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)

308 Reinforced concrete slabs and yield-line analysis
slab (or beam) depends strongly on the tension steel ratio A.lbd. For
example, if the tension steel ratio e(= A.lbd) is 3%, the effect of a 3%
compression steel is to increase the ultimate moment of resistance from
7.4bd 2 to 10.3bd 2 -an increase of about 39%. However, if l? = 0.5%,
the effect of a 0.5% compression steel (or even 3 or 4%) is practically zero!
In yield-line analysis, the tension (or compression) steel ratio rarely
exceeds 1%, values between 0.5% and 0.8% being quite common; hence
the interaction between compression and tension reinforcements is usually
negligible.
8.5 Energy dissipation for a rigid region
The use of eqn (8.4-1) requires the calculation of the normal moment mn
for each yield line and can be cumbersome where the slab is reinforced
with skew bands of reinforcement. In this section another method, called
the component vector method (after Jones and Wood [8]), is explained; in
this the total internal work is calculated as the sum of the work due to the
separate rotations of each rigid region.
Let us return briefly to Fig. 8.3-5 and eqn (8.3-6), which states that for
a yield line of length I
(mn + mns)l = mill + m212
This equation may be written in the equivalent form
(mn + mns)l = mill + m2l2 (8.5-1)
where, with reference to Fig. 8.3-5, the magnitude of the vector 11 is given
by the projection of I on the m 1 moment axis and its sense is that of the
component of mn in the direction of the m1 moment axis; that is, in Fig.
8.3-5, 1 1 is directed from P' to Q where PQ is a positive yield line-if PQ
had been a negative yield line, 1 1 would have been directed from Q to P'.
The vector 1 2 is similarly defined with reference to the m2 moment axis.
Consider a typical rigid region A bounded by the yield lines ab, be and
the simple support ac (Fig. 8.5-1). Applying eqn (8.5-1) to the yield
line ab,
(mnl + mns/)ab = (mtll + m2l2)ab (8.5-2)
where 1 1 is now the vector ab 1 and 1 2 is now the vector ab2•
For the yield line be, we have
where
11 = b1c1 and l2 = b2c2
Adding eqns (8.5-2) and (8.5-3), we have
(mnl + mns/)ab and be = mt[ (lt)ab + {lt)bc]
+ m2[ (12)ab + (12)bc]
= m 1ac1 + m2ac2
(8.5-3)