Page 460 - Design of Reinforced Masonry Structures

7.22 CHAPTER SEVEN

Substitution of Eq. (7.27) in Eq. (7.26) yields Eq. (7.28):

∆ = 1 (7.28)
++
12 3 1 + 1 + 1
∆ ∆ ∆
1 2 3
Step 4: Calculate net deflection of the perforated shear wall as follows:

∆ = ∆ gross − ∆ strip + ∆ piers (7.29)
net

Relative rigidity R = 1 (7.30)
∆
net
3. Method C: It was noted in Method A that the presence of solid portions of wall above
and below the openings is ignored; only the vertical segments between the openings are
considered. In Method C, all solid portions of the wall—piers as well as beams—are
considered for determining the wall rigidity. Each pier is assumed fixed at top and bot-
tom. The method uses the principle of determining the stiffness of a system of springs
in “parallel” or in “series.” The following assumptions are made in this procedure:
(a) Vertical segments in between the openings are considered to be in parallel, similar
to system of “springs in parallel.” The height of such a segment is equal to the
smaller of the height of the opening on each side of the segment.
(b) Segment of the wall (i.e., a beam) below the opening is assumed to be in “series”
with the piers above it.
(c) The segment of the wall above the openings is assumed to be in “series” with the
group of piers/beams below it.
(d) Rigidity of a segment of the wall containing piers and a beam (below) in the group
is calculated based on the principles of springs in “parallel” and in “series.”
(e) The rigidity of the entire wall is determined based on the springs in “series,” con-
sidering the rigidities of the beam above and the group of piers below.
The principles involved in determining the stiffness of a system of springs in “parallel”
or in series can be explained as follows:
(a) Springs in parallel concept: When a system of springs, each with a different stiffness,
is in “parallel,” all springs experience the same deformation, but the force in each
spring is proportional to its stiffness (i.e., spring force = spring stiffness times the
system displacement). Consider, for example, a system of three springs, having stiff-
nesses equal to k , k , and k . This system is subjected to a force P causing a displace-
3
1

2
ment y as shown in Fig. 7.19. Force P is shared simultaneously by all three springs.
The force in each spring equals stiffness times displacement; thus, the force in the
three springs can be expressed as k y, k y, and k y, respectively. For equilibrium, the
2
1
3
sum of these three spring forces must equal P. Thus,
P = k y + k y + k y (7.31)
1 2 3
Eq. (7.31) can be expressed as
P
= k + k + k (7.32)
y 1 2 3