F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

8.4 Energy dissipation in a yield line
Energy dissipation in a yield line 301
Figure 8.4-l(a) shows a·positive yield line ab of length I and making angles
aA and as respectively with the two axes eg and df, about which the
rigid regions A and B rotate through the small angles ()A and Os. Figure
8.4-l(b) shows a cross-section taken perpendicular to the yield line; the
angle between the rigid regions is ()nA + 0 0 s, where OnA is the component,
in the direction of the yield line, of the actual rotation ()A of the
rigid region A, and Ons is the component of Os, as shown in the vector
diagrams in Fig. 8.4-l(a). The sign convention used for the rotation
vectors needs some explanation. Since the work done on any yield line is
always positive and since Fig. 8.4-l(a) shows clearly that positive work
results if the sense of rotation of a rigid region is opposite to that of the
normal moment mn acting on that region, the usual practice is to adopt a
sign convention for rotation vectors which is opposite to that for moment
vectors. In other words, while the right-handed screw rule is used for
moment vectors, the left-handed screw rule will be used for rotation
vectors-as in Fig. 8.4-l(a).
From Fig. 8.4-1,
Therefore
energy dissipation per unit length of yield line
= mn(OnA + Ons)
= m 0 0 A cos a A + mnOs cos as
c
(b)
Fig. 8.4-1