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)

Design formulae and procedure-BS 8110 simplified stress block 119
Example 4.6-11
A bending moment M is applied to a beam section. If M > Mu of eqn
(4.6-5), show that zld is given by the following BS 8110 formula:
J = 0.5 + ~( 0.25 - ~~) (4.6-17)
where K' = Mulfcubd 2 (see eqn 4.6-5);
SOLUTION
z = lever-arm distance measured from the tension reinforcement
As to the centroid of the concrete compression block.
When M = Mu of eqn (4.6-5), xld is equal to 0.5. When M > Mu (i.e.
K > K') compression reinforcement is required, but x/d remains equal
to 0.5. Therefore, with reference to eqn (4.6-4), the equation
K' = (0.405x/d)(1 - 0.45x/d)
will give x/d = 0.5 when K' is set equal to 0.156 of eqn (4.6-5). From Fig.
4.6-1(b),
z = d - 0.45x
This expression gives the lever-arm distance z measured from the tension
steel As to the centroid of the concrete compression block, and remains
valid forM> Mu.
Eliminating x from the above two equations,
from which
(ar _ (a) + 1.111K' = o
J = 0.5 + ~(0.25- ~~)
Of course, setting K' = 0.156 in this equation (see also eqn 4.6-5) gives
z = 0.775d, which agrees with that in Fig. 4.6-1(c).
4. 7 Design formulae and procedure-BS 8110
simplified stress block (up to 30% moment
redistribution)*
As stated at the beginning of Section 4.6, the formulae and design
procedure in that section are valid for up to 10% moment redistribution.
We shall now explain how those formulae and the design procedure can be
modified for application for up to 30% moment redistribution. Of course,
the design formulae and procedure in this section, and those in Section 4.6,
both
(a) conform to BS 8110, and
* Section 4.7 may be omitted on first reading; beginners should move on to Section 4.8.