Page 263 - Design of Reinforced Masonry Structures

4.126 CHAPTER FOUR

Transformed area of steel = (n – 1)A = (27.62 – 1) (1.0) = 26.62 in. 2
s
Total area of masonry (including the transformed area of reinforcement)
= 7.63 (24) + 26.62 = 209.74 in. 2
The distance of the centroidal axis of the beam from its top face, y , is calculated as
t
7
)( )
6
y = ( . )(32412 + (26 .62 )(20 ) = 13 .015inn.
t
209 .74
The moment of inertia of the transformed section, I , is calculated as follows (using
gt
parallel-axis theorem):
3
76
−
4
2
62
)
I = (. )(24 ) 3 + ( . )(24 13 −12 ) 2 + 26.. (20 13 .015 = 10 ,277in.
7 63
)( .015
gt
12
The gross moment of inertia of the section (ignoring reinforcement) was calculated
4
in the previous example to be 8790 in. . Thus, I is greater than I by 10,277 − 8790 =
g
gt
4
1487 in. . The difference amounts to
⎛ 1487 ⎞
⎜ ⎝ 8790⎠ ⎟ ( 100 =) 16 92%
.
4.21 DEFLECTIONS OF REINFORCED
MASONRY BEAMS
4.21.1 Concept of Effective Moment of Inertia of a Reinforced
Masonry Beam
The MSJC-08 Code [4.3] does not specify a methodology for computing deflections in
masonry beams. However, Code Section 3.1.5.2 requires that “deflection calculations of
reinforced masonry members shall consider the effects of cracking and reinforcement on
member stiffness.” It states further that “flexural and shear stiffness properties assumed for
deflection calculations shall not exceed one-half of the gross section properties, unless a
cracked-section analysis is performed.”
Very little data is available on the deflections of masonry beams and related computa-
tional methods. Deflections of beams are inversely proportional to their flexural stiffness,
EI (E I for masonry beams). There is always some uncertainty about the value of modulus
m
of elasticity, E , since testing for its value is uncommon even when deflection is important
m
to the design. Deflection calculations in both concrete and masonry structures are therefore

generally approximate. Fortunately, it is uncommon for a masonry element to be limited or
sized based on deflection limitations.
Under full service loads, masonry beams, like reinforced concrete beams, develop cracks
which create uncertainty in the values of their moments of inertia. This is because the extent
of cracking varies along the span due to the variation in the bending moment along the span.
This causes a change in the position of beam’s neutral axis along the span and a consequent
change in the value of the moment of inertia of the beam along the span.
When the maximum bending moment present in the beam is equal to or less than the
cracking moment (M ), the flexural stresses in masonry are below or equal to the modulus
cr
of rupture (f ), and the beam section is assumed to remain uncracked. In that case, the full
r
cross section of the beam can be counted upon resist deflection, and the moment of inertia
of gross section (I ) can be used to calculate deflections. Under full service loads, however,
g