Page 264 - Design of Reinforced Masonry Structures

DESIGN OF REINFORCED MASONRY BEAMS 4.127

cracking develops in the tension zone of the beam. The extent and pattern of this cracking
varies along the span (closely spaced and wider cracks near the midspan where moments
are large, and fewer and narrower near the supports where moments are small), which
complicates the problem of determining the value of true moment of inertia of a cracked
beam. At the cracked sections, the moment of inertia is close to I , but in between the
cr
cracks (where the section is uncracked), the moment of inertia is perhaps closer to I . In
g
essence, the flexural stiffness of the beam, EI, varies along the span. To account for these
uncertainties, it becomes necessary to use the effective moment of inertia, I , to calculate
eff
deflections of beams under service loads.
To the extent that masonry beams behave similar to reinforced concrete beams under
service loads, the methods that have been developed for the latter may be also used for
masonry beams. For reinforced concrete members, Branson [4.23, 4.24] suggested the fol-
lowing expression for the effective moment of inertia of reinforced concrete section:
⎛ M ⎞ a ⎡ ⎛ M ⎞ a ⎤
eff ⎜
I
I = cr ⎟ I + ⎢1 − ⎜ cr ⎟ ⎥ I (4.147)
⎝ M ⎠ gt ⎢ ⎣ ⎝ M ⎠ ⎥ ⎦ cr
a
a
where I = effective moment of inertia
eff
I = moment of inertia of the cracked section transformed to masonry
cr
I = gross-transformed moment of inertia
gt
M = maximum moment in the member at the loading stage for which the moment
a
of inertia is being considered or at any other previous loading stage [4.25]
M = nominal cracking moment strength
cr
In reinforced concrete members under very high loads, the tensile force in the concrete
is insignificant compared to with that in tension reinforcement and the member approxi-
mates a completely cracked section. The effect of the tensile forces in the concrete on the
EI is referred to as tension stiffening [4.25]. A somewhat similar condition may be assumed
in masonry beams at ultimate load conditions. It is also clear from Eq. (4.146) that the
effective moment of inertia I would decrease as the loads (and hence M ) would increase.
a
eff
This means that I is larger when only the dead load is acting than when both dead and
eff
live loads are acting together.
For regions of constant moment, Branson [4.22] found the value of exponent a in
Eq. (4.147) to be 4. For simply supported beams, Branson suggested that both the tension
stiffening and the variation in flexural stiffness along the span could be accounted for by
assuming a = 3. Thus, Eq. (4.147) can be modified and expressed as Eq. (4.148) as specified
in the ACI Code [4.2], wherein, for simplicity, I has been substituted for I ignoring the
g
gt
small contribution of the tension reinforcement to the moment of inertia:
⎛ M ⎞ 3 ⎡ ⎛ M ⎞ 3 ⎤
I = cr ⎟ I + ⎢1 − ⎜ cr ⎟ I ⎥ < I (4.148)
eff ⎜
⎝ M ⎠ g ⎢ ⎣ ⎝ M ⎠ ⎥ ⎦ c cr g
a
a
where I = gross moment of inertia of masonry section (neglecting reinforcement)
g
An alternatively form of Eq. (4.148) is Eq. (4.149):
⎛ M ⎞ 3
I = I + ( I − I ) cr ⎟ < I (4.149)
cr ⎜
g
eff
cr
⎝ M ⎠ g
a
It is seen from Eq. (4.148) or Eq. (4.149) that the effective moment of inertia I is a
eff
function of the dimensionless ratio M /M which provides a transition between the upper
cr a
bounds of I and I . The current practice is to use the effective moment of inertia, I , to
g cr eff