Reinforcement limits for the rc members

1. DESIGN FOR FLEXURE –
ACI REINFORCEMENT LIMITS
ACI procedures for ensuring under-reinforced behavior of flexural members have
changed in ACI 318-02, as compared to past versions of ACI 318. The new procedure
essentially presents a unified procedure for reinforced and prestressed concrete. The
provisions may also be found in Appendix B of ACI 318-99, though there have been
slight adjustments of the φ factors (to compensate for changes in the load combinations
found in Chapter 9 under ACI 318-02).
In ACI 318-02, the same set of strength reduction factors are used for the design of
beams and columns. The φ factor is adjusted depending upon the type of failure.
Definitions
The following definitions are used in specifying φ factors. Recall that the extreme fiber
concrete compressive strain at ultimate is assumed to be .003 in all cases.
Compression-controlled section (ACI 10.3.3) - a cross-section in which the net tensile
strain in the extreme tensile steel at the ultimate stage is less than or equal to the
compression controlled strain limit (i.e. the reinforcement tensile strain at the balanced
condition, or .002 for Grade 60 steel).
Tension-controlled section (ACI 10.3.4) - a cross-section in which the net tensile strain in
the extreme tensile steel at the ultimate stage is greater than or equal to .005.
Note that the definitions are based on the reinforcement strain in the extreme level of
reinforcement, and not at the centroid of the tensile reinforcement.
Strain distributions corresponding to tension and compression controlled sections are
shown in the figure below.
From: PCA, Notes on ACI 318-99
1
> 0.005

2. We can relate these definitions to the type of failure as follows:
• Compression-controlled sections are either balanced or overreinforced
• Transition sections are somewhat underreinforced
• Tension-controlled sections are significantly underreinforced
Strain Limit for Beams (ACI 10.3.5)
ACI 318-02 requires that beams (members without a significant axial force) be designed
such that the net tensile strain in the extreme tensile steel at the ultimate stage is greater
than or equal to .004. This ensures that beams are designed to be sufficiently
underreinforced. Per ACI 318-02 definitions, beams must be either tension-controlled, or
in the transition zone between tension-controlled and compression-controlled designs.
Beams cannot be designed as compression-controlled.
Values of c/dt and a/dt for various strain conditions
Limiting c/dt or a/dt ratios corresponding to specific strain conditions are given below. In
each case, these ratios are taken from simple linear relationships of the strain at the
extreme concrete compressive fiber (0.003) and the strain in the extreme layer of tensile
reinforcement.
Values of c/dt and a/dt corresponding to the tension-controlled limit:
1375.375.
005.003.
003.
β==
+
=
tt d
a
AND
d
c
Values of c/dt and a/dt corresponding to the minimum strain limit for beams:
1429.429.
004.003.
003.
β==
+
=
tt d
a
AND
d
c
Values of c/dt and a/dt corresponding to the compression-controlled limit:
1600.600.
002.003.
003.
β==
+
=
tt d
a
AND
d
c
Strength Reduction Factors (ACI 9.3.2)
The ACI Code requires a lower φ factor (higher factor of safety) for compression-
controlled members because they exhibit brittle failures. The φ factor for tension and
compression controlled sections are 0.90 and 0.70, respectively, for members with other
than spiral transverse reinforcement. Most beams fall into this category. A linear
transition is assumed between the tension- and compression-controlled limits as shown in
the figure below.
2

3. From: PCA, Notes on ACI 318-02 (Also ACI 318-02, pg. 100)
Recall that ACI 318-02 limits the design of beams such that the strain in the extreme
layer of tension reinforcement is at least 0.004. At a strain εt = 0.004, the strength
reduction factor (for other than spiral reinforced members) is φ = 0.81. Thus, it can be
stated that the strength reduction factor for beams varies between φ = 0.81 and φ = 0.90,
depending on the strain in the extreme layer of tension reinforcement. It is generally
most economical to design beams such that the strain in the extreme layer of tension
reinforcement exceeds 0.005, with φ = 0.90.
Values of ρ/ρb for various strain conditions
For common cases, such as a rectangular cross-section with a single layer of tensile
reinforcement and no compression reinforcement, we can derive simple relationships
between the balanced reinforcement ratio and the tension- and compression-controlled
limits. The simple derivation is shown below:
yieldstermsgRearrangin
'85.'85.'85.
'85.'85.'85.
'85.
ent,reinforcemtensiononlyawithbeamrrectangulabalanced)(ororcedunderreinfanFor
11 c
y
tc
y
c
y
c
y
c
ys
c
ys
ysc
t
f
f
d
c
f
f
d
c
f
f
d
a
f
df
a
bdf
dfA
a
bf
fA
afAabfTC
ddAssume
β
ρ
β
ρρ
ρ
=⇒=⇒=⇒
=⇒=⇒=⇒=⇒=
=






=
ty
c
d
c
f
f '85. 1β
ρ
We can now substitute the values of εt corresponding to the different strain conditions.
For example, at the compression-controlled limit (which corresponds to the balanced
case) for Grade 60 reinforcement:
3

4. 







=⇒=





+
=⇒=
y
c
t
t
f
f
d
c '85.
600.600.
002.003.
003.
002.0 1β
ρε
Values of ρ/ρb corresponding to different cases can be derived by making similar
substitutions for the tension-controlled limit and the beam minimum tension strain limit,
and then comparing the results.
The table below summarizes several parameters corresponding to the different strain
conditions:
At Compression-
Controlled Limit
At Beam
Minimum
Strain Limit
Transition Zone
(Acceptable Beam
Designs)
At Tension-
Controlled Limit
Above Tension-
Controlled Limit
εt εt = 0.002 εt = 0.004 0.004 < εt < 0.005 εt = 0.005 εt > 0.005
Behavior Balanced Sufficiently
Underreinforced
Significantly
Underreinforced
Very Underreinforced Very Underreinforced
c/dt c/dt = 0.600 c/dt = 0.429 0.429 > c/dt > 0.375 c/dt = 0.375 c/dt < 0.375
a/dt a/dt = 0.600β1 a/dt = 0.429β1 0.429β1 > a/dt > 0.375β1 a/dt = 0.375β1 a/dt < 0.375β1
φ (spirals) φ =0.70 φ =0.838 0.838 < φ < 0.90
φ =0.57 + 67εt
φ=0.37 + 0.20/( c/dt )
φ =0.90 φ =0.90
φ (other) φ =0.65 φ =0.812 0.812 < φ < 0.90
φ =0.48 + 83εt
φ=0.23 + 0.25/( c/dt )
φ =0.90 φ =0.90
ρ/ρb ρ/ρb = 1.000 ρ/ρb = 0.715 0.715 > ρ/ρb > 0.625 ρ/ρb = 0.625 ρ/ρb < 0.625
Note: tabulated values assume singly-reinforced rectangular sections with one layer of Grade 60 reinforcement
Previous versions of ACI 318 (ACI 318-99 and earlier) limited the reinforcement ratio
that could be used in design to ρ/ρb < 0.75. In the table above, it can be seen that ACI
318-02 effectively limits this ratio to 0.715 for singly-reinforced rectangular sections with
one layer of reinforcement. ACI 318-02 is therefore only slightly more conservative.
However, in order to use the “standard” strength reduction factor of φ = 0.90, note that
ACI 318-02 limits designs to ρ/ρb < 0.625.
Minimum reinforcement limits for beams (ACI 10.5)
ACI 318-02 requires a minimum amount of reinforcement in beams so that beams are
prevented from failing immediately after cracking. The minimum amount of
reinforcement is given as:
db
f
db
f
f
A w
y
w
y
c
s
200'3
min, ≥=
Note that ACI 10.5.3 stipulates that the above limit need not be applied if the area of
tensile reinforcement provided is at least 33% greater than required by analysis.
4