# In my previous post on **finite element method** some matrix formations including global stiffness matrix [K]_{NPxNP} and local stiffness matrix [S]_{NFxNF} were introduced with the help of equations derived by equilibrium condition, elastic stress-strain condition and compatibility condition.

Let us try to understand what exactly are these matrices.

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**What is a Global Stiffness Matrix in Finite Element method?**

In the **finite element method (FEM)**, the global stiffness matrix represents the system of linear equations that relates the displacements of nodes in a finite element model to the applied loads and boundary conditions. It is derived from the assembly of the element stiffness matrices.

In finite element method, a complex domain is divided into a collection of smaller, simpler regions called elements. Each element has its own stiffness matrix, which describes the relationship between the nodal displacements and the internal forces within that element. The global stiffness matrix is obtained by assembling the individual element stiffness matrices into a single matrix that represents the entire system.

The global stiffness matrix is a square matrix with a size equal to the total number of **degrees of freedom** in the system. Each degree of freedom corresponds to a nodal displacement. The entries of the global stiffness matrix represent the stiffness coefficients relating the displacements of different nodes.

The assembly of the global stiffness matrix involves combining the contributions from all the element stiffness matrices. This is typically done by mapping the element stiffness matrices onto the global coordinate system using transformation matrices and then adding the corresponding entries to the appropriate locations in the global stiffness matrix.

Once the global stiffness matrix is assembled, it can be used to solve the system of equations that relates the nodal displacements to the applied loads and boundary conditions. This is typically done by applying appropriate solution techniques such as direct solvers or iterative methods to obtain the displacements of the nodes and the corresponding internal forces in the finite element model.

**What is a Local Stiffness Matrix in Finite Element method?**

In the finite element method (FEM), the local stiffness matrix represents the stiffness properties of an individual finite element. It describes the relationship between the nodal displacements and the internal forces within that element.

Each finite element in finite element method is associated with a local stiffness matrix. The size of the local stiffness matrix depends on the number of degrees of freedom associated with the element. For example, a 2D triangular element with three nodes will have a 6×6 local stiffness matrix (2 degrees of freedom per node).

The entries of the local stiffness matrix represent the stiffness coefficients that relate the displacements of different nodes within the element. These coefficients depend on the material properties, geometry, and element shape functions.

The local stiffness matrix is derived based on the element’s governing equations, which typically involve equilibrium and compatibility conditions. The specific derivation process depends on the element type and the underlying theory being used (e.g., linear elasticity, structural mechanics, etc.).

Once the local stiffness matrix is obtained, it can be used in the assembly process to construct the global stiffness matrix. The assembly involves mapping the local stiffness matrix to the global coordinate system and adding its contributions to the appropriate locations in the global stiffness matrix.

In summary, the local stiffness matrix represents the stiffness properties of an individual finite element and serves as a fundamental building block for constructing the global stiffness matrix in the finite element method.

**What is Virtual Work Principle and how is it used for Finite Element method?**

The principle of virtual work is a fundamental concept in structural mechanics that is used to analyze the equilibrium and deformation of structures. It provides a powerful tool for solving complex structural problems by relating the external loads applied to a structure to the internal forces and deformations within the structure.

The principle of virtual work is based on the idea that the work done by external loads acting on a structure is equal to the work done by the internal forces resulting from the forces is zero.

To understand the principle of virtual work, let’s consider a simple example of a loaded beam. The external loads applied to the beam, such as point loads or distributed loads, induce deformations in the beam, resulting in internal forces, such as bending moments and shear forces.

According to the principle of virtual work, if we consider an arbitrary small displacement or deformation of the beam, known as a virtual displacement, the work done by the external loads on the virtual displacement is equal to the work done by the internal forces resulting from the virtual displacement.

Mathematically, the principle of virtual work can be expressed as:

∑(δW_ext) + ∑(δW_int) = 0

Where: ∑(δW_ext) is the sum of the virtual work done by the external loads. ∑(δW_int) is the sum of the virtual work done by the internal forces.

By applying this principle, we can establish equations that relate the external loads to the internal forces and deformations of a structure. These equations can be solved to determine the unknown forces and deformations, enabling the analysis and design of complex structures.

It is important to note that the principle of virtual work assumes small displacements and linear elastic behavior of the material. It is a powerful tool for analyzing linear elastic structures but may not be applicable to structures with large displacements, non-linear materials, or systems undergoing significant plastic deformations.

Overall, the principle of virtual work provides a mathematical framework for analyzing the equilibrium and deformations of structures, allowing engineers to solve complex structural problems efficiently.

Finite Element method is a numerical method widely used for structural analysis. The principle of virtual work forms the basis of finite element method algorithms, where the discretized structure is divided into smaller elements. The virtual work principle is used to relate the nodal displacements, element stresses, and external loads in the finite element formulation.

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