When assembling a global stiffness matrix for a frame, how does the connectivity information of elements directly translate into the placement of individual element stiffness matrix terms within the global matrix?

The connectivity information of elements directly translates into the placement of individual element stiffness matrix terms within the global matrix through a systematic mapping of degrees of freedom. First, it is essential to understand what these matrices and terms represent. The global stiffness matrix (K) is a large square matrix that represents the stiffness of the entire frame structure, relating all global nodal forces to all global nodal displacements. An element stiffness matrix (k_e) is a smaller square matrix that describes the stiffness properties of a single, isolated frame element, relating the forces and moments at its ends to the corresponding displacements and rotations of those ends. Degrees of Freedom (DOFs) are the independent displacements and rotations that define the deformation of a node. For a typical two-dimensional frame, each node has three DOFs: a translation in the global X-direction, a translation in the global Y-direction, and a rotation about the global Z-axis. A single element connecting two nodes therefore possesses six DOFs (three at its 'start' node and three at its 'end' node), leading to a 6x6 element stiffness matrix. For a three-dimensional frame, each node has six DOFs (three translations and three rotations), leading to a 12x12 element stiffness matrix. The explanation below focuses on a 2D frame for simplicity, but the principle is identical for 3D. Before assembly, all DOFs in the entire frame structure are assigned unique numerical identifiers, known as global DOFs. These global DOF numbers are assigned sequentially across all nodes of the structure. For instance, node 1 might have global DOFs 1, 2, 3 (X, Y, Rotation), node 2 might have global DOFs 4, 5, 6, and so on. Connectivity information explicitly defines which global nodes each element connects. For example, an element 'A' might connect global node 'i' to global node 'j'. This connectivity is the key to mapping. For each element, its local DOFs (e.g., the first three for its 'start' node and the next three for its 'end' node) are directly mapped to the global DOFs of the nodes it connects. So, if element 'A' connects global node 'i' (with global DOFs G_i1, G_i2, G_i3) to global node 'j' (with global DOFs G_j1, G_j2, G_j3), then the element's six local DOFs will correspond to the set of global DOFs {G_i1, G_i2, G_i3, G_j1, G_j2, G_j3} in that specific order. This ordered list of global DOF numbers for an element is often called its global address vector. The actual placement of the element stiffness terms into the global stiffness matrix is performed using the direct stiffness method. The global stiffness matrix is initially assembled as a zero matrix of appropriate size (total number of global DOFs x total number of global DOFs). Then, for each element in the frame, its calculated element stiffness matrix terms are added to the global stiffness matrix. Specifically, if an element stiffness matrix term is k_pq (representing the force at element's local DOF 'p' due to unit displacement at element's local DOF 'q'), and if local DOF 'p' maps to global DOF 'M' and local DOF 'q' maps to global DOF 'N' (from the element's global address vector), then the term k_pq is added to the entry K_MN in the global stiffness matrix. This process is repeated for all terms of all element stiffness matrices. If multiple elements contribute to the same global stiffness matrix entry (i.e., they share a node and thus some global DOFs), their stiffness contributions are simply superposed (added together) into that single global matrix entry. This systematic mapping and summation ensures that the assembled global stiffness matrix accurately reflects the stiffness contribution of every part of the structure, considering their interconnections.