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)

312 Reinforced concrete slabs and yield-line analysis
From eqn (8.5-7),
energy dissipation for region A
= m [(ad) cos (70° - a) (ad) sin ~700 _ a) + 0] Nm
(where the zero within the brackets occurs because the rotation
vector has a zero projection on one of the moment axes)
= m cot (70° - a) Nm
energy dissipation for region B
= m [(ad) cos a(ad) ~in a+ o] Nm
= m cot a Nm
The work equation is therefore
m cot (70° - a) + m cot a = 8q sin 70°
which simplifies to
m = 8 sin a sin (70° - a)
q
The worst layout for the assumed yield-line pattern is when
or
d(:q) = 0 = -sin a cos (70° - a) + sin (70° - a) cos a
sin(70° - 2a) = 0 therefore a = 35°
The work equation then becomes
m = 8 sin 2 35° or q = 0.38m N/m 2
q
Comments
(a) Example 8.5-2 shows that a= 35° = ~(70°). The reader should prove
that this result is quite general: for an isotropically reinforced
triangular slab supported on two edges, the worst layout for the yield
line is when it bisects the angle between the two supported edges.
(b) In Example 8.5-2, the assumed yield-line pattern is defined by a
single variable a, so that the worst layout of the assumed pattern is
obtained by differentiating (m/q) with respect to a. But the
procedure can be extended to a yield-line pattern defined by many
variables, at. a 2, .•• a 0 • For such a case, we differentiate n times to
obtain n equations for the n a's:
--'a(7m~l q~) = =0
OUt
a(mlq)
= =0