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Modern Geomatics Technologies and Applications

Finite element method is a powerful tool for the numerical solution of a wide variety
1
of engineering problems. FEM encompasses a wide range of applications including
stress and deformation analysis for the structure of buildings, bridges, dams and
airplanes as well as problems in heat transferring, magnetic field, seepage and other
problems in fluid mechanics. In finite element method, a continuum body is divided

to some smaller geometric components such as triangles or quadrangles or
combination of them. This operation is known as partitioning and the common points

of components are called nodes. Material properties and internal stresses are defined
with regards to the unknown displacements of the corners of each component. Due to
the setting arrangement of elements next to each other, their equations are assembled

and equilibrium equations are obtained for the entire system by usage of external
forces and support conditions in node's location. These equations communicate

between the nodal forces with nodal displacements. Their constant parameters are
geometric and elastic properties of finite elements. By solving these equations, nodal
displacements, and the interior stresses are calculated.

The following relations could be written in a triangle between the values of the
displacement and coordinates of the vertices of the triangle:
u(x , y) = Ni ui + Nj uj +Nm um , v(x , y) = Ni vi + Nj vj +Nm vm (1)

N is a function of coordinates of the nodes. In the matrix form, the eqn. (1) may be
written as a straightforward matrix where N is the total displacement:
 =N d (2)
With differentiation of displacement equation and with assumption of a plane strain,

we've the following equations:
  M  (3)

1 Finite Element Method