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)

Calculation of short-term and long-term deflections (BS 8110) 175
where f, is the steel service stress as defined in eqn (5.3-1(b)).
Example 5.4-1 illustrates a useful application of eqn (5.4-2).
(c) Equation (5.4-2) becomes identical with eqn (5.4-1) if As,rcq =
As.prov (see Problem 5.6).
Example 5.4-1
With reference to the detailing rules for crack control in Fig. 5.4-1, if the
clear spacing ab exceeds the value in Table 5.4-1, what remedial actions
can be taken?
SOLUTION
Possible remedial actions include:
(a) The three main bars may be replaced by, say, four or five smaller
bars.
(b) Additional small bars may be inserted between the three bars.
According to the '0.45 rule' explained in Comment (a) to Fig. 5.4-1,
size 12 bars may be inserted between size 25 bars and so on.
(c) If the tension steel area actual provided is over and above that
required for the ultimate limit state, this will have the effect of
reducing the service stress fs as given by eqn (5.3-1(b)). Then use
eqn (5.4-2) to obtain a relaxed limit on the clear bar spacing.
(d) A full crack width calculation may be carried out using BS 8110's
procedure as explained in Section 5.6; this usually shows that the
values in Table 5.4-1 are conservative.
5.5 Calculations of short-term and long-term
deflections (BS 8110: Part 2)
(Note: In day-to-day design, deflections are controlled by a straightforward
procedure of limiting the span/depth ratio-see Section 5.3)
The difficulties concerning the calculation of deflections of concrete
beams arise from the uncertainties regarding the flexural stiffness EI and
the effects of creep and shrinkage. Before explaining how these uncertainties
are dealt with in practice, we shall first refer to the well-known
moment-area theorems (15], which express slopes and deflections in terms
of the properties of the M/ El diagram. For elastic members, the quantity
MIEI is equal to the curvature llr (see eqns 5.2-11,5.2-15 and 5.2-23);
therefore the moment-area theorems may be rephrased (more usefully) as
the curvature-area theorems [16]:
(a) The change in slope 0 between two points A and Bon a member is
equal to the area of the curvature diagram between the two points:
0 = J:(~)dx
where r is the radius of curvature of the typical element dx.
(b) The deflection L1 of point B, measured from the tangent at point A, is
equal to the moment of the curvature diagram between A and B,
taken about the point B whose deflection is sought: