Page 208 - Design of Reinforced Masonry Structures

4.72 CHAPTER FOUR

In Eq. (4.112), the value of a is unknown. By definition,
a = 0.80c (4.5a repeated)

With the above substitution, Eq. (4.112) can be written as Eq. (4.113):
⎛ ⎡ d ′ ⎞ ⎤
.
.(
08 080 ′ fbc ) + ′ ⎜ ⎢ ⎢ 1− ⎟ ε E − 080 ′ f m ⎥ − Af = 0 (4.113)
.
A
⎣ c ⎦
m s ⎝ ⎠ m s sy
Equation (4.113) can be simplified and written as Eq. (4.114):
.
.
064 ′ fbc 2 + ′ A ε E c − ′ A ε E d ′ − 080 ′′ −f A c A fc = 0 (4.114)
s sy
m
m
s
s
s
s m
s m
Equation (4.114) can be expressed as a quadric in c:
f A
(.064 ′ fb )c 2 + ( ′ A ε E − A f − . 080 ′′ − ′ A εε Ed′ = 0 (4.115)
)c
m s m s s y m s s m s
2
Equation (4.115) is a quadratic of the form: Ax + Bx + C = 0, which can be solved for x
[= c in Eq. (4.115)]:
−± B − 4 AC
2
B
x = (4.116)
2 A
Note that Eq. (4.115) has two roots of x which are given by Eq. (4.116); 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
s
n
by summing up moments due to C and C about T:
s
m
a ⎞
−
M = C ⎜ ⎛ d − ⎟ + Cd d′) (4.117)
(
n m ⎝ ⎠ 2 s
Example 4.25 presents calculations for determining the nominal strength of doubly rein-
forced beam based on the aforederived relationships.
It is noted that reducing the stress in compression reinforcement by 0 80 ′ f in
.
m
Eq. (4.110) is a conservative approach, and is not necessary for practical purposes as
ignoring this quantity would not affect the results significantly. However, to maintain
accuracy of results, Eq. (4.110) would be used in this book to calculate C .
s
.
If one wishes to ignore the 0 80 ′ f term from Eq. (4.110), the force in compression
m
reinforcement can be expressed as

C = A′ ⎜ ⎢ ⎛ ⎡ 1 − d′ ⎞ ⎟ ε E s ⎥ ⎤
⎣ ⎦
s s ⎝ c ⎠ m (4.118)
Equation (4.118) should also be used to determine C if the neutral axis is so located
s
that c = d′ in which case the strain in compression reinforcement would be zero (so that
C = 0), or if c < d′ in which case the strain in the compression reinforcement would be
s
tensile and the force in the compression force would be tensile. In such cases, Eq. (4.113)
would then be simplified to Eq. (4.119):
⎛ d ′ ⎞
.
A
08 080 ′ fbc ) + ′ ⎜ 1− ⎟ ε E − A f = 0 (4.119)
.(
m s ⎝ c ⎠ m s s y