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)

310 Reinforced concrete slabs and yield-line analysis
Note:
ae1 II m 1 axis
ae2 II m 2 axis
/R•ae
• ab•bc•cd•de
Fig. 8.5-2
sense as that of ()A· From statement (b) of Johansen's stepped yield
criterion, eqn (8.5-7) can be extended to slabs with several bands of
reinforcement, though in practice designers rarely use more than two
bands of reinforcement near the bottom face and two bands near the top
face.
Example 8.5-1
With reference to eqn (8.5-7), explain how the projection of ()A on a
moment axis may be found.
SOLUTION
Consider the typical rigid region A in Fig. 8.5-3, bounded by the positive
yield lines ab, be, cd, de and by the axis of rotation ae. Consider a unit
virtual deflection at, say, point d. Then
1
()A = dd'
where dd' is measured in a direction normal to the axis of rotation. The
magnitude of the projection of() A on the m 1 moment axis is() A cos a where
a is the angle between the m 1 axis and the axis of rotation. Therefore
()A cos a = (d~')(~~:) = d~ 1
That is, considering a unit deflection at any typical point d on the rigid
region A, the magnitude of the projection of ()A on the m 1 moment axis is
lldd 1 where dd 1 is the distance from d to the axis of rotation measured in a
direction perpendicular to the m 1 moment axis.
Similarly, for the same unit deflection at d, the magnitude of the
projection of ()A on the m 2 moment axis is 1/dd2, where the point d2 is as
shown in Fig. 8.5-3. (Note: for clarity, the m 2 moment axis in Fig. 8.5-3
has been shown different from that in Fig. 8.5-2.)