Page 145 - Design of Reinforced Masonry Structures

DESIGN OF REINFORCED MASONRY BEAMS 4.9

Equating C = T from Eqs. (4.7) and (4.8) (for horizontal equilibrium), respectively,
yields
.
080 ′ fab = A f (4.8)
m s y
The depth a of the compression zone of concrete is obtained from Eq. (4.8):
Af
a = sy (4.9)
.
080 fb ′ m
The nominal strength of the beam, M , equals the moment provided by the C-T couple.
n
Its magnitude can be determined by taking moment of the compressive stress resultant C
about the line of action of T. Thus,
⎛
M = C d − a ⎞ (4.10)
n ⎝ ⎠ 2

Substitution for C from Eq. (4.6) in the above equation yields
⎛ a ⎞
′
M = 080. f ab d − (4.11)
n m ⎝ ⎠ 2
Alternatively, taking moment of T about the line of action of C and substituting T = A f
s y
from Eq. (4.7), the nominal strength can be expressed as
⎛ a ⎞
M = T d − ⎠ 2
⎝
n
⎛ a ⎞ (4.12)
= Af ⎝ d − ⎠ 2
sy
Multiplying both sides of Eq. (4.12) by the strength reduction factor, f, we obtain

φM = φA f ⎛ d − a ⎞ (4.13)
n s y ⎝ ⎠ 2

Equations (4.11) and (4.13) are based on principles of statics, and are used to determine

the nominal strength of beams of rectangular cross sections. In practical beams, the amount
of reinforcing steel is so limited (intentionally) that it yields before the onset of crushing
of concrete. Therefore, Eq. (4.13) is used as the basic equation for determining the flexural
strength of rectangular beams.
Equation (4.13) is based on the premise that the steel reinforcement has yielded, the
validity of which must be checked. This is done by considering the compatibility of strains
from the strain distribution diagram at the ultimate conditions (Fig. 4.4). For a member of
effective depth d, and having neutral axis located at a distance c from the extreme compres-
sion fibers, we have (from similar triangles)
ε dc
−
s = (4.14)
ε mu c