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7/27/2019 P-m Characteristics of Reinforced Concrete Sections
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P-M CHARACTERISTICS OF REINFORCED CONCRETE SECTIONS
A Thesis
Presented tothe Graduate School of
Clemson University
In Partial Fulfillment
of the Requirements for the DegreeMaster of Science
Civil Engineering
by
Paul W. Johnson III
December 2008
Accepted by:Dr. Patrick Fortney, Committee Chair
Dr. Scott Schiff
Dr. Bryant Nielson
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ABSTRACT
This manuscript presents two parametric studies which were performed to
evaluate a provision which originated in the 1994 Uniform Building Code (UBC). This
provision states that if gravity-induced axial loads carried by a reinforced concrete
member are greater than 35% of the pure axial load-carrying capacity (Po) of that
member, then the member cannot be used as part of the lateral force resisting system
(LRFS). Along with the UBC provision, the Structural Engineers Association of
Californias (SEAOC) Blue Book states that the value of 0.35Po corresponds to the level
of axial load representing a balanced state of strain, i.e., the balanced point of the axial
load-moment (P-M) interaction. It is also generally accepted that the balanced point on a
P-M interaction is located at the point of maximum moment.
To evaluate these assertions, two parametric studies were performed. In the first
study, square column cross-sections were analyzed while the second study considered
rectangular wall cross-sections. Concrete compressive strength and reinforcement ratios
were considered as variables. Two noteworthy observations are drawn from the results of
the studies: depending on reinforcing schemes and reinforcement ratios; (1) the balanced
point on a P-M is not always located at a value of 0.35 Po; (2) the balanced point is not
necessarily located at the point of maximum moment it may lay above or below the
point of maximum moment, or may be located approximately at the point of maximum
moment depending on cross sectional and material properties, as well as reinforcement
scheme.
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TABLE OF CONTENTS
Page
TITLE PAGE .................................................................................................................... i
ABSTRACT ..................................................................................................................... ii
LIST OF TABLES .......................................................................................................... vi
LIST OF FIGURES ...................................................................................................... viii
CHAPTER
I. INTRODUCTION ........................................................................................ 1
Background .............................................................................................. 2
Research Significance .............................................................................. 9P-M Interactions..................................................................................... 10
II. P-M CHARACTERISTICS OF SQUAREREINFORCED CONCRETE CROSS-SECTIONS ............................. 15
Parameters of Study ............................................................................... 15
Maximum Moment Relationship with Balanced Point .......................... 16Balanced Point Axial Load Relative to Po ............................................. 20
Observations and Recommendations ..................................................... 21
III. P-M CHARACTERISTICS OF RECTANGULAR
REINFORCED CONCRETE CROSS-SECTIONS ............................. 32
Parameters of Study ............................................................................... 32
Maximum Moment Relationship with Balanced Point .......................... 33
Balanced Point Axial Load Relative to Po ............................................. 35
Axial Load at Tensile Strain Limit Relative to Po ................................. 36
Constructing Design Based Interaction Diagrams ................................. 37Observations and Recommendations ..................................................... 40
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Table of Contents (Continued)
Page
IV. SUMMARY OF FINDINGS ....................................................................... 53
V. CONCLUSIONS AND RECOMMENDATIONS ...................................... 57
APPENDICES ............................................................................................................... 60
A: P-M characteristics of square sections
A1 2.5 inch cover ................................................................................ 61
A2 Minimum cover ........................................................................... 147A3.1 3 bars on each face ................................................................... 233
A3.2 4 bars on each face ................................................................... 242A3.3 5 bars on each face ................................................................... 251A3.4 6 bars on each face ................................................................... 260
B: P-M characteristics of rectangular sectionsB1 Increase reinforcement ratio by decreasing
bar spacing ...................................................................................... 269
B2 Increase reinforcement ratio by increasingbar size ............................................................................................ 386
C: Axial load component ratios of rectangular sections
C1 Increase reinforcement ratio by decreasingbar spacing ...................................................................................... 399
C2 Increase reinforcement ratio by increasing
bar size ............................................................................................ 516
D: Summary tables of square sections
D1 2.5 inch cover .............................................................................. 529D2 Minimum cover ........................................................................... 567
D3.1 3 bars on each face ................................................................... 605
D3.2 4 bars on each face ................................................................... 633
D3.3 5 bars on each face ................................................................... 659
D3.4 6 bars on each face ................................................................... 681
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Table of Contents (Continued)
Page
E: Summary tables for rectangular sections
E1 Increase reinforcement ratio by decreasingbar spacing ...................................................................................... 700
E2 Increase reinforcement ratio by increasingbar size ............................................................................................ 795
REFERENCES ...................................................................................................... 817
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LIST OF TABLES
Table Page
1-1 Difference in location of NA and PNA........................................................ 12
2-1 Variables of parametric study ...................................................................... 23
2-2 24 in square doubly reinforced column summary using3 ksi concrete ........................................................................................ 24
2-3 24 in square doubly reinforced column summary using
4 ksi concrete ........................................................................................ 24
2-4 24 in square doubly reinforced column summary using6 ksi concrete ........................................................................................ 24
2-5 24 in square column summary with 6 bars on each face
using 3 ksi concrete ............................................................................... 25
2-6 24 in square column summary with 6 bars on each face
using 4 ksi concrete ............................................................................... 25
2-7 24 in square column summary with 6 bars on each face
using 6 ksi concrete ............................................................................... 26
3-1A Variables of parametric study increasing
reinforcement ratio by decreasing bar spacing ...................................... 42
3-1B Variables of parametric study increasing
reinforcement ratio by increasing bar size ............................................. 42
3-2 8 in thick x 20 ft long wall summary using 3 ksi
concrete .................................................................................................. 43
3-3 8 in thick x 20 ft long wall summary using 4 ksi
concrete .................................................................................................. 43
3-4 8 in thick x 20 ft long wall summary using 6 ksiconcrete .................................................................................................. 43
3-5 14 in thick x 10 ft long wall summary using 3 ksiconcrete .................................................................................................. 44
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List of Tables (Continued)
Page
3-6 14 in thick x 10 ft long wall summary using 4 ksiconcrete .................................................................................................. 44
3-7 14 in thick x 10 ft long wall summary using 6 ksi
concrete .................................................................................................. 45
4-1 Summary of range of axial load component ratios for
column sections ...................................................................................... 55
4-2 Summary of range of axial load component ratios forwall sections ........................................................................................... 55
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LIST OF FIGURES
Figure Page
1-1 Eccentricity at the balanced point ................................................................ 12
1-2 Representative stress, strain, and force distributions ................................... 13
1-3 Representative P-M interactions .................................................................. 14
2-1 Representative cross-sections considered in
parametric study ..................................................................................... 26
2-2 P-M interaction of a 24 in square doubly reinforced
column using 3 ksi concrete ................................................................... 27
2-3 P-M interaction of a 24 in square doubly reinforced
column using 4 ksi concrete ................................................................... 27
2-4 P-M interaction of a 24 in square doubly reinforced
column using 6 ksi concrete ................................................................... 28
2-5 P-M interaction of a 12 in square doubly reinforced
column using 6 ksi concrete ................................................................... 28
2-6 P-M interaction of a 10 in square column with 3 barson each face using 3 ksi concrete ........................................................... 29
2-7 P-M interaction of a 10 in square column with 4 barson each face using 6 ksi concrete ........................................................... 29
2-8 P-M interaction of a 24 in square column with 6 barson each face using 3 ksi concrete ........................................................... 30
2-9 P-M interaction of a 24 in square column with 6 bars
on each face using 4 ksi concrete ........................................................... 30
2-10 P-M interaction of a 24 in square column with 6 bars
on each face using 6 ksi concrete ........................................................... 31
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List of Figures (Continued)
Page
3-1 Representative cross section considered in parametric
study ....................................................................................................... 45
3-2 P-M interaction of a 20 ft x 8 in wall using 3 ksiconcrete ................................................................................................. 46
3-3 P-M interaction of a 20 ft x 8 in wall using 4 ksiconcrete ................................................................................................. 46
3-4 P-M interaction of a 20 ft x 8 in wall using 6 ksi
concrete ................................................................................................. 47
3-5 P-M interaction of a 10 ft x 14 in wall using 3 ksiconcrete ................................................................................................. 47
3-6 P-M interaction of a 10 ft x 14 in wall using 4 ksi
concrete ................................................................................................. 48
3-7 P-M interaction of a 10 ft x 14 in wall using 6 ksi
concrete ................................................................................................. 48
3-8 Axial load component ratios of a 20 ft x 8 in wall using
3 ksi concrete ........................................................................................ 49
3-9 Axial load component ratios of a 20 ft x 8 in wall using
4 ksi concrete ........................................................................................ 49
3-10 Axial load component ratios of a 20 ft x 8 in wall using
6 ksi concrete ........................................................................................ 50
3-11 Axial load component ratios of a 10 ft x 14 in wall using
3 ksi concrete ........................................................................................ 50
3-12 Axial load component ratios of a 10 ft x 14 in wall using
4 ksi concrete ........................................................................................ 51
3-13 Axial load component ratios of a 10 ft x 14 in wall using6 ksi concrete ........................................................................................ 51
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List of Figures (Continued)
Page
3-14 Design P-M for a 20 ft x 8 in wall using 3 ksi concrete .............................. 52
3-15 Design P-M for a 10 ft x 14 in wall using 3 ksi concrete ............................ 52
4-1 P-M interaction diagram of a 10 ft x 14 in wall using
3 ksi concrete with = 0.61% ................................................................ 56
4-2 P-M interaction diagram of a 10 ft x 14 in wall using
3 ksi concrete with = 6.97% ................................................................ 56
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CHAPTER ONE
INTRODUCTION
Axial load-moment interaction diagrams (P-M) are used as a design aid by
engineers to ensure that a reinforced concrete beam-column has sufficient capacity to
carry design axial loads and moments, as well as ensuring sufficient ductility in the case
of seismic design. This manuscript provides a two-part parametric study investigating the
properties of P-M interactions; the first part of the study investigated column cross-
sections, while the latter investigated wall cross-sections. The purpose of this study was
to investigate the following two goals: (1) determine what percent of the pure axial load
carrying capacity (Po) of a reinforced concrete member corresponds with the axial load at
a balanced state of strain (Pb), and (2) investigate the commonly adopted assertion that
the balanced state of strain is approximately located at the point of maximum moment.
When constructing a P-M, three critical coordinates are usually of interest: (1) the
axial compressive capacity of the section when loaded with zero eccentricity (Po); (2) the
pure bending capacity of the section (Mo); and (3) the balanced point which represents a
balanced state of strain (Mb, Pb). A balanced state of strain is defined as the point on a P-
M interaction where the extreme fiber concrete compressive strain reaches its maximum
useable strain while simultaneously the extreme layer of tension steel reaches yield strain.
ACI 318 (2008) assumes the maximum useable concrete compressive strain to be 0.003.
At the balanced point, there is an axial load (Pb) and a corresponding moment (Mb).
Many textbooks, references, and, specifications note that the balanced state of strain
occurs at the location of maximum moment on the P-M curve: This paper presents the
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results of a parametric study, in which square cross-sections as well as rectangular wall
cross-sections are evaluated, with the intention of demonstrating that this is not always
the case. This paper also presents findings pertaining to the level of axial load
corresponding to a balanced state of strain relative to the pure axial load-carrying
capacity.
Background
The magnitude of axial load imposed on a reinforced concrete (RC) column or
wall is an important parameter to consider when ductile behavior is critical. For example,
when RC members carrying gravity-induced loads are suddenly subjected to lateral loads,
axial load-moment interaction may have an impact on a members capability to maintain
gravity load-carrying capacity. This may be especially critical if the applied lateral loads
induce increases in axial demand beyond that of gravity loads as in the case of coupled
core wall systems or columns in moment frames. It is generally accepted that gravity-
induced axial loads on RC members should be kept within an acceptable range if that
member is to be considered to participate in the lateral force resisting system (LFRS).
General practice in ductile design is to limit the design gravity loads to less than
the axial load corresponding to a balanced state of strain, i.e., below the balanced point
on an axial load-moment interaction (P-M) surface. A provision originating in the
Uniform Building Code (UBC 1994) required gravity-induced axial loads to be kept
below 35% of the pure nominal axial load-carrying capacity, 0.35Po, of the member if
that member is to be considered part of the LFRS. This design concept has crept into
general design of RC vertical members regardless of whether that member is a column or
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wall. Research into the origin of this axial load limit lead to commentary provided by the
Structural Engineers Association of California (SEAOC) in the Blue Book (1999) which
states that the 0.35Po limit is approximately the axial load corresponding to a balanced
state of strain. Thus, the intention of the original UBC limit is to keep axial loads below
an axial load corresponding to a balanced state of strain where ductile behavior is
required as in the case of lateral loads resulting from seismic events.
Axial load-moment interaction (P-M) space is used to evaluate sufficient design
of an RC member subjected to combined axial-moment loading. It is generally accepted
that the balanced point (balanced state of strain) is located at the point of maximum
moment on the P-M surface. Notable reinforced concrete design textbooks and reference
materials (Park and Paulay 1975; MacGregor and Wight 2004; Nilson et al. 2004; PCI
2004; PCA Notes on ACI 318 2008; McCormac 2006; Nawy 2006) support this accepted
notion; in these reference materials, the assertion that the balanced point occurs at the
point of maximum moment is supported. However, it should be noted that singly- or
doubly-reinforced cross-sections with low reinforcement ratios are predominantly used as
discussion points concerning the construction and understanding of the P-M
characteristics of RC members.
Personal correspondence with educators and practicing engineers participating in
the RC design community further support that the general thinking is that the balanced
point can be associated with the location of maximum moment on the P-M surface;
regardless of cross-sectional properties. Conversely, the author of this paper proposes
that, although the balanced point coincides with the point of maximum moment for many
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of the simple RC cross-sections generally used as discussion points in reference
materials, cross-sectional and material properties impact P-M characteristics.
Furthermore, regardless of whether or not the balanced point coincides with point of
maximum moment, cross-sectional and material properties have a significant impact on
the location of the balanced point and point of maximum moment relative the pure axial
load-carrying capacity. In fact, even for simple RC cross-sections, concrete compressive
strength and reinforcement ratio may produce balanced point axial loads as low as
-0.01% of the pure axial load-carrying capacity of the section, demonstrating that the
cross section must be in net tension in order to achieve a balanced state of strain.
Thus, if the balanced point is not necessarily located at the point of maximum
moment, the question is raised, As cross-sectional properties are changed, how is the
differential change in maximum moment relative to Po related to the differential change
in the balanced point location relative to Po? With this question in mind, a parametric
study was composed to evaluate the ratios of the axial load corresponding to maximum
moment, Pmm, and axial load at balanced point, Pb, relative to the pure axial load-carrying
capacity, Po (Pmm/Po and Pb/Po, respectively).
In order to explain why the axial load at the balanced point deviates from the
maximum moment, Pb and Pmm respectively, or the value of 0.35Po, mathematical
expressions have been provided. First, the discussion will be started assuming we are
using a constant 2.5 in cover from the face of concrete to the centroid of the longitudinal
reinforcement. Using the same cross-sectional dimensions as the column sections
investigated as part of this thesis (as will be discussed in Chapter 2), a plot can be
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generated explaining the difference between the location of the neutral axis (NA), and the
plastic neutral axis (PNA). Table 1-1 has been created to summarize this data, where Cb
represents the depth to the NA at a balanced state of strain, and drepresents the distance
from the top of the section to the centroid of the bottom layer of steel. Finally, if the
distance between the NA and PNA is a negative value, the NA is located above the PNA
and if a positive value is reported, the NA is located below the PNA.
From Table 1-1, we can plot the values of distance, which should be noted that
this distance represents the amount of eccentricity on the section, against the h/dratio for
each of the sections. Figure 1-1 shows this plot. It should be noted that at an h/dratio of
1.20 the eccentricity has a value of zero; this would indicate that the NA and PNA are
located at the same point, thus the axial load acts through both points. Other than noting
the point of zero eccentricity at an h/d ratio of 1.20 there are three other noteworthy
points: (1) when the NA is located above the PNA, or when h/dis less than 1.20, the net
internal axial load reduces the internal moment, (2) when the NA is located below the
PNA, or when h/d is greater than 1.20, the applied axial load increases the internal
moment, and finally, (3) when the NA is located at the PNA, or at an h/dratio of 1.20, the
net internal axial load has no effect on the internal moment. However, the question still
remains, what is the exact location of maximum moment. If a mathematical expression
can be written to describe the internal moment generated in a section, a simple derivative
can be taken and set to zero to find the maximum value along the P-M curve. To develop
an equation for internal moment, Figure 1-2 has been provided. This Figure shows a
representative cross section as well as its respective strain, stress, and force distribution.
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Referring to Figures 1-1 and 1-2, it is now possible to write an expression which
describes the internal moment of the section. First an equation for the applied axial force
will be written. Once this equation is developed, each of the components in the axial
force equation is simply multiplied by their respective moment arm to write a moment
equation. And finally as mentioned previously, a derivative of the moment can be taken.
Below are the equations which represent this process with the moment equation
developed by taking moments about the top of the section. In the following equation,
( )', , 0.85sC i sC i cA f f is the sum of all the compression steel forces (accounting for
displaced concrete); , ,sT i sT iA f is the sum of all the tension steel forces; '0.85 c wf b a is the
resultant concrete compressive force:
( )
( )
' '
, , , ,
' '
, , , ,
0.85 0.85
( ) 0.85 0.852 2
( ) 2
sC i sC i c c w sT i sT i
top sC i sC i c i c w sT i sT i i
top
P A f f f b a A f
a hM P A f f d f b a A f d P c
d h
M P cdP
= +
= +
=
Therefore, it can be seen that the maximum moment occurs at a location of half of
the section depth minus the distance to the neutral axis. So, it can be stated that when the
neutral axis is located at a distance of half the section depth, moment is maximum.
However, it should also be noted that since all of the sections in this study are symmetric
and have symmetric reinforcement, the plastic neutral axis is half the section depth.
Therefore, the internal moment is maximum when the neutral axis coincides with the
plastic neutral axis. Using the assumptions made in this study that the maximum usable
concrete strain is set at 0.003 and grade 60 reinforcing steel is used, and the compression
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strain limit per ACI 318-08 is 0.002, the balanced point is located at a value of c = 3d/5.
With this substitution, and noting that the only possibility for the axial load at the
balanced point to equal the axial load at maximum moment, their respective neutral axis
depths must be the same, or:
3
5 2
61.2
5
b
d hc c
h
d
= = =
= =
Notice that a value of 1.20 is achieved which agrees with Figure 1-1. So, what affects the
axial load at the balanced point? The location of the neutral axis is fixed for a given cross
section, therefore, a general equation for the axial load can be written as:
( )
( )
' '
, , , ,
1
' '
, , 1 , ,
0.85 0.85
0.85 0.85
sC i sC i c c w sT i sT i
sC i sC i c c w sT i sT i
P A f f f b a A f
a c
P A f f f b c A f
= +
=
= +
And with the balanced point located at c = 3d/5:
( )
( )
' '
, , 1 , ,
' '
, , 1 , ,
30.85 0.85
5
0.85 0.51
sC i sC i c c w sT i sT i
sC i sC i c c w sT i sT i
dP A f f f b A f
P A f f f b d A f
= +
= +
So, three scenarios can be investigated based on this study: (1) all variables stay
the same except for changes in concrete compressive strength, (2) for the same concrete
strength, the reinforcement ratio is changed by increasing bar size, and (3) for the same
concrete strength, the reinforcement ratio is changed by changing the number of bars.
This discussion was developed based on the case of having a constant 2.5 in cover,
however, is valid for any cover value. If Case 1 is examined, it is determined that the
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forces in the steel are unchanged, however, the concrete strength, and concrete strength
factor, 1, are changed. This would result in the following equation for the change in axial
load:
' '
, 1, , 1,0.85f i w c f f c i iP P P b c f f = =
In the above equation, the subscripts on the concrete strength and the concrete
strength factor represent the final and initial conditions. For 3 and 4 ksi concrete, the
concrete strength factor is the same. So, the only change in axial load is due to fc. For 5
and 6 ksi concrete, the change in axial load is due to both fc and 1. For the second
case where concrete strength is unchanged, and the reinforcement ratio is changed by
increasing just the bar size it should be recognized that the resultant concrete compressive
force is unchanged, the state of strain in each layer of steel is for the most part unchanged
although the cover to the longitudinal steel may vary slightly as the bar size changes, and
the number of compressive steel forces and tensile steel forces are unchanged. The final
case is similar to the second case. For this case, the concrete strength remains constant
while the reinforcement ratio is changed by increasing the number of bars in the section.
Like the second scenario, the resultant concrete compressive force is unchanged. Because
the number of bars on each face changes, the state of strain in each layer of steel is
changed due to the fact that there is reinforcement on all four faces. This results in
changes in stresses and forces in the steel. Therefore, the number of steel compressive
forces and tensile forces change. Since the number of bars on each face is now changing,
the value ofPmm/Po behaves more erratically. It is possible that when the number of bars
on each face change, that a bar may be very close to the neutral axis and would have very
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little impact on the internal resisting moment. However, it is also possible that when the
number of bars changes, multiple bars move further from the neutral axis generating a
significantly higher internal moment. So, depending on the number of bars used, an
additional layer of steel could potentially have little impact based on its location relative
to the neutral axis. Therefore, for the second and third scenarios, the change in axial load
can be written as:
( )', , , ,0.85sC i sC i c sT i sT iP A f f A f =
These two cases now become more difficult than the first scenario as the state of
strain is changing. The impact of changing the number of bars to change the
reinforcement ratio is further complicated by considering displaced concrete. With
changes in reinforcement ratios, the number of layers of steel in compression or tension
could change. Therefore, we have the following arguments to consider based on the state
of strain:
'
'
0
0 .85
.85
s y s y
s y s s
y s s s c
s y s y c
f f
f Eif
f E f
f f f
=
< =
= =
Research Significance
Cross-sectional properties play an important role in P-M characteristics related to
ductile design. The study presented in this manuscript provides insight into P-M
characteristics of both square RC cross-sections as well as rectangular wall RC cross-
sections, which will allow designers to more accurately define the anticipated behavior of
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RC sections; especially, relative to the amount of ductile space available within the P-M
design space. Furthermore, it is shown that assuming the balanced point is located at
maximum moment, and that the axial load component at the balanced point is 35% of Po,
could lead to either an overly-conservative or unsafe design, depending on cross-sectional
properties and concrete strength.
P-M Interactions
When constructing an axial load-moment interaction (P-M) diagram, five points
at a minimum should be considered; (1) the pure axial load-carrying capacity (no
eccentricity), Po; (2) the pure moment capacity (no axial load), Mo; (3) the point of
maximum moment, (Mmm, Pmm); (4) the balanced point (Mb, Pb); and (5) the point
defining the boundary for tension-controlled limit state, (Mt, Pt). With these five points, a
conservative interaction space can be defined (see Figure 1-3). However, providing
intermediate points above and below the balanced point provides a more accurate
accounting of the interaction space, and always increases the interaction area. Figure 1-3
shows a representative P-M interaction where many intermediate points were used to
construct the interaction surface. This study used a method where many intermediate
points were used. This was done to more accurately describe the interaction surface as
well as to more accurately locate the point of maximum moment and the pure moment
capacity of the sections considered as locating these points are an iterative process,
therefore the more points used, the more accurate the surface is.
Figure 1-3 shows a representative P-M interaction space with critical points and
areas identified. As can be seen in Figure 1-3, three critical lines are drawn from the
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origin to the point of maximum moment, the balanced point, and the tension-controlled
boundary. It should be noted that in Figure 1-3, the balanced point is shown at a higher
level of axial load than that of the point of maximum moment it is important to
recognize that depending on cross-sectional properties, the balanced point may be located
either above or below the point of maximum moment. Referring to the line that connects
the origin and the balanced point, the region of the interaction space above this inclined
line is considered to be compression-controlled (i.e., failure through concrete crushing
non-ductile); the region below the inclined line representing the tension-controlled
boundary is the tension-controlled region (i.e., failure through tension steel yielding
ductile). The area between the line from the origin to the balanced point and the line from
the origin to the tensile strain limit is known as the transition zone (refer to the strength
reduction requirements prescribed in ACI 318-08).
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Table 1-1:Difference in location between NA and PNA
Size (in) d (in) h/d Cb (in)Distance Between NA
and PNA (in)10 7.5 1.333 4.5 -0.50
12 9.5 1.263 5.7 -0.30
14 11.5 1.217 6.9 -0.10
16 13.5 1.185 8.1 0.10
18 15.5 1.161 9.3 0.30
20 17.5 1.143 10.5 0.50
22 19.5 1.128 11.7 0.70
24 21.5 1.116 12.9 0.90
Figure 1-1:
Eccentricity at the balanced point
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.10 1.15 1.20 1.25 1.30 1.35
DistanceBetweenNAandPNA
h/d Ratio
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(a) (b)
(c) (d)
Figure 1-2:Representative stress, strain, and force distributions
(a) representative cross-section, (b) strain distribution,
(c) stress distribution, (d) force distribution
c
e = e
e = 0.003
s y
c
Cross-Section Strain Distribution
0.85 f'
a
f
f
f
s3
s2
s1
fs4
Stress Distribution
Cc
Pb
F
F
F
F
s3
s2
s1
s4
Mb
Force Distribution
c
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Figure 1-3: Representative P-M interactions
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CHAPTER TWO
P-M CHARACTERISTICS OF SQUARE REINFORCED CONCRETE
CROSS-SECTIONS
Parameters of Study
Four types of square column cross-sections were evaluated in this study: (1)
doubly-reinforced cross-sections with a constant 2.5 in distance from the face of the
section to the bar centroid; (2) doubly-reinforced sections where the distance from the
face of the section to the bar centroid is 2 in (assuming 1.5 in clear cover from the face of
the concrete to the tie plus a #4 tie) plus half the longitudinal bar diameter; this method
will be referred to as the minimum cover method; (3) cross-sections with bars on each
face where reinforcement ratios were varied by changes in bar sizes; and (4) cross-
sections with bars on each face where reinforcement ratios were varied by
increasing/decreasing the number of bars on each face. Where reinforcement ratios were
varied by changing the number of bars on each face of the section, sections with 3, 4, 5,
and 6 bars on each face were considered. The purpose for investigating both the 2.5 in
cover and the 2 in plus half bar diameter cover was to see what difference, if any, small
changes in cover distance has on the P-M characteristics for a given cross section.
Concrete compressive strength was varied ranging from 3 to 6 ksi inclusive. Figure 2-1
shows representative sketches of the doubly-reinforced and multi-layer reinforced square
sections considered in this study.
While constructing the P-M interactions 60 ksi reinforcing steel was assumed.
Maximum useable concrete compressive strain, for normal weight concrete, per ACI
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318-08 was taken as 0.003, the axial load was assumed to act at the plastic neutral axis,
and a rectangular equivalent concrete compression block with a compressive stress of
0.85fc, was assumed. Displaced concrete was taken into account for any layers of steel in
compression.
Table 2-1 provides a summary of the variables considered for this study.
Referring to Table 2-1, it is evident that the study presented in this paper is rather
extensive; a total of 2,660 square cross-sections were evaluated resulting in 808 different
families of P-M interactions. A family of interactions is defined as a group of P-M
interactions for a given geometric cross-section and concrete compressive strength where
the members of the family are varying reinforcement ratios. Certain P-M families are
presented in the body of this thesis to support the discussion held. However, the
appendices contain summary tables and P-M diagrams for all of the families in this study.
Maximum Moment Relationship with Balance Point
Figures 2-2, 2-3, and 2-4 provide families of P-M interactions for 24 in square
doubly-reinforced cross-sections for 3, 4, and 6 ksi concrete, respectively. These three
Figures represent sections using the minimum cover method. For all three families, the
reinforcement ratios range from 1 to 7%. Referring to the Figures, it can be observed that
the balanced point coincides with the location of maximum moment regardless of
reinforcement ratio or concrete compressive strength.
Tables 2-2, 2-3, and 2-4 summarize the results of the 24 in square sections
corresponding to the P-M families shown in Figures 2-2, 2-3, and 2-4. Note that the Pb/Po
and Pmm/Po ratios are the same for all reinforcement ratios for each of the respective
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sections. However, this is not the case for all doubly-reinforced sections. An example of
this is shown in Figure 2-5 which shows a family of P-Ms for a 12 in square section with
6 ksi concrete using the constant 2.5 in cover method. For the doubly-reinforced sections
with the constant 2.5 in distance to the bar centroid, it is noted that for the smaller
sections investigated (10, 12, and 14 in sections) the values ofPmm are above Pb for low
reinforcement ratios. It should also be noted that the difference between these values is
larger in the smaller cross-sections. This difference between axial loads at maximum
moment and balanced point values is also affected by concrete compressive strength; the
larger the concrete strength, the larger the difference. For doubly-reinforced sections
larger than 14 in, values of Pmm and Pb are coincident at all concrete strengths and
reinforcement ratios investigated. The values for Pb vary nearly linearly with changes in
reinforcement ratio, and the values ofPb tend to decrease as is increased for all cross-
sections in this family. However, the rate of decrease in Pb becomes less as cross-
sectional dimensions are increased. For square sections 16 in and larger, the axial
component of the balanced point is nearly unchanged and is approximately horizontal. It
should be noted that bar size will have no effect on the values of Pb and Pmm as long as
the reinforcement ratios and bar locations remain the same. The reason for this is the
location of the bar in the cross-section is unchanged; therefore the force generated by the
layer of steel is unchanged (a characteristic driven by the fact that the state of strain is
unchanged at the balanced point).
For doubly-reinforced cross-sections where the bar is located at 2 in plus half the
bar diameter there are some notable differences, however, there were also many
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similarities. For instance, Pmm values are higher than Pb in the small cross-sections at low
reinforcement ratios. Again, the sections larger than 14 in have identical values of Pmm
and Pb regardless of reinforcement ratio and concrete strength. The behavior of the
doubly-reinforced sections, using this method in regard to the location of Pb, is still for
the most part similar to that of the doubly-reinforced sections using a constant 2.5 in
cover; however, Pb does not necessarily vary linearly for these sections. This is because
the location of the centroid of the bar changes depending on the bar size that is used.
With this in mind, if the two methods, (1) constant 2.5 in distance to the bar centroid
from face of section and (2) 2 in cover plus half the bar diameter, are compared for
otherwise identical sections, the interaction diagrams for sections with bar sizes smaller
than #8s will have increased moment capacity as the cover on the bar is less than 2.5 in
thus increasing the moment arm. The opposite is true for bar sizes above #8s; the
moment capacity is decreased as the cover on the bar is 2.5 in decreasing the moment
arm. Sections where #8 bars are used are identical. The amount of cover on the bar also
affects the locations ofPb and Pmm. When the bar size is increased, the axial component
ofPb and Pmm decreases. Tables 2-2, 2-3, and 2-4 summarize the Pb/Po and Pmm/Po ratios
for the 24 in square doubly-reinforced cross-sections.
The next set of families investigated were members with reinforcing bars on all
four faces. The first set has 3 bars on each face. For a 10 in square section, the axial load
components, Pmm and Pb, are negative for sections with high reinforcement ratios and 3
ksi concrete strength (e.g., 8% ratio with #9 bars). Thus, the balanced state of strain for
this situation can only occur when there is a net tensile load on the section (see Figure 2-
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6). Again, values of Pmm are higher than values of Pb at low reinforcement ratios.
However, as the cross-sectional dimensions increase, this phenomenon only occurs at
reinforcement ratios less than 1%.
When 4 bars on each face were considered, results similar to the 3 bars on each
face were encountered. For low concrete compressive strengths, the values ofPb and Pmm
are the same. However, as shown in Figure 2-7, as concrete strength increases, the values
of Pmm begin to increase to levels above Pb for reinforcement ratios as high as 3.5% in the
10 in section. However, as cross-sectional area increases, higher values of Pmm,
compared to Pb, are only evident when the reinforcement ratio is below 1%.
As the sections become more heavily reinforced and five bars and six bars on
each face are used, values for Pmm fall below Pb for low concrete compressive strengths
and high reinforcement ratios. For sections with four or less layers of steel, high
reinforcement ratios produced identical values of Pb and Pmm. For high concrete
compressive strengths and low reinforcement ratios, Pb values are above the Pmm values
for all sections with more than two bars on each face. Figures 2-8, 2-9, and 2-10 show
P-M families for a 24 in square sections with 6 bars on each face for 3, 4, and 6 ksi
concrete, respectively. The actual shape of the P-M interactions begins to change at the
location of maximum moment. The P-M interaction is nearly vertical between Pmm and
Pb. These trends occur when both 5 and 6 bars on each face are used. When 6 bars on
each face are used, the discrepancies between Pmm and Pb values become more
exaggerated. For heavily reinforced sections with a 3 ksi concrete strength (see Figure 2-
8) the deviation ofPb from Pmm can be as much as 167.7%. The interaction diagrams in
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Figures 2-8, 2-9 and 2-10 show significant differences between Pb and Pmm for a large
range of reinforcement ratios and different concrete strengths. The plots also show the
effect the concrete compressive strength has on the values ofPb and Pmm. As concrete
strength increases, the difference between Pmm and Pb decreases.
Balance Point Axial Load Relative to Po
For doubly-reinforced sections, it is apparent that the balance point is generally
located well below 0.35Po (see Tables 2-2, 2-3, and 2-4). For these sections, there is a
range of values ofPb/Po from 0.16 to 0.39 (see Table 2-3) and can go as low as -0.01 for
some column sections. Only sections with reinforcement ratios of 1% and 2% were
within 5% of 0.35Po. For doubly-reinforced sections with 3 ksi concrete, and
reinforcement ratios equal to or greater than 7%, the balanced point axial loads are as
small as 16% ofPo; a difference of nearly 54% from the 35% value prescribed by the
UBC provision.
For the sections with 6 bars on each face, similar results are encountered (see
Tables 2-5, 2-6, and 2-7). The Pb/Po ratios for these sections range between 0.43 and
0.16. As in the case for doubly-reinforced sections, the more highly reinforced the section
is, the larger the difference between the assumed value of 0.35Po and the actual location
of the balance point. Again, when evaluating the 35% assumption, only sections with
small reinforcement ratios (2% or less) have Pb/Po ratios of approximately 35%.
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Observations and Recommendations
This paper has provided a variety of reinforced concrete sections and their
respective P-M Interactions. It has been demonstrated that assuming that the axial load
corresponding to a balanced state of strain coincides with the point of maximum moment
and that that balanced axial load is 35% of the pure axial load-carrying capacity of the
member could potentially lead to either an overly conservative or unsafe design where
ductility is a concern. Examples describing the severity of this have been provided in
Chapter 4 of this paper. The majority of the Pb/Po ratios fall well below the approximate
value prescribed by the UBC provision; using the approximate value rather than an actual
value ofPb could result in a more brittle or even unsafe design. The authors recommend
using a calculated value for Pb rather than a fixed approximate value represented as a
percentage ofPo.
The commonly adopted notion that the balance point coincides with the maximum
moment should be disregarded. Even though this is the case for the majority of simple
reinforced concrete (RC) column sections, it is not true for all sections. The results of the
parametric study presented in this paper show that cross-sectional and material properties
have a significant impact on the relationships between axial load components of balanced
point and maximum moment relative to the pure axial load-carrying capacity of a RC
section. The resulting P-M families for the square RC sections considered in this study
can be powerful design aids for RC beam-column structural members.
Another separate study presented in this paper considered rectangular wall cross-
sections using the same set of parameters and variables with the addition to varying wall
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length to wall thickness ratios. The results of that portion of the study are presented in
Chapter 3 of this paper. In regard to wall sections, the relationship between Po, Pb, Pb,
and Pthave much more noticeable differences.
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Table 2-1: Variables of parametric study
Section Doubly-Reinforced 3 Bars on Each Face 4 Bars on Each Face
Size (b) Bar Size Bar Size Bar Size
(in) (%) US (#) (%) US (#) (%) US (#)
10 1.10 to 6.25 3 to 11 0.88 to 7.99 3 to 9 1.33 to 5.30 3 to 6
12 0.85 to 6.51 3 to 14 0.61 to 8.67 3 to 11 0.92 to 8.33 3 to 9
14 0.90 to 8.17 3 to 18 0.45 to 9.19 3 to 14 0.68 to 9.56 3 to 11
16 1.23 to 7.03 3 to 18 0.35 to 12.5 3 to 18 0.52 to 10.6 3 to 14
18 0.95 to 7.41 3 to 18 0.27 to 9.88 3 to 18 0.41 to 8.34 3 to 14
20 0.90 to 8.00 3 to 18 0.22 to 8.00 3 to 18 0.33 to 12.0 3 to 18
22 0.83 to 6.61 3 to 18 0.18 to 6.61 3 to 18 0.27 to 9.92 3 to 18
24 0.88 to 6.95 3 to 18 0.15 to 5.56 3 to 18 0.23 to 8.34 3 to 18
Table 2-1: Continued
Section 5 Bars on Each Face 6 Bars on Each Face
Size (b) Bar Size Bar Size
(in) (%) US (#) (%) US (#)
10 1.77 to 4.91 3 to 5 2.21 to 3.93 3 to 4
12 1.23 to 6.68 3 to 7 1.53 to 4.26 3 to 5
14 0.90 to 6.41 3 to 8 1.13 to 6.14 3 to 7
16 0.69 to 7.92 3 to 10 0.86 to 6.14 3 to 8
18 0.55 to 7.71 3 to 11 0.68 to 7.82 3 to 10
20 0.44 to 9.00 3 to 14 0.55 to 7.81 3 to 1122 0.37 to 7.44 3 to 14 0.46 to 6.45 3 to 11
24 0.31 to 6.25 3 to 14 0.38 to 7.82 3 to 14
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Table 2-2:24 in square doubly reinforced column using 3 ksi concrete
Bar SizeQty-Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)Pb/Po Pmm/Po
26-3 1.00 1,799 664 664 7,684 0.37 0.37
13-6 1.99 2,129 651 651 10,860 0.31 0.31
7-10 3.08 2,488 635 635 14,112 0.26 0.26
5-14 3.91 2,762 623 623 16,451 0.23 0.23
4-18 5.56 3,308 602 602 21,012 0.18 0.18
5-18 6.95 3,767 592 592 25,181 0.16 0.16
*Bar data shown represents number and size of bars on each face
Table 2-3:
24 in square doubly reinforced column using 4 ksi concrete
Bar Size
Qty-Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)
Pb/Po Pmm/Po
26-3 1.00 2,283 886 886 9,118 0.39 0.39
13-6 1.99 2,609 868 868 12,269 0.33 0.33
7-10 3.08 2,962 847 847 15,494 0.29 0.29
5-14 3.91 3,233 830 830 17,813 0.26 0.26
4-18 5.56 3,770 802 802 22,338 0.21 0.21
5-18 6.95 4,223 789 789 26,476 0.19 0.19
*Bar data shown represents number and size of bars on each face
Table 2-4:
24 in square doubly reinforced column using 6 ksi concrete
Bar Size
Qty-Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)Pb/Po Pmm/Po
26-3 1.00 3,253 1,170 1,170 11,722 0.36 0.36
13-6 1.99 3,568 1,146 1,146 14,811 0.32 0.32
7-10 3.08 3,911 1,116 1,116 17,967 0.29 0.29
5-14 3.91 4,173 1,092 1,092 20,234 0.26 0.26
4-18 5.56 4,695 1,052 1,052 24,668 0.22 0.22
5-18 6.95 5,134 1,032 1,032 28,746 0.20 0.20*Bar data shown represents number and size of bars on each face
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Table 2-5:24 in square column with 6 bars on each face using 3 ksi concrete
Bar Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)Pb/Po Pmm/Po
3 0.38 1,596 674 674 5,248 0.42 0.42
4 0.68 1,694 674 674 5,915 0.40 0.40
5 1.07 1,821 674 674 6,765 0.37 0.37
6 1.53 1,976 674 674 7,792 0.34 0.34
7 2.09 2,160 673 673 8,994 0.31 0.31
8 2.73 2,371 673 673 10,366 0.28 0.28
9 3.47 2,617 672 627 11,947 0.26 0.24
10 4.40 2,924 670 520 13,911 0.23 0.18
11 5.42 3,263 668 336 16,075 0.21 0.10
14 7.82 4,055 659 232 21,066 0.16 0.06
Table 2-6:
24 in square column with 6 bars on each face using 4 ksi concrete
Bar Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)Pb/Po Pmm/Po
3 0.38 2,083 897 897 6,699 0.43 0.43
4 0.68 2,181 895 895 7,360 0.41 0.41
5 1.07 2,306 894 894 8,202 0.39 0.39
6 1.53 2,459 892 892 9,221 0.36 0.36
7 2.09 2,639 890 890 10,412 0.34 0.34
8 2.73 2,847 887 887 11,773 0.31 0.31
9 3.47 3,090 884 884 13,337 0.29 0.29
10 4.40 3,392 879 848 15,270 0.26 0.25
11 5.42 3,726 873 755 17,383 0.23 0.20
14 7.82 4,507 857 392 22,297 0.19 0.09
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Table 2-7:24 in square column with 6 bars on each face using 6 ksi concrete
Bar Size
(%)
Po
(k)
Pb
(k)
Pmm
(k)
Mmax
(k-in)Pb/Po Pmm/Po
3 0.38 3,059 1,185 1,412 9,485 0.39 0.46
4 0.68 3,153 1,181 1,337 10,035 0.38 0.42
5 1.07 3,274 1,177 1,177 10,805 0.36 0.36
6 1.53 3,423 1,172 1,172 11,802 0.34 0.34
7 2.09 3,598 1,166 1,166 12,969 0.32 0.32
8 2.73 3,800 1,159 1,159 14,302 0.31 0.31
9 3.47 4,035 1,151 1,151 15,835 0.29 0.29
10 4.40 4,329 1,141 1,141 17,730 0.26 0.26
11 5.42 4,652 1,129 1,129 19,793 0.24 0.24
14 7.82 5,409 1,099 1,099 24,530 0.20 0.20
db
b
TOP STEEL AREA
BOTTOM STEEL AREA
#4 ENCLOSED HOOP
db
b
NUMBER OF BARS ONEACH FACE VARIES
#4 ENCLOSED HOOP
BAR SIZE AND
(a) Typical doubly-reinforced section (b) Typical section with bars on four faces
Figure 2-1: Representative square cross-sections considered in parametric study
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Figure 2-2:P-M interaction of a 24 in square doubly reinforced column for fc of 3 ksi
Figure 2-3:
P-M interaction of a 24 in square doubly reinforced column for fc of 4 ksi
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Figure 2-4:P-M interaction of a 24 in square doubly reinforced column for fc of 6 ksi
Figure 2-5:
P-M interaction of a 12 in square doubly reinforced column for fc of 6 ksi
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Figure 2-6:P-M interaction of a 10 in square column with 3 bars on each face for fc of 3 ksi
Figure 2-7:
P-M interaction of a 10 in square column with 4 bars on each face for fc of 6 ksi
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Figure 2-8:P-M interaction of a 24 in square column with 6 bars on each face for fc of 3 ksi
Figure 2-9:
P-M interaction of a 24 in square column with 6 bars on each face for fc of 4 ksi
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increased by decreasing bar spacing while holding bar size constant, whereas Table 3-1b
summarizes parameters for increasing the reinforcement ratio by increasing the bar size
and holding bar spacing constant. When referring to these two tables, it becomes apparent
that the study offered in this paper is quite extensive; a total of 2,368 rectangular wall
cross-sections were analyzed which resulted in 512 different P-M interaction families, or
a group of P-M interactions for a given geometric cross-section and concrete compressive
strength where the members of the family are the varying reinforcement ratios. Since it is
not feasible to present a comprehensive set of results for this study, specific P-M families
are presented to sustain the discussion held in an effort to convey the significant findings.
Maximum Moment Relationship with Balanced Point
Figures 3-2, 3-3, and 3-4 provide families of P-M interactions for an 8 in thick
wall which is 20 ft in length, and Figures 3-5, 3-6, and 3-7 provide P-M Interactions for a
10 ft long wall 14 in thick. Each set of diagrams embody concrete compressive strengths
of 3, 4, and 6 ksi, respectively. Both sets of diagrams were assembled using the method
where the reinforcement ratio, , is increased by increasing the bars size; not changing
the number of bars in each mat of reinforcement. When examining the figures, it is
observed that the balanced point only rarely coincides with the location of maximum
moment. It should also be noted that as the reinforcement ratio is increased, the axial
component of the balanced point increases while the axial load corresponding with
maximum moment decreases. One other noteworthy point is that as the concrete
compressive strength increases, the point where the balanced point corresponds with
maximum moment happens at higher reinforcement ratios. Therefore, should someone
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assume that the balanced point occurs at the point of maximum moment, there can be
significant conservatism in the design when the point of maximum moment is assumed to
coincide with the balanced point and is calculated as 0.35Po, especially as the
reinforcement ratio increases (except for the few sections with low reinforcement ratios
when high compressive concrete strengths are used and Pb is actually less than Pmm which
could potentially lead to a non-ductile section). To further illustrate this point, examples
have been provided in Chapter 4 of this paper.
Tables 3-2, 3-3, and 3-4 provide summaries of the sections corresponding with the
P-M interactions shown in Figures 3-2, 3-3, and 3-4 while tables 3-5, 3-6, and 3-7
summarizes the data in regard to Figures 3-5, 3-6, and 3-7. As shown in the tables, it is
seen that the ratios ofPb/Po and Pmm/Po very seldom equal each other. This behavior is
typical of all the wall sections considered in this study. For cross-sections using relatively
low concrete strengths, for example 3 ksi, it can generally be stated that the value ofPb
will always be larger than that ofPmm. As the concrete strength increases, it is common
to see values ofPb less than that ofPmm for low reinforcement ratios, regardless of the
cross-sectional aspect ratio. When this situation is the case, should an engineer make the
assumption that the balanced point is at the point of maximum moment, the cross-section
could prove to be less ductile than anticipated when assuming Pb corresponds with the
point of maximum moment.
To further illustrate the difference in location of the balanced point compared to
the maximum moment, Figures 3-8 through 3-13 have been provided. These Figures plot
the ratios Pb/Po, Pmm/Po, and Pt/Po against reinforcement ratio, where Pt is the level of
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axial load corresponding to a strain of 0.005, per ACI 318-08, in the extreme tension steel
layer (the defining boundary of the tension-controlled region). Figures 3-8 through 3-10
represent a 20 ft long wall 8 in thick, while Figures 3-11 through 3-13 illustrate a 10 ft
long wall which is 14 in thick. These plots graphically show the trends previously
discussed for Pb/Po and Pmm/Po. At low concrete strengths, as in Figure 3-8 and Figure
3-11, values of Pb are always greater than Pmm regardless of the reinforcement ratio.
When the concrete strength is increased to 6 ksi, as shown in Figure 3-10 and Figure
3-13, Pb starts out below Pmm at low reinforcement ratios. As the reinforcement ratio
increases, the values get closer together until they intersect. As the reinforcement ratio is
increased beyond the location where the balanced point equals the point of maximum
moment, the section exhibits the same behavior of sections with lower concrete strengths;
as the reinforcement ratio is increased, the value ofPmm begins to decrease much more
rapidly than Pb.
Balanced Point Axial Load Relative to Po
When comparing the UBC provision to the data gathered from this study, it is
evident that for rectangular cross-sections the balance point is generally located above
0.35Po. This can be seen in Tables 3-2, 3-3, and 3-4 which is representative of a 20 foot
long cross-section which is 8 inches thick, and again in Tables 3-5, 3-6, and 3-7 which
illustrate a wall section 10 feet in length and measuring 14 inches thick. In the two sets of
tables, values of Pb/Po range from 0.48 to 0.25. For smaller reinforcement ratios, the
lower values on this range of Pb/Po of 0.48 to 0.25 would be just over 0.30. Thus, the
lower values ofPb/Po are not far from the approximate value of 0.35Po; however, if this
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This outlined method creates a basic yet accurate way for designers to accurately
construct the design P-M space. It removes the tedious calculations required to develop
various P-M curves over and over should the first design not be adequate as well as the
need to use computer programs to estimate these curves. Rather, curves have already
been supplied and axial load components have been provided to quickly construct a
design interaction. Figures 3-14 and 3-15 provide two examples using this method. The
major difference between the aforementioned method and the method currently used is
the fact that there may be some difference between Pb and Pmm which is ignored in the
current method. Currently, engineers generally use a three point method to quickly
construct a P-M using only Po, Pmm (which is assumed to act coincident with Pb) andMo.
As previously mentioned, this could be a conservative method; however, it could also
provide an unsafe design, depending on the properties of the section, if it is assumed the
section is ductile below the balanced point when assuming the balanced point acts at the
point of maximum moment.
Figure 3-14 shows the design space for a 20 ft long by 8 in wide wall while Figure
3-15 shows the design space for a 10 ft long wall which is 14 in wide. For Figure 3-14,
Table 3-2 will provide the data required to construct the design space. In the first
example, we will first select a trial section as is mentioned in step one above; we will
explore a reinforcement ratio of 1.38%. Step two requires that table 3-2 is entered and
values for Pb, Pmm, and Pt which returns values of 3,348 k, 2,619 k, and 1,130 k,
respectively. The next step requires the engineer to compute the pure axial capacity of the
section, Pn,Max, which is simply 0.52Po for tied sections and Po is simply another value
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taken from the table, and found to be 7,998 k which yields a value of Pn,Max of 4,159 k.
Once these values are known, the axial load component values can be plotted on the
theoretical curve and the Pn,Max limit is known. Straight lines are then drawn to connect
the origin to each of the component values. The design P-M is now ready to be
constructed. Values above the balance point, Pb, are simply the theoretical curve scaled
by a factor of 0.65, while points below the tensile strain limit, Pt, are simply the
theoretical curve scaled by a factor of 0.9. The region between these two curves
represents the transition zone of strength reduction and can conservatively be constructed
by connecting the two curves just constructed with a straight line. Finally, the curve is
now ready to be used for design. Design loads are simply plotted on the curve and if they
fall within the design space, the design is acceptable; otherwise, another section may
need to be considered. If ductility is a concern, the design demands must fall beneath the
region defined by the balanced point (Mb, Pb).
When developing the design P-M shown in Figure 3-16, we will use a
reinforcement ratio of 2.43% where Pb, Pmm, and Pt are found to be 2,467 kips, 1,691
kips, and 674 kips respectively (see Table 3-5). Similar to the previous example, the
maximum permitted axial load, Pn,Max, can be computed for this section by taking
0.52Powhich yields 5,724 k for Figure 3-15. From this point, this example is the same as
the first. The values are plotted on the theoretical curve, and the design curve is
constructed. Once the design curve is constructed, the design loads can be plotted, and
the section can be verified as adequate, or unsafe.
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Observations and Recommendations
This paper has provided a variety of reinforced concrete sections and their
respective P-M Interactions. It has been demonstrated that assuming that the axial load
corresponding to a balanced state of strain coincides with the point of maximum moment
and that that balanced axial load is 35% of the pure axial load-carrying capacity of the
member could potentially lead to either an overly conservative or unsafe design. The
majority of the Pb/Po ratios were much higher than the approximate value prescribed by
the UBC provision; using the approximate value rather than an actual value of Pb could
result in a much more conservative design. Conversely, it has also been demonstrated that
the balanced point axial load can be significantly lower than 0.35 Po which could lead to
a design with insufficient ductility. The author recommends using a calculated value for
Pb rather than a fixed approximate value represented as a percentage ofPo.
Another important interpretation that comes from this study, is that regardless of
aspect ratio, the axial load component ratios are nearly identical for a given reinforcement
ratio. For instance, if we are to re-visit Table 3-2 and Table 3-5, we see two sections with
aspect ratios of 30 and 8.57 respectively. When number 9 bars are used, a reinforcement
ratio of 3.12% was used for the section in Table 3-2 while the section in Table 3-5 has a
reinforcement ratio 3.09%. Despite the small disparity in reinforcement ratio and the
huge difference in aspect ratio, the axial load component ratios are comparable. When
Table 3-2 is considered, Pb/Po, Pt/Po, and Pmm/Po values are recorded as 0.36, 0.07, and
0.22 respectively, while Table 3-5 has values of 0.35, 0.07, and 0.21.
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The commonly adopted notion that the balance point coincides with the maximum
moment should be disregarded. The results of the parametric study presented in this paper
show that cross-sectional and material properties have a significant impact on the
relationships between axial load components of balanced point and maximum moment
relative to the pure axial load-carrying capacity of a reinforced concrete (RC) section.
The resulting P-M families for the rectangular RC sections considered in this study can
be powerful design aids.
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Table 3-2:
8 in thick x 20 foot long wall using 3 ksi concrete
BarSize
(%)
Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.35 5,277 2,507 1,447 0.48 0.45 0.27
3 0.61 5,573 2,551 1,365 0.46 0.41 0.24
3 0.96 5,954 2,609 1,259 0.44 0.37 0.21
3 1.38 6,419 2,679 1,130 0.42 0.33 0.18
3 1.88 6,969 2,762 976 0.40 0.29 0.14
3 2.45 7,603 2,858 799 0.38 0.25 0.11
3 3.12 8,341 2,969 593 0.36 0.22 0.07
3 3.96 9,263 3,108 335 0.34 0.17 0.04
Table 3-3:
8 in thick x 20 foot long wall using 4 ksi concrete
Bar
Size
(%)
Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.35 6,903 3,319 1,963 0.48 0.46 0.28
4 0.61 7,195 3,361 1,879 0.47 0.43 0.26
5 0.96 7,570 3,415 1,770 0.45 0.39 0.23
6 1.38 8,028 3,481 1,638 0.43 0.37 0.20
7 1.88 8,570 3,559 1,482 0.42 0.33 0.17
8 2.45 9,195 3,649 1,301 0.40 0.29 0.14
9 3.12 9,922 3,753 1,091 0.38 0.25 0.11
10 3.96 10,830 3,883 828 0.36 0.22 0.08
Table 3-4:
8 in thick x 20 foot long wall using 6 ksi concrete
Bar
Size
(%)Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.35 10,156 4,369 2,628 0.43 0.47 0.26
4 0.61 10,439 4,405 2,541 0.42 0.45 0.24
5 0.96 10,803 4,452 2,428 0.41 0.41 0.23
6 1.38 11,247 4,509 2,291 0.40 0.38 0.20
7 1.88 11,773 4,577 2,128 0.39 0.36 0.18
8 2.45 12,379 4,656 1,940 0.38 0.34 0.16
9 3.12 13,084 4,746 1,722 0.36 0.30 0.13
10 3.96 13,965 4,860 1,449 0.35 0.26 0.10
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Table 3-5:
14 in thick x 10 foot long wall using 3 ksi concrete
BarSize
(%)
Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.34 4,614 2,178 1,257 0.47 0.45 0.27
4 0.61 4,871 2,215 1,183 0.46 0.41 0.24
5 0.95 5,201 2,262 1,088 0.44 0.37 0.21
6 1.37 5,604 2,320 971 0.41 0.34 0.17
7 1.86 6,080 2,388 833 0.39 0.28 0.14
8 2.43 6,630 2,467 674 0.37 0.26 0.10
9 3.09 7,269 2,557 488 0.35 0.21 0.07
10 3.92 8,068 2,671 254 0.33 0.17 0.03
11 4.83 8,949 2,795 -4 0.31 0.14 0.00
14 6.97 11,009 3,084 -611 0.28 0.07 -0.0618 12.38 16,236 3,805 -2,170 0.23 0.00 -0.13
Table 3-6:
14 in thick x 10 foot long wall using 4 ksi concrete
Bar
Size
(%)
Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.34 6,037 2,885 1,705 0.48 0.46 0.28
4 0.61 6,290 2,919 1,629 0.46 0.43 0.26
5 0.95 6,615 2,963 1,532 0.45 0.40 0.236 1.37 7,012 3,017 1,413 0.43 0.36 0.20
7 1.86 7,482 3,081 1,272 0.41 0.33 0.17
8 2.43 8,024 3,154 1,109 0.39 0.28 0.14
9 3.09 8,653 3,239 919 0.37 0.26 0.11
10 3.92 9,440 3,345 681 0.35 0.22 0.07
11 4.83 10,308 3,462 418 0.34 0.18 0.04
14 6.97 12,338 3,732 -202 0.30 0.13 -0.02
18 12.38 17,487 4,408 -1,792 0.25 0.02 -0.10
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Table 3-7:
14 in thick x 10 foot long wall using 6 ksi concrete
BarSize
(%)
Po
(k)
Pb
(k)
Pt
(k)Pb/Po Pmm/Po Pt/Po
3 0.34 8,883 3,798 2,283 0.43 0.47 0.26
4 0.61 9,129 3,827 2,205 0.42 0.45 0.24
5 0.95 9,444 3,865 2,103 0.41 0.42 0.22
6 1.37 9,829 3,912 1,980 0.40 0.39 0.20
7 1.86 10,285 3,967 1,833 0.39 0.36 0.18
8 2.43 10,810 4,030 1,663 0.37 0.33 0.15
9 3.09 11,421 4,104 1,466 0.36 0.31 0.13
10 3.92 12,184 4,196 1,218 0.34 0.27 0.10
11 4.83 13,026 4,297 945 0.33 0.24 0.07
14 6.97 14,995 4,531 301 0.30 0.18 0.0218 12.38 19,990 5,115 -1,350 0.26 0.08 -0.07
Figure 3-1: Representative cross-sections considered in parametric study
b
h
#4 HORIZONTALREINFORCEMENT
NUMBER OF BARS ONEACH FACE VARIES
BAR SIZE AND
d
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Figure 3-2:P-M interaction of a 20 ft x 8 in wall for fc of 3 ksi
Figure 3-3:
P-M interaction of a 20 ft x 8 in wall for fc of 4 ksi
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Figure 3-4:P-M interaction of a 20 ft x 8 in wall for fc of 6 ksi
Figure 3-5:
P-M interaction of a 10 ft x 14 in wall for fc of 3 ksi
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Figure 3-8:Axial load component ratios of a 20 ft x 8 in wall for fc of 3 ksi
Figure 3-9:
Axial load component ratios of a 20 ft x 8 in wall for fc of 4 ksi
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Figure 3-12:Axial load component ratios of a 10 ft x 14 in wall for fc of 4 ksi
Figure 3-13:
Axial load component ratios of a 10 ft x 14 in wall for fc of 6 ksi
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Figure 3-14:Design P-M for a 20 ft x 8 in wall for fc of 3 ksi
Figure 3-15:
Design P-M for a 10 ft x 14 in wall for fc of 3 ksi
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Table 4-1:
Summary of range of axial load component ratios for column sections
Family Pb/Po Pmm/Po Pt/Po
2.5 Inch Cover 0.03 To 0.38 0.03 To 0.44 -0.17 To 0.24
Minimum Cover 0.00 To 0.39 0.00 To 0.42 -0.22 To 0.24
3 Bars on Each Face -0.01 To 0.45 -0.01 To 0.48 -0.32 To 0.28
4 Bars on Each Face 0.05 To 0.44 0.05 To 0.48 -0.23 To 0.27
5 Bars on Each Face 0.09 To 0.44 0.05 To 0.47 -0.19 To 0.26
6 Bars on Each Face 0.14 To 0.42 0.06 To 0.47 -0.12 To 0.26
Table 4-2:
Summary of range of axial load component ratios for wall sections
Family Pb/Po Pmm/Po Pt/Po
Increase reinforcementratio by decreasing spacing
0.27 To 0.45 0.04 To 0.43 -0.08 To 0.23
Increase reinforcement
ratio by increasing bar size0.23 To 0.48 -0.01 To 0.48 -0.13 To 0.28
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CHAPTER FIVE
CONCLUSIONS AND RECOMMENDATIONS
The goal of this paper was to investigate three points: (1) should the 1994 UBC
provision, which states that if the gravity induced axial loads are greater than 0.35Po, be
accepted for all reinforced concrete sections, (2) should the SEAOC Blue Book provision
stating that the balanced point is approximately located at 0.35Po be accepted, and finally
(3) is the commonly adopted notion that the balanced point is approximately located at
the point of maximum moment generally accurate. This paper has provided a variety of
reinforced concrete sections and their respective P-M Interactions to verify or dismiss
these three points.
The UBC provision was examined by compiling 1,320 P-M interaction families
and recording Pb/Po values for every section analyzed in each of the families. When
column sections are considered, the parametric study shows that for a majority of these
sections, the value ofPb/Po falls well below 0.35Po. When this is the case, a brittle or
even unsafe design could be encountered resulting in a section not having the ductility it
was thought to have when being designed. However, when wall sections are considered,
the opposite was generally the case; for the majority of these sections, values of Pb/Po
were commonly found to be above the balance point. In this case, using the UBC
provision could provide an overly conservative design. Therefore, it is suggested that
rather than having a potentially unsafe or overly conservative design, it is recommended
that a calculated value for Pb/Po be used to aid in determining what amount of axial load
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can be applied to a section to attain the degree of ductility required in each specific
design rather than using a fixed approximate value represented as a percentage of Po to
determine this.
The second point to be investigated is the provision provided by the SEAOC Blue
Book. This provision states the balanced state of strain occurs at approximately 0.35Po.
As mentioned above when considering the UBC provision, it was determined that the
balanced point is not necessarily located at 0.35Po; for column sections, the balanced
point is by and large located below this value while for wall sections the balanced point is
above this value. Therefore, like the UBC provision, the SEAOC Blue book provision
should also be disregarded. Rather than risk using an unsafe design, it is recommended
that a true location be determined for the balanced point.
The last goal of this study was to investigate the idea that the location of
maximum moment coincides with the location of the balanced point. The results of the
parametric study presented in this paper show that cross-sectional and material properties
have a significant impact on the relationships between axial load components of balanced
point and maximum moment relative to the pure axial load-carrying capacity of a
reinforced concrete (RC) section. Although the majority of column sections stay true to
this notion, it is not true for all column sections. When wall sections are considered, this
statement can be far from accurate. Therefore, this commonly adopted notion should also
be disregarded.
Following the body of this thesis are appendices which include summary tables,
P-M interaction diagrams, and axial load component ratio plots. However, axial load
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component ratios for column sections have not been included. This is because for column
sections the sizes of the P-M families are quite small; each family may only contain one
or two family members. Because of the small family sizes, it is hard to recognize any
trends that may develop. Therefore, since the summary tables and P-M diagrams are
provided, the component plots have been omitted. Appendix A contains the P-M
diagrams for square sections while Appendix B shows the P-M diagrams for the
rectangular sections. Following is Appendix C which contains the axial load component
ratio plots for the rectangular sections. Lastly, Appendix D and E show the summary
tables for column sections and wall sections respectively.