Finite Element Method Fundamentals: From Mesh to Global Stiffness Matrix in Static Stress Analysis

This article explains the finite element method (FEM) for static stress analysis. It covers how a structure is discretized into small elements, how element stiffness matrices are formed, how global stiffness is assembled, and how boundary conditions and external loads yield nodal displacements. It also outlines how displacements lead to strains and stresses and highlights the role of software in managing the computations.

Introduction to the finite element method

The finite element method is a powerful numerical approach used when analytic solutions are impractical due to complex geometry, loading, or material behavior. In this context, FEM is applied to static linear elastic stress analysis, where the aim is to determine stresses, strains, and displacements within a solid body. Internal stresses arise to maintain equilibrium across any material volume, and the FEM provides a practical framework to satisfy equilibrium over a finite set of elements and nodes.

Discretization and mesh generation

The body is split into many small elements connected at nodes. This discretization reduces a continuous equilibrium problem to a finite set of equations that can be solved numerically. The collection of nodes and elements is called a mesh. Depending on the problem, different element shapes can be used, such as triangular surface elements for complex geometries or solid elements for 3D bodies. The choice of element type and mesh density affects accuracy and computational cost.

Element types and order

Elements come in several varieties: surface elements (2D, thin bodies), solid elements (3D), and line elements (1D). Within surface and solid elements, first order (linear) and second order (with mid-side nodes) formulations exist, offering different accuracies. For example, plane stress surface elements may be used for a thin bracket with loading in a single plane, while more complex 3D shapes may require solid elements or line elements for simplification.

Degrees of freedom and nodal displacement

For each element, a local displacement vector u contains the nodal displacements and rotations. In a 2D beam element, there are 6 degrees of freedom per element, corresponding to translations and rotations at the element’s nodes. In 3D, each node has 6 DOF, while shell and solid elements have their own DOF counts. The overall goal is to determine the displacement at every node in the mesh, which then allows calculation of strains and stresses.

Stiffness matrices and the Hooke relation

The relationship between nodal forces and displacements is governed by the element stiffness matrix k: F = kU. The stiffness matrix encodes how much each node will displace under a given set of forces and moments. The matrix is square, with dimensions equal to the total number of element degrees of freedom. For a 2D beam element, the stiffness matrix has a characteristic layout that reflects the element’s mechanical behavior, including axial, bending, shear, and torsional responses depending on the element type.

Global assembly

To analyze the entire structure, the individual element stiffness matrices are assembled into a global stiffness matrix. This process accounts for how elements connect at shared nodes, so terms corresponding to the same degree of freedom are added (continuity). The global matrix is typically sparse and banded, and for linear elastic problems, it is also symmetric. Different mesh configurations produce different global stiffness matrices, which in turn change the predicted displacements.

Boundary conditions and solving

External loads are applied through force vectors, and boundary conditions fix certain displacements at selected nodes. With these, the global equilibrium equation KU = F is solved for the nodal displacements. In practice, engineers avoid direct matrix inversion and use iterative solvers such as conjugate gradient methods to exploit the sparsity of K, enabling analysis of systems with large numbers of degrees of freedom.

Post-processing and outputs

Once displacements are known, strains and stresses throughout the mesh can be computed. Modern FEM analyses can involve hundreds of thousands of degrees of freedom, which would be impractical by hand, underscoring the importance of specialized software. Post-processing also includes model validation and interpretation to ensure results are physically meaningful for design decisions.

Derivation of stiffness and weak forms

Element stiffness can be derived via several methods. The direct method uses the element's equilibrium equations, often involving differential equations that describe the governing physics. For general problems, weak forms provide a more versatile framework by converting differential equations into integral forms. Two common weak form approaches are the principle of minimum potential energy and the Galerkin method. Both lead to the same stiffness matrix for stress analysis problems and set the stage for interpolating field variables inside the element through shape functions.

From problem definition to results

The typical FEM workflow begins with a clear problem definition including material properties, loads, and boundary conditions. Then the body is discretized, the stiffness matrices are defined and assembled, external loads and boundary conditions are applied, and the global system is solved to obtain nodal displacements. After that, stresses, strains and other field variables are extracted, and the model undergoes post-processing and validation. Software handles the heavy numerical tasks, while the engineer ensures the problem is well-posed, the mesh is appropriate, and the results are correctly interpreted.

Conclusion

This introduction outlines the fundamentals of the finite element method for static linear elastic analysis, highlighting discretization, element types, stiffness matrices, assembly, solving strategies, and result interpretation. It provides a foundation for more advanced topics such as discretization refinement, solver performance, and post-processing techniques.