Carryover Moment is defined as moment developed or induced at one end due to a moment at anor end. It is useful in calculating stiffness factor and moment distribution for a particular joint.

Joint O is connected to four-member OA, OB, OC, and OD where far end is supported by vertical guided roller, fixed support, free end, and fixed support respectively.
Stiffness for OA is \(\frac{EI}{L} \)
Stiffness for OB is \(\frac{4EI}{L} \)
Stiffness for OC is 0
Stiffness for OD is \(\frac{4EI}{L} \)
Therefore stiffness for joint O is sum of stiffness for all members connected to it.
Stiffness for joint O is \(\frac{EI}{L}+0+\frac{4EI}{L}+\frac{4EI}{L}=\frac{9EI}{L} \)
Distribution factor for member OC=\(\frac{Stiffness \, of \, member \, OC}{Stiffness \, of \, rigid \, joint}=\frac{0}{\frac{9EI}{L}} \)
=0

Joint O is connected to four-member OA, OB, OC, and OD where far end is supported by vertical guided roller, fixed support, free end, and fixed support respectively.
Stiffness for OA is \(\frac{EI}{L} \)
Stiffness for OB is \(\frac{4EI}{L} \)
Stiffness for OC is 0
Stiffness for OD is \(\frac{4EI}{L} \)
Therefore stiffness for joint O is sum of stiffness for all members connected to it.
Stiffness for joint O is \(\frac{EI}{L}+0+\frac{4EI}{L}+\frac{4EI}{L}=\frac{9EI}{L} \)
Distribution factor for member OB=\(\frac{Stiffness \, of \, member \, OB}{Stiffness \, of \, rigid \, joint}=\frac{\frac{4EI}{L}}{\frac{9EI}{L}}=\frac{4}{9} \)

As free end cannot resist any moment nor can support any vertical or horizontal reactions. End support B cannot take any portion of moment applied at end A. Thus, stiffness for member would be zero.

Joint O is connected to four-member OA, OB, and OD where far end is supported by vertical guided roller, fixed support, and hinged support respectively.
Stiffness for OA is \(\frac{EI}{L} \)
Stiffness for OB is \(\frac{3EI}{L} \)
Stiffness for OC is \(\frac{4EI}{L} \)
Therefore stiffness for joint O is sum of stiffness for all members connected to it.
Stiffness for joint O is \(\frac{EI}{L}+\frac{4EI}{L}+\frac{3EI}{L}=\frac{8EI}{L} \)
Distribution factor for member OC=\(\frac{Stiffness \, of \, member \,}{Stiffness \, of \, rigid \, joint}=\frac{\frac{3EI}{L}}{\frac{8EI}{L}}=\frac{3}{8} \)

Moment Distribution method developed by Hardy Cross to analyze indeterminate structures like beams and rigid jointed frame with internal hinges also.

Distribution factor is ratio in which moment sharing capacity of various members meeting at a rigid joint.

When a structural member of uniform section is subjected to a moment at one end only, n moment required so as to rotate that end to produce a unit slope, is called stiffness of member. Stiffness is basically a measure of ability of member to resist rotational deformation.

Since horizontal roller support does not provide any restraint to vertical loading if moment is applied at that end. The end will simply slide down and hence would not resisting any portion of applied moment.