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)

176 Reinforced concrete beams-the serviceability limit states
where x is measured from B.
Comments
The above results were first given the name curvature-area theorems in
the first edition of the book, published in 1975; a formal proof using the
principle of virtual work was given subsequently [16).
For estimating the deflections of concrete structures, the curvature-area
theorems have distinct advantages over the conventional moment-area
theorems:
(a) Unlike the moment-area theorems, the curvature-area theorems
express the purely geometrical relations between the slopes, 0, the
deflections Ll and the curvatures 1/r. Since the relations are purely
geometrical, their validity is independent of the mechanical properties
of the materials. That is, the curvature-area theorems are
equally applicable irrespective of whether the structure is elastic or
plastic or elasto-plastic.
(b) Unlike the moment-area theorems, the curvature-area theorems
can be used even where the deformations are caused by other effects
than bending moments, e.g. by shrinkage and creep. Once the
curvatures are known, the slopes and deflections are completely
defined by the curvature-area theorems; whether the curvatures
have been caused by bending moments or by shrinkage and creep
does not affect the results.
Example 5.5-1
Figure 5.5-1 shows the curvature diagrams for a beam of uniform flexural
rigidity El, acted on by various loadings. Determine the midspan
deflection a in each case.
SOLUTION
(a) Referring to the curvature diagram for beam (a),
shaded area = ! (1) (1..) = 1 (1..)
2 2 rm 4 rm
Moment of shaded area about left support
= £(,:)(~) = f~(,:)
From the curvature-area theorem, this is the deflection of the left
support from the tangent at midspan, and is numerically equal to the
midspan deflection a. Therefore
a = 12z2(t)
(b) Similarly,
a= (~~,:)ell) = ii 2 (t)