The fundamental equation governing linear static problems in Finite Element Analysis (FEA) tools like Abaqus is KU = F, where the global stiffness matrix (K) multiplied by the unknown displacement vector (U) equals the external force vector (F). This equation captures the complete physical behavior of the system, including material stiffness, geometric properties of the physical domain, applied loads, and boundary conditions.
The global stiffness matrix (K) is formed by combining the stiffness properties of all individual elements, each defined by its local stiffness matrix (k). This assembled matrix represents the overall structural response to external loads, determining how the structure deforms under applied forces. By accounting for the connectivity between elements and the physical properties of the material and geometry, it provides a complete mathematical model of the structure’s stiffness.
In this article, let’s discuss how a global stiffness matrix (K) can be assembled from local stiffness matrices using a simple 1D bar problem as discussed below.
Problem Definition
As shown in the figure below, we have one dimensional bar of 3 mm in length. The bar has square cross-section with an area of 1mm2.
The bar is divided into three 1-D elements and four nodes as depicted in the figure. Each element is 1mm long and has two nodes. The nodes (2) and (3) are shared by two adjacent elements whereas the nodes (1) and (4) belong to only one element each. Since the elements are 1D, they have only one degree of freedom at each node. Hence the overall unknown degrees of freedom for this 1D problem are 4. If the physical domain is multi-dimensional, then the total unknown degrees of freedom can be calculated as,
Total DOF = Number of nodes * Number of DOF per node
Calculating Elemental Stiffness Matrix
The first step in building global stiffness matrix is to calculate element level stiffness matrices. The element level stiffness matrices are calculated using shape functions and their first derivatives. As explained in our previous blog here, the linear shape functions for the 1D bar elements are derived to be
Here, x2 – x1 represents the length of each element which is equal to 1 in the current problem. Therefore, the above shape function can be simplified as,
N1 corresponds to the shape function of the element at its local node 1 whereas N2 corresponds to the shape function of the element at its local node 2.
The derivatives of the shape functions can be calculated as,
Now, we have to calculate these derivatives for each element in the physical domain.
In the above equations, the number inside [ ] represents the global element number. As the elements have same length, the first derivatives of shape functions for all elements are equal. Care must be taken when the elements have different lengths which would lead to different first derivative values of shape functions for each element.
For the 1D bar problem, the element stiffness matrix is given as,
The stiffness matrices for the elements can be calculated using the above equation as follows,
Building Global Stiffness Matrix
The above elemental stiffness matrices are used to calculate the global stiffness matrix. The global stiffness matrix is a square matrix of size equal to total number of DOF in the problem domain. So, the size of global matrix in the current problem is 4.
The first step to building the global stiffness matrix is to identify the numbering of local and global degrees of freedom. For element 1, the local DOF numbering is 1 and 2 and the global DOF numbering is 1 and 2 as well. So, we will fill the global matrix as shown below.
For element 2, the local DOF numbering is still 1 and 2 and the global DOF numbering is 2 and 3 as the second node is shared between elements 1 and 2. So, we will fill the global matrix as shown below. If the matrix position is already occupied, we will add the new entry to the old one.
For element 3, the local DOF numbering is still 1 and 2 and the global DOF numbering is 3 and 4 as the third node is shared between elements 2 and 3. So, we will fill the global matrix as shown below. If the matrix position is already occupied, we will add the new entry to the old one.
Hence the overall global matrix ca be calculated as,
Constructing the global stiffness matrix becomes increasingly challenging as the problem size grows and when higher-order elements are used. However, understanding the global and local node and degree of freedom (DOF) numbering patterns can greatly help in the process. If you are using commercial FEA software, building of the global stiffness matrix for your FEA problem is handled by the software.
Final Thoughts
The global stiffness matrix is one of the fundamental components of FEA, which combines the stiffness contributions of all individual elements into a single matrix, that governs how the entire physical system behaves under load. Hopefully, this article has provided a basic understanding on how to build global stiffness matrix from element stiffness matrix and it might just help you understand how your next model is actually working?
If you are interested in building global stiffness matrix for your FEA problem using Abaqus, don’t hesitate to reach out to our expert stiffness matrix building team!
