this case, the small strain matrix can be written in vector form as a function of the nodal degrees of freedom vector as follows: For a general plane strain or plane stress linear elastic material, the vector representation of the stress matrix is related to the vector representation of the small strain matrix by a symmetric material coefficients matrix : Utilizing the principle of virtual work, the virtual displacement field and the associated virtual strain field can have the following forms with being an arbitrary “virtual” vector of degrees of freedom of the system: where The internal virtual work integral term for this particular element will have the form: where the integration is done on each term in the matrix and is performed over the volume (which in the 2D case would be an integration over the area multiplied by the thickness) of the element. The local stiffness matrix has dimensions and has the form: The external virtual work integral term for this particular element will have the form: Where is the traction vectors on the surface of the element, is the mass density, and is the body forces vector on the element. The integration in the first term is done over the element surface while in the second term is done over the element volume. The local nodal forces vector will components and will have the form: 