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)

104 Reinforced concrete beams-the ultimate limit state
(1) its length to the face of the support plus half its effective depth;
or , where it forms the end of a continuous beam,
(2) the length to the centre of the support.
4.6 Design formulae and procedure- BS 8110
simplified stress block
As an alternative to the parabolic-rectangular stress block of Fig. 4.4-3,
BS 8110 permits design calculations to be based on the simplified
rectangular stress block of Fig. 4.4-5. In this section we shall:
(1) derive the basic design formulae from first principles (Section 4.6(a));
(2) explain how to design from first principles (Section 4.6(b));
(3) explain the design procedure of the I.Struct.E. Manual [14]-the
manual's formulae will be derived and the technical background
explained (Section 4.6(c)).
(Note: BS 8110 permits up to 30% moment redistribution. Design formulae
and procedure allowing for moment redistribution will be given in Section
4. 7, while the technical fundamentals of moment redistribution will be
explained in Section 4.9. For the time being the reader need only note that
all the formulae and design procedures in this section can be used for up to
10% moment redistribution.)
4.6(a) Derivation of design formulae
BS 8110 intends that, where the simplified stress block is used, x/ d should
not exceed 0.5. Consider the beam section in Fig. 4.6-1(a). As Example
4.6-1 will show, for xl d values up to 0.5 the tension reinforcement is
bound to reach the design strength of 0.87fy at the ultimate limit state.
Therefore, the forces are as shown in Fig. 4.6-1(b), from which
r
lbl
d
L •A·•
Cross section
(a)
Fig. 4.6-1
0·45fcu
r-1_t
r 0·45x
0<9x 0-405 fcubx
.LI--~
Z=d-0·45x
&::1---+J
0·87(y As
Forces
(b)
0·45fcu
r--1_t
1 0·225d
0·45d 0·2 fcubd
j_t--__,
Z=0·775d
_j
0·87fy As
Forces ( x = d/2 )
(c)