Introduction to FEA

Discretization

A continuum with infinite DOF → dividing into finite elements with finite DOF and connected by nodes → Solving governed equation by nodes only → Interpolation desired variables between every two nodes

Explain in a mathematical way:

Partial differential equations → linear algebraic equations for unknowns at nodes → filed quantity within element is described by interpolation function → filed quantity over the entire structure domain is approximated element by element in piecewise function

Force method → to establish flexibility matrix [f]

Displacement method → to establish stiffness matrix [K]

The FEM is conventionally a displacement method. So the main objective is to construct the stiffness matrix for the structure before the solving the linear algebraic equations

The steps of FEM:

Element, Assembly, Reduction, Solution

Considering 1D element:

1D element acts like a spring, the stiffness can be derived from the force-displacement relationship:
$$
f=(\frac{AE}{L})q=kq
$$
Thus, for a 1D elements with 2 end nodes, the element force-displacement relation can be written as:
$$
f_1 = k(q_1-q_2)\
f_2=-k(q_1-q_2)\
\Rightarrow
\left{\begin{matrix}
f_1\f_2
\end{matrix}\right}
=
\begin{bmatrix}
k, -k\
-k, k
\end{bmatrix}
\left{\begin{matrix}
q_1\q_2
\end{matrix}\right}\
\Rightarrow
\left{\begin{matrix}f\end{matrix}\right}=[k^e]\left{\begin{matrix}q\end{matrix}\right}
$$
Then, the element nodal force vector, element stiffness matrix and element nodal displacement vector are noticeable.

If we consider one more element connected in series, the force-displacement relations can be written as:

For ease of understanding, we mark this way: the superscripts a and b denote the different sides of the nodes, the subscripts 1, 2, and 3 denote the serial number of the nodes, while the subscripts of stiffness k denote the elements between the two nodes.
$$
f_1^b=k_{12}q_1^b-k_{12}q_2^a\
f_2^a+f_2^b=-k_{12}q_1^b+(k_{12}+k_{23})q_2^a-k_{23}q_3^a\
f_3^a=-k_{12}q_2^b+k_{23}q_3^a\
$$
Since these two elements share a common node where the total force is also shared at, then some basic local-global relations can be determined (For ease of expression):
$$
Q_1=q_1^b,Q_2=q_2^a=q_2^b,Q_3=q_3^a\
F_1=f_1^b,F_2=f_2^a+f_2^b,F_3=f_3^a
$$
Thus, rewrite the relations and transform into linear algebraic way:
$$
\left{\begin{matrix}
F_1\F_2\F_3
\end{matrix}\right}
=
\begin{bmatrix}
k_{12} & -k_{12} & 0\
-k_{12} & k_{12}+k_{23} & -k_{23}\
0 & -k_{23}{} & k_{23}
\end{bmatrix}
\left{\begin{matrix}
Q_1\Q_2\Q_3
\end{matrix}\right}\
\Rightarrow
\underbrace{force \ vector}=\underbrace{[K]}{stiffness\ matrix}\underbrace_{disp.\ vector}
$$
In the process of expressing the force-displacement relationship with linear algebra, the construction of the stiffness matrix is actually like assembling the separate stiffness matrix of each connected element. Such stiffness matrix clearly shows the connectivity between elements.

After constructing the force-displacement relation, how to solve it? During the solution, we can reduce the matrix system by imposing boundary conditions and loading conditions,

For above 1D element examples, there are other elements like surface of solid element. However, no matter which kind of element, they only affect the stiffness matrix and the corresponding assembly.