Degree of Element (DOF) of Element in Finite Element Method (FEM)

 1-D element

• Rod     -    2 DOF/ node (U_x, θ_x) 
• Bar      -    6 DOF / node (U_x, U_y, U_z , θ_x, θ_y, θ_z ) 
• Beam   -    6 DOF / node (U_x, U_y, U_z , θ_x, θ_y, θ_z ) 
• Pipe     -    6 DOF / node (U_x, U_y, U_z , θ_x, θ_y, θ_z ) 
• Axisymmetric shell   -    3 DOF / node (U_x, U_z, θ_y ) 

2-D element

• Plane Stress     -    2 DOF / node (U_x, U_y ) 
• Plane Strain     -    2 DOF / node (U_x, U_y ) 
• Plate                 -    3 DOF / node ( U_z , θ_x, θ_y ) 
• Membrane        -    3 DOF / node (U_x, U_y, θ_z ) 
• Thin Shell        -    6 DOF / node (U_x, U_y, U_z , θ_x, θ_y, θ_z ) 
• Axisymmetric Solid    -    2 DOF / node (U_x, U_y ) 

3-D element

• Tetra                    -    3 DOF / node (U_x, U_y, U_z  ) 
• Penta or Wedge   -    3 DOF / node (U_x, U_y, U_z  ) 
• Hexa or brick       -    3 DOF / node (U_x, U_y, U_z  ) 
• Pyramid               -    3 DOF / node (U_x, U_y, U_z  ) 

Degree of Freedom of Element

 Firstly let's study the degree of freedom, what is the degree of freedom (DOF)?

Degree of freedom is defined as the minimum number of parameters (Coordinates, motion, temp, pressure, etc ) required to define the position of any entity completely in space.

For Example: if we make a point in 2D space (XY-plane), it is required to define the coordinate of the point in the X and Y direction to specify a location, So, the point has 2 degrees of freedom in 2D space.

If the same point is in 3D space. then it is required to define the coordinate of the point in the X, Y, and Z direction to specify a location. So, Point has 3 degrees of freedom in 3D space.

Now, If we take a line in 2D space, we have to define the rotation angle of the line with respect to either X-axis or Y-axis, along with the X and Y direction coordinate of either endpoint. So, any point on the line in 2D space will have 3 Degree of freedom (DOF).

Lastly, If we take a line in 3D space, we have to define the rotation angle of the line with respect to all three axes (X, Y, and Z-Axis) along with all three coordinates of either endpoint. So, any point on the line in 3D space will have 6 Degree of freedom.

So, In space (3D environment) every node has 6 degrees of freedom(DOF) - 3 translational (U_x, U_y, U_z) and 3 rotational ( θ_x, θ_y, θ_z ). 

Degree of freedom (DOF) in FEM plays an important role. If we increase one element, then 6 DOF will increase. 

            Number of Nodes     =    10

            dof of per node        =    6

            Total DOF                =    60

For the Degree of freedom of different elements Click Here.