P-m Characteristics of Reinforced Concrete Sections

7/27/2019 P-m Characteristics of Reinforced Concrete Sections

1/69

i

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


2/69

ii

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.


3/69

iii

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


4/69

iv

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





5/69

v

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


6/69

vi

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





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


concrete .................................................................................................. 43

3-4 8 in thick x 20 ft long wall summary using 6 ksiconcrete .................................................................................................. 43



7/69

vii

List of Tables (Continued)

Page



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


8/69

viii

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







2-6 P-M interaction of a 10 in square column with 3 barson each face using 3 ksi concrete ........................................................... 29



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


9/69

ix

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


3-4 P-M interaction of a 20 ft x 8 in wall using 6 ksi

concrete ................................................................................................. 47



concrete ................................................................................................. 48


concrete ................................................................................................. 48

3-8 Axial load component ratios of a 20 ft x 8 in wall using

3 ksi concrete ........................................................................................ 49


4 ksi concrete ........................................................................................ 49


6 ksi concrete ........................................................................................ 50


3 ksi concrete ........................................................................................ 50


4 ksi concrete ........................................................................................ 51

3-13 Axial load component ratios of a 10 ft x 14 in wall using6 ksi concrete ........................................................................................ 51


10/69

x

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


11/69

1

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


12/69

2

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


13/69

3

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


14/69

4

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


15/69

5

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.


16/69

6

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


17/69

7

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



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



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


18/69

8

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


19/69

9

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


20/69

10

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


21/69

11

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).


22/69

12

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


23/69

13

(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


24/69

14

Figure 1-3: Representative P-M interactions


25/69

15

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


26/69

16

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


27/69

17

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


28/69

18

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-


29/69

19

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


30/69

20

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%.


31/69

21

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


32/69

22

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.


33/69

23

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


34/69

24

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


35/69

25

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


36/69

26

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


37/69

27

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


38/69

28

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


39/69

29

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


40/69

30

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


41/69


42/69


43/69

33

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


44/69

34

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


45/69

35

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


46/69


47/69


48/69

38

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


49/69

39

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.


50/69

40

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.


51/69

41

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.


52/69


53/69

43

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:


Bar

Size

(%)

Po

(k)

Pb

(k)

Pt


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:


Bar

Size

(%)Po

(k)

Pb

(k)

Pt


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


54/69

44

Table 3-5:


BarSize

(%)

Po

(k)

Pb

(k)

Pt


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:


Bar

Size

(%)

Po

(k)

Pb

(k)

Pt


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


55/69

45

Table 3-7:


BarSize

(%)

Po

(k)

Pb

(k)

Pt


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


56/69

46

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


57/69

47

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


58/69


59/69

49

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


60/69


61/69

51

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


62/69

52

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


63/69


64/69


65/69

55

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



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


66/69


67/69

57

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


68/69

58

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


69/69

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.

Date post:	14-Apr-2018
Category:	Documents
Upload:	gouthamibandari91
View:	218 times
Download:	0 times

P-m Characteristics of Reinforced Concrete Sections

Documents