Rigid body

The space model consists of nodes (e.g. node 1, 5, 9 etc.) and bars (e.g. bar 1, 2, 5 etc.). Nodes are symbolic geometric points where bars end.

In general, each node has 6 displacements (6 degrees of freedom), three translations δ_x, δ_y, δ_z and three rotations φ_x, φ_y, φ_z. The objective of the analysis is the calculation of the 6 displacements for all nodes.

Bars are used for modelling structural elements, beams (e.g. beam b1 is modelled with bar 5) and columns (e.g. column c1 is modelled with bar 1). Thus modelling structural elements of insignificant cross-sections with linear bar elements does not take into consideration two factors:

(a) The eccentricities of the beam-to-column supports and

(b) The increased stiffnesses around the nodes at beam-to-column junction.

These effects are taken into consideration, using the technique of rigid bodies (rigid offsets). Slave nodes are, therefore, created at the intersection point of the beam axis with the column side face (e.g. node 6) and an ideal rigid body is formed between nodes 5 and 6.

Provided the 6 displacements of the master node are δ_x, δ_y, δ_z, φ_x,φ_y, φ_z, the corresponding δ_x,s, δ_y,s, δ_z,s, φ _x,s, φ_y,s, φ_z,sof the slave node are:

φ _x,s =φ_x

φ _y,s =φ_y

φ _z,s =φ_z

δ _x,s =δ_x-s_y·φ_z (given s_z=0)

δ _y,s =δ_y+s_x·φ_z (given s_z=0)

δ _z,s =δ_z-s_x·φ_y+s_y·φ_x

In a floor consisting of 20 master and 14 slave nodes, the number of the unknown displacements (degrees of freedom) is equal to 20x6=120.