I would like to check how the support condition specified in CAESAR II 4.4 analysis works? Since we define the support by assigning a node, I would like to know if the support is fixed in space or just moves along the pipe.(Eg. support in the form of a concrete sleeper modelled in CAESAR II )

If you model a support, say a "Y" (vertical) support for example, and you don't specify a CNODE and you leave the stiffness field blank, then you have defined a rigid vertical support. The associated node is fixed in space.

If you specify a stiffness on this "Y" support, less than the rigid default, then the node may experience displacement in the "Y" direction, depending on the pipe loads.

If you specify a CNODE, then the restrained node is fixed to the CNODE in the "Y" direction. Neither is fixed in space so both can move as dictated by the piping loads.

To add to Rich's description, a Y support at node 10 simply tells CAESAR that if node 10 tries to move up or down, it will be resisted by a stiffness. This will be the mathematical condition relating to that direction of movement even if the node has moved 10m or 1E10m in the X direction. You could say that, in that sense, the restraint moves with the node, but I encourage you to consider the restraint as an infinite plane that constrains movement according to the specified direction.