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)

296 Reinforced concrete slabs and yield-line analysis
mns' = (Asfy COS 1>t)Z
= m 1s cos 1> 1 (from the expression for m 1 s above)
ffl 05S' = (AJy sin 1>t)Z
= m 1s sin 1>t
Since from Fig. 8.3-2, sis' = cos 1>~> these equations may be written as
mn = fflt cos 2 1> 1 (8.3-1)
fflns = fflt sin 1>t cos 1>t (8.3-2)
Note that in these equations 1> 1 is the acute angle measured anticlockwise
from the yield line to the moment axis. In Fig. 8.3-3(a), mn and mns are
represented by the conventional double-headed arrow moment vectors, the
direction of the arrow being that of the advance of a right-handed screw
turned in the same sense as the moment.
The reader should now verify that if the yield line makes an acute angle
(j> 1 measured clockwise from yield line to moment axis as in Fig. 8.3-3(b),
then eqns (8.3-1) and (8.3-2) become
mn = m 1 cos 2 1> 1 (as in eqn 8.3-1) (8.3-3)
mns = -ml sin 1>t cos 1>t (8.3-4)
We shall now return to Johansen's stepped yield criterion, which states
that:
(a) The yield line may be considered to be divided into small steps with
sides respectively parallel and perpendicular to the reinforcement.
On the sides parallel to the reinforcement (that is, on those sides
perpendicular to the moment axis, such as a'b, b'c ... e'f in Fig.
8.3-4) there is neither normal nor twisting moment. On the sides
perpendicular to the reinforcement (i.e. parallel to the moment axis,
such as aa', bb' ... ff' in Fig. 8.3-4) there is only a normal moment
Fig. 8.3-3
(a) {b)