Page 323 - Design of Reinforced Masonry Structures

COLUMNS 5.43

With the above substitution, Eq. (5.30) can be written as Eq. (5.32):
⎡ ( ) ⎤
′ d
.
080 ′ f ( 08 ) + ′ A 1− ε E − 080 ′ f − − Af = 0 (5.32, 4.113 repeated)
.
. c b
m s ⎣ ⎢ c m s m ⎦ ⎥ sy
Equation (5.32) can be simplified and written as Eq. (5.33):
=
064 ′ fbc 2 + ′ A ε E c − ′ A ε E d ′− 080 ′′ −f A c A f c = 0 (5.33, 4.114 repeated)
.
.
s
m
s
s m
s m
m
s y
s
Equation (5.33) can be expressed as a quadric in c:
)c
f A
(.064 ′ fb )c 2 + ( ′ A ε E − A f − . 080 ′′ − ′ A ε Ed′ = 0 (5.34, 4.115 repeated)
m st m s s y m s s sm s
2
Equation (5.34) is a quadratic of the form: Ax + Bx + C = 0, which can be solved
for x [= c in Eq. (5.34)]:
−± B − 4 AC
2
B
x = (5.35, 4.116 repeated)
2 A
where x = c
A = 064. f b ′ m
B = A′ε E − A f − 080. f A′
m
s
s y
sm
s
C =− A′ε E d′
sm s
Note that Eq. (5.34) has two roots of x which are given by Eq. (5.35); the negative root
should be ignored as it has no significance in this problem. Once c is known, a = 0.8c, and
quantities C and C are easily determined. Finally, the magnitude of M can be determined
m
n
s
by summing up moments due to C and C about T:
m
s
m( )
(
−
M = C d − a + Cd d′) (5.36, 4.117 repeated)
n
2 s
It is noted that the strain in compression steel may be small, zero, or even tensile,
depending on the distance of neutral axis from the compression reinforcement. If the
neutral axis passes through the centroid of compression reinforcement (c = d′), the strain
in compression reinforcement would be zero so that C would be zero. In case the strain
s
in compression reinforcement is tensile (c < d′), the force in the compression reinforce-
ment would be tensile, and C should be determined from Eq. (5.26).
s
3. The balanced condition
The balanced condition is defined as a load condition such that the maximum

compressive strain in masonry (e = 0.0025 for concrete masonry and 0.0035 for clay
m
masonry) and tensile strain in steel (= 0.00207 for Grade 60 steel) occur simultane-
ously. The location of neutral axis (distance c from the compression face) and the strain
in compression reinforcement ( ′ ε ) are determined, respectively, from Eqs. (5.37) and
s
(5.24), based on the strain distribution diagram shown in Fig. 5.12. Thus,
⎛ ε ⎞
m
c = ⎜ ⎝ε m + ε y ⎠ ⎟ d (5.37)
( )
ε s ′ = 1 − ′ d ε m (5.24 repeated)
c