Page 158 - Design of Reinforced Masonry Structures

4.22 CHAPTER FOUR

Eq. (4.10). We define reinforcement ratio r (rho) as the ratio of the area of reinforcement
to the cross-sectional area of the beam, bd:
ρ = A s (4.37)
bd
so that
A = rbd (4.38)
s
where r = the reinforcement ratio.
Substitution of Eq. (4.38) into Eq. (4.9) yields
f ρ y ⎛ d ⎞
a = ⎠ (4.39)
f ′ ⎝ 080.
m
Now, define
ρ f
ω = y (4.40)
′ f
m
where the quantity w (omega) is called the mechanical reinforcement ratio or the reinforc-
ing index. With the substitution of Eq. (4.40) into Eq. (4.39), we can write
ω d
a = (4.41)
080
.
Substitution for a from Eq. (4.41) into Eq. (4.11) yields the nominal strength of a reinforced
masonry beam:
⎛ a ⎞
′
M = 080. f ab d − ⎠ 2 (4.11 repeated)
⎝
m
n
⎛ ω ⎞
M = ′ 2 ⎜ 1 −
f bd ω
⎠
n m ⎝ 20 80) ⎟
(.
or
)
f bd ω
M = ′ m 2 (1 − 0 625ω (4.42)
.
n
Multiplying both the sides of Eq. 4.42 with the strength reduction factor f, we obtain
′
φM = φ f bd 2 ω − 0 625 ω) (4.43)
n m
.
(
1
where φM = the design strength of a beam. For design purposes,
n
φM ≥ M (4.3 repeated)
n u
Equation (4.43) can be used to determine the nominal strength M of a rectangular beam.
n
Equation (4.43) shows that the nominal strength of a reinforced masonry beam, M , depends
n
only on the cross section and material properties of a beam. Examples 4.6 and 4.7 illustrate
the application of Eq. 4.43.
For design purposes, Eq. 4.43 can be expressed as
φM
φ f ′
n = [ ω 1 0 625−( . ω] (4.44)
bd 2 m