Civil Engineering Reference
In-Depth Information
φ
Figure 3.15
Bi-linear M-
curves for various precentages of tension steel
φ
u
can be calculated according to Equations (3.121) and (3.124),
respectively, for singly reinforced beams. In the case of doubly reinforced beams, these two
curvatures can be calculated according to the procedures described in Section 3.2.5.1.
The ultimate curvature
The two curvatures
φ
y
and
of a reinforced concrete mem-
ber is strongly affected by the percentages of steel. Five singly reinforced beams with steel
percentages varying from 0.8 to 3.0% are analyzed according to Equations (3.121)-(3.124)
and their bilinear
M
φ
u
, and therefore the ductility ratio
µ
f
c
bd
2
-
, which has
units of radian per unit length, has been made nondimensional by multiplying by the effective
depth
d
. The moment
M
has also been made nondimensional by dividing by the parameter
f
c
bd
2
. In calculating these curves a more realistic ultimate strain
/
φ
d
curves are plotted in Figure 3.15. The curvature
φ
ε
u
of 0.004 is assumed
rather than the conservative ACI value of 0.003. The normal material properties and geometric
location of steel bars are selected, namely,
f
c
=
27.6 MPa (4000 psi),
f
y
=
413 MPa (60 000
psi),
n
=
E
s
/
E
c
=
8 and a concrete cover (center of steel bar to surface) of 0
.
1
d
. Figure
3.15 shows clearly that the ultimate rotation
φ
u
d
decreases rapidly with the increase of the
tension steel percentage,
ρ
. The ductility ratio
µ
decreases from 8.0 for
ρ
=
0.8% to 1.7 for
ρ
=
3.0%.
The ductility of a bending member can be enhanced by the addition of compression steel as
shown by the nondimensional
M
f
c
bd
2
d
curves in Figure 3.16. The curves for the doubly
reinforced beams in this figure are calculated by the procedures discussed in Section 3.2.5.1.
For the three beams with tensile steel percentage
/
−
φ
ρ
of 3%, the compression steel percentages
ρ
of 0, 1 and 2% will give ultimate rotations
φ
d
of 0.0064, 0.0096 and 0.0193, respectively.
The corresponding ductility ratio
increases in the sequence of 1.7, 2.5 and 5.4. For the two
beams with tension steel percentages
µ
ρ
ρ
of 1%, an increase of compression steel percentage
