Page 152 - Design of Reinforced Masonry Structures

4.16 CHAPTER FOUR

desired number greater than 1.0 (so that ae > e ). Since the value of a (hence the strain in
y
y
steel reinforcement) is based on strain gradient consistent with ultimate strain in masonry,
it is possible to derive a mathematical relationship between c/d ratio and strains in masonry
and reinforcement at ultimate conditions. From similar triangles in Fig. 4.5
c ε
= mu (4.28)
d ε mu +αε y
Values of c/d ratio to be satisfied consistent with the desired value of tension reinforce-
ment strain factor a for a given set of design condition can be determined from Eq. (4.28)
for the specific value of e .
mu
a. Concrete Masonry: For a concrete masonry beam, e = 0.0025, e = 0.002 (Grade 60
mu
y
reinforcement), and a = 1.5 (assuming M / V d ≥ 1, see Table 4.2). Substitution of these
u
u v
values in Eq. (4.28) yields
.
c 0 0025
.
= = 0 454 (4.29)
(
.
.
.
d 0 0025 +1 5 0 002)
Equation (4.29) states that for c/d ≤ 0.454, the value of a in Eq. (4.28) would be greater
than 1.5. Therefore, the condition that would indicate a ≥ 1.5 (the desired condition) can
be stated as
c
≤ 0 454 (4.30)
.
d
Equation (4.30) states that for all values of c/d > 0.454, the value of a would be less than
1.5 in a concrete masonry beams having Grade 60 tension reinforcement, that is, the
strain in the tension reinforcement would be less than 1.5 times its yield value.
Equations similar to Eq. (4.30) can be derived for Grade 40 reinforcement following
the above procedure. The value of yield strain for Grade 40 reinforcement was calcu-
lated to be 0.00138. Thus from Eq. (4.28), we have for concrete masonry,
c 0 0025
.
= = 0 547 (4.31)
.
. ( .
.
d 0 0025 +1 5 0 00138)
Equation (4.31) states that for all values of c/d > 0.547, the value of a would be less than
1.5 in clay masonry beams having Grade 40 tension reinforcement. Thus, for the tension
reinforcement strain factor a to be equal to or greater than 1.5 in a concrete masonry
beam having Grade 40 tension reinforcement, the condition given by Eq. (4.32) should
be satisfied:
c
.
d ≤ 0 547 (4.32)
The significance of Eqs. (4.30) and (4.32) lies in the fact that they can be used (without
calculating strain in the tension reinforcement) to determine if the strain in tension rein-
forcement is equal to greater than 1.5 times the yield strain, which is a critical condition
for an acceptable beam design.
b. Clay Masonry: The values of c/d ratios for clay masonry beams can be derived by sub-
stituting e . = 0.0035 instead of 0.0025 in Eq. (4.29):
mu
.
c 0 0035
= = 0 538 (4.33)
.
d 0 0035 +1 5 0 002)
.
(
.
.