7.1.2 Work done in a plastic hinge
Let us consider the internal energy in the plastic hinge from Fig. 7.5(a):



(7.11) 
where is the work done by external forces (see Appendix B.1).
The work done by external forces at Sign Convention 1 (see Fig. 7.5(a))







(7.12) 
The work done by external forces at Sign Convention 2 (see Fig. 7.5(b))







(7.13) 
The incremental rotation
of the plastic hinge is given by the difference
between rotations of the member ends at joint.
It is necessary to check the compatibility of the rotations related to the hinge
with the sign of the plastic moment of Eq. (7.13).
Considering that
and
, the work of boundary forces at the member ends at joint is



(7.14) 
where



(7.15) 
Figure 7.5:
Work done in a plastic hinge
[Sign Convention 1]
 
[Sign Convention 2]

The boundary forces work at the member end
and the boundary forces work
at the plastic hinge boundary, at a similar angle of rotation, , have different signs (see Fig. 7.5(b)):







(7.16) 
The sum of the amounts of work done by internal and external forces is zero (see Appendix B.1 and Eq. (B.8)), or



(7.17) 
and



(7.18) 
The sum of the amounts of work done by boundary forces of members at the plastic hinge



(7.19) 
andres
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