# The Finite Element Method (FEM) is a numerical technique used in structural analysis to estimate the behavior of complex structures under various loading conditions. The method involves dividing the structure into smaller and simpler parts called finite elements, and analyzing each element independently.

To begin, the structure is discretized into a mesh of finite elements, and each element is represented by a set of equations that describes its behavior. These equations are then combined to form a system of equations that describes the behavior of the entire structure. This system of equations can then be solved numerically to determine the response of the structure to a given set of loading conditions.

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The Finite Element Method is commonly employed in the analysis and design of structures in various fields such as civil, mechanical, aerospace, and biomedical engineering. It is especially useful for analyzing structures with complex geometries or material properties that are difficult to evaluate using traditional analytical methods.

Moreover, the Finite Element Method can handle non-linear and time-varying behavior of structures, as well as different types of boundary conditions, such as fixed, pinned, or roller supports.

Overall, the Finite Element Method is a powerful tool for structural analysis that allows engineers to simulate the behavior of complex structures and optimize their design for a given set of conditions.

## In general, the following set of equations are commonly used in structural analysis using Finite element method:

**Equilibrium equation:**This equation ensures that the forces acting on an element are in equilibrium. It is given by: {P}_{NPX1}= [A]_{NPXNF}*{F}_{NFX1}

where {P} is the External force matrix {F}=Internal force matrix , [A]_{NPXNF= }Equilibrium matrix

NP=Number of degrees of freedom of the structure , NF= Number of members/elements

2. **Elastic stress-strain equation:** {F}_{NFX1}=[S]_{NFXNF}*{e}_{NFX1}

where [S] is the Local Stiffness matrix of the element ,

{e}= Elongation/Shortening of the element or internal deformation matrix

3. **Compatibility equation:** It is given by:{e}_{NFX1} = [B]_{NFXNP}*{x}_{NPX1}

where {x} is the joint displacements matrix of the structure , [B]_{NFXNP= }Deformation matrix

We know, {P} = [A]*{F} = [A]*[S]*{e} = [A]*[S]*[B]*{x}

Again from virtual work principle (Work done by external forces = Work done by internal forces),we can prove that, [B]= [A]^{T}= Transpose of matrix [A]

{P}= [A]*[S]*[B]*{x}=[A]*[S]*[A]^{T}*{x}

{P}=[K]*{x} , where[K]=[A]*[S]*[A]^{T}

[K]_{NPXNP}= Overall Global Stiffness Matrix

The solution of this system of equations gives the nodal displacements, strains, and stresses in the element, which can be used to calculate other quantities of interest such as deflections, reactions, and forces.

Let’s see the formation of [A] and [B] matrices with the help of an example.

Refer my second post on **finite element method**.

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