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)

298 Reinforced concrete slabs and yield-line analysis
r
~2
Moment
axes
Fig. 8.3-5
m" = m1 cos2 cjJ1 + m2 cos2 cjJ2 (8.3-7)
fflns = m1 sin ifJ1 cos ifJ1 - m2 sin ifJ2 cos ifJ2 (8.3-8)
If the two bands of reinforcement are perpendicular to each other, the
slab is said to be orthotropically reinforced; in this case, the m 1 and m 2
moment axes are mutually perpendicular and the angles cjJ 1 and cjJ 2 are
complementary. Hence eqns (8.3-7) and (8.3-8) reduce to
2 A . 2 A
m 0 = m 1 cos ., 1 + m2 sm ., 1
fflns = (m 1 -
m2) sin ifJ1 cos ifJ1
(8.3-9)
(8.3-10)
If, in an orthotropically reinforced slab, the yield moments m 1 and m 2
are equal (say m 1 = m 2 = m), the slab is said to be isotropically reinforced;
eqns (8.3-9) and (8.3-10) further reduce to
(8.3-11)
(8.3-12)
That is, in an isotropically reinforced slab the normal moment m" on a
yield line has the same value whatever the orientation of the yield line; and
the twisting moment mns is always zero. It should be noted that the definition
of an isotropically reinforced slab is based on the yield moments m 1
and m 2 being equal. If a slab is reinforced in two mutually perpendicular
directions with two identical bands of reinforcement laid one on top of
the other, the yield moments m 1 and m 2 are not strictly equal-because of
the difference in the effective depths for the two bands of reinforcement.
However, in yield-line analysis, such a small difference is ignored and the
slab is regarded as isotropically reinforced; the yield moment m is
calculated on the basis of the average effective depth for the two layers