Curvature ductility of reinforced and prestressed concrete columns
by Bruce Alan Suprenant
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Civil Engineering
Montana State University
© Copyright by Bruce Alan Suprenant (1984)
Abstract:
Engineers are concerned with the survival of reinforced and prestressed concrete columns during
earthquakes. The prediction of column survival can be deduced from moment-curvature curves of the
column section. An analytical approach is incorporated into a computer model. The computer program
is based on assumed stress-strain relations for confined and unconfined concrete, nonprestressed and
prestressing steel. The results of studies on reinforced and prestressed concrete columns indicate that
reinforced concrete columns may be designed to resist earthquakes, while prestressed concrete columns
may not.
The initial reduction in moment capacity, after concrete cover spalling, of a prestressed concrete
column could be as much as 50%. Analyses indicate that the bond between concrete and prestressing
strand after concrete cover spalling is not critical. CURVATURE DUCTILITY OF REINFORCED AND
PRESTRESSED CONCRETE COLUMNS
by
Bruce Alan Suprenant
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Civil Engineering
MONTANA STATE UNIVERSITY
Bozeman, Montana
June 1984
© 1985
BRUCE ALAN SUPRENANT
All Rights Reserved
APPROVAL
o f a thesis submitted by
Bruce Alan Suprenant
This thesis has been read by each member of the thesis committee and has been found
to be satisfactory regarding content, English usage, format, citation, bibliographic style,
and consistency, and is ready for submission to the College o f Graduate Studies.
Date
Chairperson, Graduate Committee
Approved for the Major Department
Date
Head, Major Department
Approved for the College o f Graduate Studies
M X w Date
IItY
Graduate Dean
STATEMENT OF PERMISSION TO USE
In presenting this ti}$sis in partial fulfillmept qf thp requirements for a doctoral degree
at Montana State University, I agree that the Library shall make it available to borrowers
under rules o f the Library. I further agree that copying of this thesis is allowable only for
scholarly purposes, consistent with “fair use” as prescribed in .the U.S. Copyright IrSWRequests for extensive pppying or reproduction o f this thesis;should be referred to Uni­
versity Microfilms International, 300 North Zepb Road, Ann Arbor, Michigan 48106, to
whom I haVp granted “the exclusive right to reproduce and distribute copies pf the disser­
tation in and frbip nriorpfilm and thp right to reproduce and distribute by abstract ip any
format,’’
Any thesis would be difficult to finish without the support o f many people. I would
like to thank my parents, Jack and Betty, for their encouragement and support over many
years and my wife, Susan, for her patience. Also, my little one, Ashley, for sleeping
through the nights.
V
TABLE OF CONTENTS
Page
APPROVAL..............................................................................................................................
ii
STATEMENT OF PERMISSION TO USE............................................................................
iii
DEDICATION...................................................................................
iv
TABLE OF CONTENTS..................."......................................................................................
v
LIST OF TABLES.......................................
LIST OF FIGURES................................................................
ABSTRACT.................
vii
viii
x
Chapter
I
BACKGROUND..........................................................................! ...........................
Introduction........................................................................................................
Ductility for Seismic Loading..........................................................................
Existing Code Requirements for Special Transverse Steel
in Columns for Seismic Loading...................................................................
Determining Transverse Steel Content............................................................
Derivation of Moment-Curvature Curves.......................................................
Assumptions.................................................................................................
Analysis o f S ectio n s...................................................................................
Models............................................................................................................
Stress-Strain Curve for Confined C oncrete.......................................
Stress-Strain Curve for Unconfined Concrete ...................................
Stress-Strain Relationship for Nonprestressed Steel.............. ...........•
Stress-Strain Relationship for Prestressing S te e l...........................-.
II
I
I
2
10
13
16
16
16
20
20
23
24
27
ANALYTICAL S T U D Y ..........................................................................................
30
Introduction.......................................................................................................
Confinement......................... '............'........................................................
Non-Prestressed Steel Strain Hardening.....................'............................
Longitudinal Steel Content.......................................................................
Concrete Cover............................................................................................
30
33
35
36
37
vi
TABLE OF CONTENTS—Continued
Page
Prestressed Concrete..........................................................................................
Confinement.................................................................................................
Axial L o a d ...................................................................................................
Bond Compatibility Factor.............................................
Concrete Cover............................................................................................
Improving Curvature Ductilityof Prestressed Columns.........................
38
38
39
40
41
41
III
ENGINEERING EXAMPLES.................................................................................
43
IV
CONCLUSIONS AND RECOMMENDATIONS...................................................
46
Reinforced Concrete Columns..........................................................................
For Prestressed Concrete C olum ns.................................................................
49
49
REFERENCES....................... ' .................................................................................................
50
APPENDICES................................
56
Appendix A —Computer Generated M-0 C urves................................ ....................
Appendix B —Program Documentation and Listing................................................
57
77
vii
LIST OF TABLES
Tables
Page
1. Relationship Between Z and Hoop Steel..............................................................
33
2. Moment Reduction and Curvature Ductility ........................................................
37
3. Moment Reduction and Curvature D uctility................
39
4. Moment Reduction...................................................................................................
41
5. Column Characteristics.......................................................................................
42
viii
LIST OF FIGURES
Figures
Page
1. Relation between displacement and curvature d u ctility.................. 4
2. Building structures under seismic loading and possible mechanisms
5
3. Dissipation of energy in prestressed and reinforced concrete members...........
9
4. Idealized moment-curvature hysteresis loops for structural concrete
system s...............................■.......................................................................................
11
I
5. Envelope curve for concrete..........
..........................................................
15
6 . Envelope curve for steel . ........................................................................................
15
7. Theoretical moment-curvature determination.....................................................
18
8 . Stress-strain relations for copcrete..........................................................................
21
9. Stress-strain relation for non-prestressed steel.................................. ..................
25
10. Stress-strain relations for prestressing s te e l..........................................................
28
11.
31
Column properties.....................................
12. General M-0 curve for reinforced concrete colum ns.........................................
47
13. General M-0 curve for prestressed concretecolumns. .......................................
47
Appendix Figures
58
15. Effect o f confining hoop steel; P = 0.20 f 'c A g ...................................................
59
16. Effect o f confining hoop steel; P = 0.10 f 'c A g .......................................
60
17. Effect of confining hoop, steel; P = 0.30 f 'c A g ...................................................
61
18.
12
62
19. Effect o f steel strain hardening; P = 0.20,f'c Ag, Z = 12 ...................................
63
Effect
o f steel strain hardening; P = 0.10 f'c Ag,
Z
=
.
14. Effect o f confining hoop steel.................................................................................
ix
Figures
'
Page
20. Effect o f steel strain hardening; P = 0.30 f'c Ag, Z = 1 2 ....................................
64
21.
of longitudinal steel content; Z = 1 2 . . ...............................................
65
22. Effect of confinement and varying steel percentages;
P = 0.20 f 'c A g ............................................-. ..................... : ..................................
66
23. Effect of concrete cover; P = 0.10 f'c Ag..........................................................
67
24. Effect of concrete cover; P - 0.20 f 'c Ag...............................................................
6§
25. Effeqt of concrete cover; P = 0.30 f 'c Ag. , . ........................................................
69
26. Effect of confinement; P = 0.20 f'c A g .................................................................
70
27. Effect of axial load; Z = 12 ......................................................................................
'71
Effect
28. Effect o f bond compatibility factor; P = 0.20 f'c Ag,. Z = 12....................
72
29. Effect pf concrete cover; P = 0.20 f'c Ag, Z = 1 2 ...............................................
73
30. Effect o f non-prestressed steel and hoop steel; P = 0.20 f 'c A g ........................
74
3 1. Ductility of concrete column Life Sciences Building, Bozeman, M T..............
75
32. Ductility of prestressed bridge piles.......................................................................
76
ABSTRACT
Engineers are concerned with the survival o f reinforced and prestressed concrete col­
umns during earthquakes. The prediction o f column survival can be deduced from momentcurvature curves o f the column section. An analytical approach is incorporated into a com­
puter model. The computer program is based on assumed stress-strain relations for confined
and unconfined concrete, nonprestressed and prestressing steel. The results of studies on
reinforced and prestressed concrete columns indicate that reinforced concrete columns
may be designed to resist earthquakes, while prestressed concrete columns may not.
The initial reduction in moment capacity, after concrete cover spalling, of a prestressed
concrete column could be as much as 50%. Analyses indicate that the bond between con­
crete and prestressing strand after concrete cover spalling is not critical.
I
CHAPTER I
BACKGROUND
Introduction
The ACI Code [10,11] requires the consideration o f strength and .serviceability in
reinforced concrete design. Ductility is considered to be of secondary importance and is
detailed into a structure, rather than designed. Because the ACI Code [10,11] regulates or
delegates ductility to a secondary level, many engineers have been confused by the impor­
tance of ductility for a structure or member and its implication in the selection o f lateral
load levels for earthquake design.
Dynamic analyses of structures responding to ground motions recorded during severe
earthquakes have been performed [35,58]. These show that the theoretical elastic response
inertia loads are much greater than the lateral loads recommended by design codes [ 10,11,
46,59] for earthquake loading. It is well documented [9,24,30,37] that structures designed
to the lateral load levels o f design codes have survived severe earthquakes. This post earth­
quake survival has been attributed mainly to the ability of ductile structures to dissipate
energy by post-elastic deformations. Although other energy dissipaters such as damping
and soil-structure interaction help provide a reduced response, ductility o f reinforced con­
crete members is considered to be the most important energy dissipater [35].
This thesis presents a theoretical study o f the ductility o f square confined reinforced
and prestressed concrete column sections under monotonic loading. The parameters which
were investigated to determine their effect on the ductility o f reinforced and prestressed
2
concrete columns are: axial load, longitudinal steel content, confinement, bond compati­
bility factor, concrete cover, and non-prestressed steel.
Ductility for Seismic Loading
A measure of the ductility o f structures with regard to seismic loading is the displace­
ment ductility factor defined as Au/Ay , where Au is the lateral deflection at the end of the
post-elastic range and Ay is the lateral deflection at first yield [37]. The displacement duc­
tility factor required in design may be estimated on the basis of the ratio o f elastic response
load to code load and typical values for displacement ductility factors usually range from
three to five [9,56].
A rotational ductility factor for members has been calculated by some dynamic
analyses as ©u/©y , where ©u = maximum rotation of end o f member and ©y -,rotation at
end o f member at first yield. The design engineer, however, would benefit by information
concerning the member section behavior, expressed by the curvature ductility factor
0u/0 . Where 0U f= maximum curvature at the section and <j>y = curvature o f the section at
first yield. For the designer, required 0u/0 y values are a more important index for ductility
demand than the displacement or rotational ductility factors. This is because once yielding
has commenced in a structure the deformations concentrate at plastic hinge positions and
further displacement occurs mainly by curvature in the plastic hinge region [37]. Thus the
required 0u/0y ratio will be larger than the ©u/ 0 y ratio, and the ©u/ 0 y ratio will be greater
than the Au/Ay ratio.
The relationship between curvature ductility and displacement ductility hgs been
illustrated with reference to a cantilever column with a lateral load at the end by Park amf
Paulay [37]. By using well known curvature-area theorems, the lateral deflection at the
3
column top for first yield and ultimate deformation can be determined. Therefore, the dis­
placement ductility factor may be related to the curvature displacement factor as shown in
Figure I.
For multistory frames, the 0u/0y ratio required o f members, designed according to
present code seismic loading, has not yet been clearly established. The relationship between
displacement and curvature ductilities can be complex [37]. Plastic hinges forming in
beams and columns do so at different levels o f axial and lateral load. Park and Paulay [37]
have attempted to deduce curvature ductility demand of multistory frames using static
collapse mechanisms.
The sequence o f plastic hinge development in structures will influence the curvature
ductility demand. Nonlinear dynamic analyses [37] have indicated that ductility demand
concentrates in the weak parts o f structures. The weak portion may act as a plastic hinge,
enabling the remaining structure to respond elastically, or requiring the first hinge, i,e.,
weak portion, to achieve hig]i curvature ductilities. Park and Paulay [37] have illustrated
this by examining static collapse mechanism. Frame and shear walls which can be used for
seismic resistance are shown in Figure 2. Possible mechanisms which could form due to
flexural yielding and formation o f plastic hinges are also shown in Figure 2. A column
sidesway mechanism can form, if yielding commences in the columns at only one level,
this would indicate that the columns of all other levels are stronger or carry less load, Such
a mechanism requires yery larger curvature ductility demands on the plastic hinges of the
critical level [3 7 ], particularly for tall buildings. If, however, the yielding commences in
the beams before the columns, a beam sidesway mechanism will develop [3 7 ]. This mecha­
nism makes more moderate demands on the curvature ductility at the plastic hinges in the
beams and column bases. This is basically due to the greater number of hinges required to
)
4
Member with ultimate curvature reached: (a) member, (b) bending moment diagram,
(c) curvature diagram.
22
Au =
,
^
+ (0U - 0y) 2p (2 -0.52p )
3 2 P (^ L )
2'
K =
C2 (u - 1)
<t>y ~ 3 2p (2 - 0.52p)
Curvature Ductility versus Displacement Ductility
0.02
0.05
0.07
0.10
0.15
0.20
0.25
VS
Il
3
2p/2
68
28
21
15
11
8
7
51
35
18
22
15
8
16
11
12
8
8
7
6
6
5
3
4
4
3
2
6
5
Figure I. Relation between displacement and curvature ductility.
3
3
5
•Plastic
hinge
Frame
Cantilever shear wall
and mechanism
Column s i d e s w a y
mechanism
Beam s i d e s w a y
mechanism
Coupled shear walls and
mechanism
Figure 2. Building structures under seismic loading and possible mechanisms [37].
6
form in the beam mechanism. A beam sidesway mechanism is the preferred mode of inelas­
tic deformation. Hence most codes [ 10,11,30,59] require a strong column-weak beam
approach when designing moment resistant frames.
Higher modes o f vibration in the actual dynamic situation influence the moment pat­
tern and it has been found [37] that plastic hinges in beams move up the frame in waves
involving a few levels at a time. For static collapse mechanism involving a plastic hinge at
the base, the curvature ductility demand for a given displacement ductility factor depends
very much on the plastic hinge length. Recent experimental information is available [17,
38,50] on plastic hinge lengths for reinforced concrete columns confined by hoop and
spiral reinforcement.
The static collapse mechanisms of Figure 2 are idealized in that they involve behavior
under code type static loading. Due mainly to the effects of higher modes of vibrations,
the actual dynamic situation is different, but a review o f Figure 2 will give the designer a
reasonable perception o f the collapse mechanism.
The strong column-weak beam design concept attempts to have plastic hinges form in
the beams rather than the columns. The seismic provisions o f ACI 318-77 [10] and ACI
318-83 [11] require that at beam-column connections the sum o f the moment strengths of
the column should exceed the sum of the moment strengths of the beams along each
principal plane at the connections. Park and Paulay [37] have shown that this code require­
ment will not prevent plastic hinges formation in the columns for three reasons:
(a) The beam input moment may be considerably higher than calculated because of
statistical variations in steel yield strength and the onset o f steel strain-hardening.
(b) Points o f inflection may occur well away from the mid-height o f columns at vari­
ous stages during an earthquake [1 9 ], thus requiring a column strength to prevent
hinge formation which is much greater than required by ACI 318-77 and ACI
318 - 83 .
7
(c) Biaxial bending resulting from an arbitrary ground motion will generally reduce
the flexural strength o f a column [3 4 ].
It is evident that column flexural strengths greater than the ACI requirement; would be
needed if plastic hinges in columns are to be avoided. However, column hinging may occur.
A column sidesway mechanism, besides requiring large curvature demands on the sec­
tion, will also require an engineer with considerable ingenuity, since the straightening and
repair o f the columns will prove to be difficult. A recent example o f damage concentrating
mainly in one story of a structure is tl>e Olive View Hospital which suffered considerable
damage because o f the 1971 S^n Fernando earthquake [24]. The permanent lateral dis-.
placement of the structure after the earthquake, which was about 2 ft, resulted almost
entirely' from the deformations in the first story columns, Because o f the extensive ejapage, the Olive View Hospital was demplished.
As an example demonstrating the necessity o f analyzing column curvature ductility,
consider the shock-absorbing soft story concept for multistory earthquake structures
proposed by Fintel and Khan [1 9 ]. This concept is based on controlling the lateral forces
that occur in the structure during an earthquake by forcing all the inelastic deformation to
a soft story. This concept requires extensive yielding in the columns and therefore requires
the columns to sustain large post-elastic deformations. It appears that the “soft story”
concept should be implemented in conjunction with appropriate consideration to fixture
repair.
The “Guide Specifications for Seismic Design o f Highway Bridges, 1983” [2Q] indi­
cates that bridge piers should be designed for a required curvature ductility. This is the first
code to imply that curvature ductility is an important requirement in design.
In the preceding discussion, the difference between reinforced or prestressed concrete
ductility was not considered. While the ductility o f reinforced concrete has been the sub­
ject o f numerous experimental and theoretical studies [6,13,16,17,21,22,25,30,|35,36,39,
8
41,42,43,44,45,52], the ductility of prestressed concrete has received little attention [9,
51]. Because of the lack of experimental and theoretical studies, prestressed concrete in
primary seismic resistant elements such as shear walls and frames has not met with the
same code acceptance as reinforced concrete.
Therg is a lack of detailed building code provisions for the seismic design of pre­
stressed concrete. For example, ACI 318-83 [11] contains special provisions for the
seismic design of reinforced conrete structures but does not have corresponding provisions
for prestressed concrete. This is also true o f the Uniform Building Code [5 9 ], the SEAOC
Code [46], and the seismic; design provisions o f the Applied Technology Council [44].
Lin [30] presented results o f a dynamic computer analysis o f a 19 story prestressed
concrete apartment building. The building was first designed using the equivalent static
earthquake forces specified by the 1961 Uniform Building Code. The elastic dynamic
response of the structure tp the North-South component of the 1940 El Centro earthquake
was calculated. The El Centro earthquake produced forces and displacements about five
times the Code values, This indicates the need for prestressed structures to be capable of
developing large post-elastic deformations or be designed for greater equivalent static lateral
loads.
The area under moment-curvature curves for prestressed and reinforced concrete give
an indication o f their energy absorption capacities. Despeyroux [18] and Candy [18]
established very important differences between the behavior of reinforced and prestressed
concrete. Candy illustrated the differences by using Figure 3. Cyclic loading tests have
given curves for the two materials approximately as drawn. The shaded area represents the
energy dissipation; the area up to the dashed line represent the energy absorbed. Thus
although the energy absorbed by a prestressed member and a reinforced concrete member
9
PRESTRESSED
CONCRETE
energy d i s s i p a t e d - ^ ™
energy absorbed
T:UVv.V.y
m
y
!REINFORCED
CONCRETE
energy
dissipated
energy
absorbed.
LOAD
,A
DEFORMATION
Figure 3. Dissipation o f energy in prestressed and reinforced concrete members [18].
10
may be the same, much more energy would be dissipated in the reinforced concrete mem­
ber [35]. Candy concluded that the reduction in response caused by plastic strain is much
smaller in prestressed concrete.
Figures 4 (a), 4 (b), and 4 (c) show idealized moment-curvature hysteresis lopps for
cyclic loaded prestressed and reinforced concrete members [58]. Nonlinear dynamic analy­
ses o f single degree o f freedom systems responding to severe earthquakes have shown that
the maximum displacement of a prestressed concrete system is on the average 1.3 times
that o f a reinforced concrete system with the same code design strength, viscous damping
ratio, and initial stiffness [37]. The r$ioment-curvature loops of prestressed concrete mem­
bers can be “fattened” to reduce the displacement response, if non-prestressed steel is
added tp the member to provide energy dissipation, Figurp 4 (c) shows a typical momentcurvature loop for partially prestressed concrete.
Althoygh the. displacement response of prestressefi and reinforced concrete members
are different, the possibility o f yielding occurring at column ends makes it important tp
ensure that the colymns are capable of behaving in a ductile manyer. Hence, for reinforced
and prestressed concrete columns adequate transverse steel in the form o f hoops should be
provided at the potential plastic hinge regions.
■ Existing Code Requirements for Special Transverse Steel
in Columns for Seismic Loading
The ACI [10,11], SEAOC [46], and ATC [44] codes contain recommendations for
the amount qf special transverse reinforcement required in the enfls o f column? when duc­
tile moment-resiting frames are to be designed for seismic loading. All columns o f ductile
moment-resisting frames designed according to the SfEAOC code are required tp havp trans­
verse reinforcement. ACI 318-77 requires special transverse reinforcement only when the
column load exceeds 0.4 ^Pb.
11
(a) Prestressed Concrete System
(b) Reinforced Concrete System
(c) Partially Prestressed Concrete System
Figure 4. Idealized moment-curvature hysteresis loops for structural concrete systems
[58].
12
For circular spiral steel the ratio o f volume o f spiral reinforcement to total volume o f
core is given as:
0.45
fV
A
CD
fy
but not less than 0.12 f'c/fyWhen rectangular hoop reinforcement is used without supplementary cross ties, the
required area of a bar leg is given as:
Ash " 0.225 CllSj1
- I
rc_
fy~
( 2)
Equation (I) was derived from the requirement that the strength o f an axially loaded
spiral column after the concrete cover has spalled should at least equal the strength before
spalling. Based on test results achieved by Richart, Brandtzaez, and Brown [4 7 ], it was
assumed that when the concrete cover spalls, the spiral steel yields apd exerts a radial pres­
sure on the concrete core, This pressure increases the compressive strength pf the porp by
approximately 4.1 times the radial pressure. Assuming that rectangular hoops are only *50
percent as effective as circular spirals as confining reinforcement, Equation (2) can be
derived.
ACI 3 1§r83 [11] proposes two new equations for spiral and loop confinement of
columns. For spiral reinforcement*
Ps = 0.12 f'c/fyh
(3)
and for hoop reinforcement not less than Equations 4 or 5
Ash = 0.3 (shcf'e/fyh)[(A g/A ch)-r- 1]
(4)
Ash = 0.12 shcf'c/fyh.
(?)
The philosophy of preserving the ultimate strength of axially loaded columns after
spalling has been discussed [37]. A more appropriate criteria wpuld emphasize the ultb
mate deformations and limited amount of moment reduction of concrete columns. The
13
confinement afforded by the transverse steel o f Equations (I), (2), (3) or (4 ) will result in
improved column behavior but these equations may not accurately represent the amount
o f confining steel necessary to achieve the required ultimate curvatures under earthquake
loading. Roy and Sozen [48] indicated that confinement by rectangular hoops may not
cause any strength increase, thus the philosophy behind the derivation, o f Equation (2)
may never be realized in actual practice.
The.previous discussion has not been related to prestressed columns. While investi­
gations on confining steel for prestressed columns is in progress, current recommendations
[10,11,46,59] for prestressed columns are the same as those mentioned for reinforced
concrete columns due to a lack of information.
Determining Transverse Steel Content
An assessment o f th$ quantity Qf special transverse steel required in columns for seis­
mic design should take into account:
1. Required curvature ductility factor, TuZyy
2 . Column axial load level, P/Pb
3. Rate o f loading, e
4. Longitudinal steel content, p
5. Effective prestress, ese
6 . Stress-strain curve o f concrete
7. Stress-strain curve o f steel
A theoretical analysis requires a stress-strain Curve for concrete confined by transverse
reinforcement and a stress-strain curve for steel which includes strain hardening, Ideally,
the effects o f cyclic loading should be considered, but the increase in expense and com­
plexity of analysis reduces its usefulness. The complexity o f the analysis would increase for
prestressed concrete column because o f the scarcity o f experimental results.
14
Park and Sampsdn [39] have been advocating the use o f monotonic mOment-curvature
analysis to assess the expected ductility of columns subjected to earthquake loading. In
order to indicate the acceptability of this practice, the behavior o f concrete and steel
materials under cyclic loading must be considered.
Sinha, Gerstle, and Tulin [ 53 ] were the first investigators to indicate that the mono­
tonic stress strain curve for unconfined concrete is an envelope curve for the cyclic stress
strain characteristics of unconfined concrete. Figure 5 shows this result. The investigators
[53] showed that thi§ was also true for idealized behavior of the reinforcing (see Figure 6).
Karsan and Jirsa [25] confirmed the results of Sirha, Gerstle, and Tulin for unconfined
concrete.
Experimental results for an unconfined concrete section, concrete and steel acting
together, indicated that the monotonic moment-curvature is an. envelope curve for cyclic
moment-curvature curves [35,37].
Park and Sampson [39] were the first to assume that the behavior o f confined con-- .
Crete would follow the same general characteristics of unconfined concrete. They assumed
that the monotonic moment curvature curve was an envelope curve for cyclic momentcurvature curves for confined concrete. Just recently, investigators from the University o f
New Zealand have proven this behavior, for both square-confined and spirally-confined
reinforced concrete columns [38,43].
Some work [50] has been performed to indicate that the monotonic stress-strain
curve o f confined concrete at high strain rates will also be an envelope for high strain rate
cyclic loading. These results have yet to be totally confirmed.
There appears to be enough evidence [8,35] to suggest that the monotonic moment
curvature curve for prestressed concrete will be an envelope curve for cycling momentcurvature curves o f prestressed concrete columns. Therefore, the assessment o f maximum
15
Figure 5. Envelope curve for concrete [53].
Figure 6 . Envelope curve for steel [53].
16
curvature ductility for prestressed concrete column sections subjected to earthquake will
be based on the same practice as reinforced concrete [37] monotonic loading.
Derivation of Moment-Curvature Curves
Assumptions
For the analysis o f monient-curvature characteristics o f reinforce^ and prestrpssed
concrete column sections under monotonic loading, the following assumptions will be
made:
1. Plane sections before bending remain plane aftqr bending.
2. The stress-strain curve for nonpregtressed and prestressed reinforcing is known,
3. The tensile strength o f concrete is ignored.
4. The stress-strain qurve for biaxially and triaxially confined concrete in compres­
sion is known.
5. The stress-strain curve for unconfined concrete cover follows the stress-strain
curve for confined concrete at strains less than 0.004. For concrete strains greater
than 0.004, all the concrete cover is assumed to have spalled.
6 . Buckling of longitudinal steel in compression is assumed to be prevented by the
close spacing o f transverse reinforcement.
7. Short-term loading only is considered.
8- The prestressing steel is initially bonded to the concrete.
Analysis of Sections
Theoretical moment-curvature curves for reinforced and prestfessefl concrete sections
under flexure and axial load can be derived on the basis of the assumptions presented
earlier. The derivation here was originally presented by Park and Paulay [37]. The curva­
tures associated with bending moment and axial load are determined using the presented
17
assumptions and requirements o f strain compatibility and equilibrium o f forces. In this
analysis only bending about the principal axes will be considered. For a given concrete
strain in the extreme compression fiber, ecm, and neutral axis depth, c, the steel strains
esl >es2> es3>• • ■eSn can
determined from the strain diagram.
The steel stresses fg l , fs2, fs3, . . ., fsn corresponding to strains es l , es2, e ^ , , , . esn
can be found from the stress-strain curve for steel. The effective strain in the stepl due to
prestressing is added to the strains imposed by the loading. Then the steel forces S1, S? ,
S 3 , . . . , S can be determined from the steel stress and reinforcing areas,
'I
The distribution o f compressive concrete stress on a section can be determined from
the strain diagram and stress-strain curve for concrete. For any given concrete strain
,
the concrete compressive force C„ and its position can be defined by parameters a and 7,
Y
where
Cc = af'c be
( 6)
acting at g. distance 7 c from the extreme compression fiber- The mean stress factor a and
the centroid factor 7 (see Figure 7b) for any strain ecm can be determined for rectangular
sections from stress-strain relationships:
/0
fcdec
(7)
Jq ecfcdec
( 8)
-V
ecm/,
The force equilibrium equations can be written as (see Figure 7c)
n
P = osf'c be + Z fsi Agi
i=l
(9)
18
Stress/, A
Strain c,
Strain ec
—
Neutral axis
•
Elevation
Neutral
#3
Section
Strain
Stress
Internal
forces
External
actions
(cl
Figure 7. Theoretical moment-curvature determination, (a) Steel in tension and compres­
sion. (b) Concrete in compression, (c) Section with strain, stress, and force dis­
tribution [58].
19
h
n
M = a f c b c ( - - 7c) + 2
1
i=l
h
fsi Asi ( - - dp
^
( 10)
where
a = mpan stress factor
7 = centroid factor
f'c - concrete stress
b = width o f concrete section
c = neutral axis depth
fgi = steel stress o f bar i
Asi = steel area o f bar i
h = depth of concrete section
di = depth to centroid o f reinforcing i from extreme fiber in compression
The Ci^Yftthre can then be fqiiftd a@
i pm
( H)
The theoretical moment curvature relationship for a given axial load may be deter­
mined by incrementing the concrete strain, ecm. For e^ h value o f e
, the neutral axis
depth, c, that satisfies force equilibriuih is found by iteration using the secant m ethod The
internal forces and neutral axis depth which satisfy equilibrium are then used to determine
the mqment, M, and curvature, 0 . By carrying out the calculation for a range o f ecm values,
the moment-curvature can be plotted.
A computer program was developed to calculate the moment-curyature response pf
square rectangular reinforced and prestressed concrete sections under monotonic logfling.
The program is described in Appendix B.
20
Models
Stress-Strain Curve for Confined Concrete. Because transverse reinforcement provides
passive confinement it would seem reasonable to expect confined and unconfined stressstrain curves to be approximately the same until the transverse reinforcement becomes
effective.
Chan [16], Blume et al. [9], Baker and Amarakone [3], Roy and Sozen [48],
Soliman and Yu [54], Sargin et al. [49], and Kent and Park [26] have proposed stressstrain relationships for concrete confined by rectangular hoops. On the basis of existing
experimental evidence, Kent and Park [26] proposed the stress-strain curve in Figure 8 for
concrete confined by rectangular hoops. At the present state o f the art, Kent and Park’s
stress-strain curve is the most acceptable.
A discussion o f the assumptions made in the derivation o f Kent and Park’s curve will
follow the analytical description o f the curve. As shown in the figure, there are three
regions:
Region AB: ec < 0.002
fC = f 'c ■
where
f
( 12)
= concrete stress, psi
f ' - concrete cylinder strength, psi
ec = concrete strain, in/in
Region BC:
0.002 < ec < e20c
fc = f'c '[I - Z (ec - 0 .002)]
where
(13)
(14)
ton 9
Confined
concrete
inconfined concrete
Figure 8. Stress-strain relations for concrete [58].
22
3 + Q.P02 f 'c
eSOu
f'c - 1000
^SOh
Ps
(15)
( 16)
= ratip o f volume o f transverse reinforcement to volume o f epnprete core
measured fo putside o f hoops
b" = width o f confined core measured tp outside o f hoops
Sll - spacing o f hopps
Region CD:
G20c < ec
fc = 0-2 f 'c.
(17)
where
G2Qcf
concrete strain at 0.2 o f maximum stress on descending branch pf stressstrain curve.
Region AB
The expression fqr the ascending part o f the purve follows the gpnprally acppptpO
secpnd degree parabola fpr unconfined concrete and assumes that the transverse reinforce­
ment has no effect on the shape o f this part of the curve or the strain at rqaximum stress.
Investigations [6,52] indicate that transverse reinforcement may increase the maximum
concrete compressive stress peyond f 'c . However, this increase may be small, and in Rov
and Sozen’s tests [4 8 ], no increase in strength was found. Therefore, it is conservatively
assumed that the maximum stress reached by the confined concrete is the cylinder strength,
m
• •
Region BC
The falling branch is assumed to be linear and its slope is specified by (he strain when
the concrete stress has fallen to 50% of the maximum stresses as suggested by Roy and
23
Sozen [46]. The parameter Z defines the slope o f the assumed linear falling branch. Equa­
tion for e50u takes into account the effect o f concrete strength on the slope o f the falling
branch o f unconfined concrete, high strength concrete being more brittle than low-strength
concrete. Equation for C501l gives the additional ductility due to transverse reinforcement
and was derived from the experimental results of three investigations [7,48,54].
Region CD
It assumed that concrete can sustain a stress o f 0.2 f 'c from C20c to infinity. Yamashiro and Siess [63] had previously used the same assumption and Barnard [4] has shown
that concrete can sustain some stress at indefinitely large strains. Roy and Sozen [48]
found that with rectangular hoops as confinement, load deformation curves extended well
beyond concrete strains o f 5%.
Stress-Strain Curve for Unconfined Concrete. For simplicity the stress-strain curve
for unconfined concrete in compression is assumed to follow the curve. for confined
concrete at strains less than 0.004. This assumption is important when considering con­
crete covering transverse reinforcement. At large strains it is likely that the unconfined
concrete outside the hoops (the concrete cover) will spall away. Because the spalling
process occurs gradually, it is difficult to determine the strain at which spalling of the
concrete cover commences. This assumption of the ineffectiveness o f the concrete cover
has also been made by Baker and Amarakone [3 ], at strains greater than 0.0035, and by
Blume [9] at strains greater than 0.004, Some investigators have ignored spalling of
concrete cover at high strains. Kent and Park [26] assumed that at strains greater than
0.004, the concrete cover carries no stress.
In this analysis, once the extreme compression fiber reaches a strain of 0.004, it will
be assumed that all the unconfined concrete (concrete cover) spalls, even the concrete
cover near the neutral axis. This assumption was based on:
24
(a) tpst results [22,23,33,38] indicating that th$ presence o f hi^h quantities of step]
hopps precipitate concrete cover spalling.
(b) that concrete cover will most likely become ineffective under several reversals of
high intensity loading such as woulcf occur if subjected to earthquake loading.
StressrStrain Relationship for Nonprestressed Steel.
The strpssrstrain chamcteristips
o f longitudinal reinforping steel including the effects of strain hardening arp rpquirp^ for
thp determination of the moment-curvature relationship of reinforced concrete members.
The stress-strain curves for longitudinal steel in tension antj compression are assumed fo bp
identical. This is considered to be a reasonable assumptions although strain hardening may
occur at a lower strain in compression steel.
While the stress-strain curve for the longitudinal steel is important, foe same inforr
mgtion for the confining steel does not appear to be very useful. Kpnt and Park [2f?] copld
find no experimental evidence to ipdipate that the strength of foe transverse reinforcement
had an effect on the stress-strain curve of the concrete. This is reflected in thp parameter Z,
where the strength o f the transverse reinforcement is neglected in Equation 14. Therefore,
the stress-strain characteristics o f the transverse reinforcement are pot reqpirpd.
Figure 9 shows the stress-strain curve for steel subjected to monotopic loading which,
will be uspd ip this investigation. It is assumed that foe curve is composed o f a straight line
up tq the yield strain, a straight Iinp representing foe yield plateau to commencement of
' strain hardening, and a curved line beyond strain hardening. Tire curved portiop from com­
mencement at strain hardening to ultimate strain is similar tp an expression proposed jy
Burns and Sipss [15] but modifiep by Kent and Park [26].
Thp stress-strain curve is in three parts as follpws:
Region ABj
es < es < ey
( 18)
I
D
to
LA
Figure 9. Stress-strain relations for non-prestressed steel [5 8 ].
26
where
f - steel stress
es = steel strain '
ey = yield strain
Bs ? modylus o f elasticity
Region BCh
ey < es < esh
V f y
where ' f
(19)
= yield stress
esh “ steel strain at beginning o f strain hardening
Region CD:
esh < es < esu
,
where m
r m < y f* H 2
y L60 (es - eSh) + ?
+
OOrdi I)2 - 60r - I
% u/fy)
' ISr2
r = Psu ^ ^sh
(e. -r e,h) (60-m)
r .
-
,
2 (?Pr + l)?
( 20 ).
(2D
(2?)
Esu - steel strain at ultimate stress
fsu = ultimate steel stress
The parameters for the stress-strain curve were chosen'based on information prpvided
by Burns and Siess [1 5 ], Kent and Park [2 6 ], and Mir%a and MacGregor [3 2 ]. The param­
eters for Grade 60 steel are:
Es = 29,200 ksi
esu = 0. 1$
esh y 0.016
fgu
f
110.8 ksi
= 60 ksi
27
Stress-Strain Relationship for Prestressing Steel. The stress^strain relation for prestressing steel in tension was derived by Blakeley and Park [ 8] and illustrated in Figure IQ.
The stress-stram relation has been used by various investigators [8,58]. The relation com­
prises three regions which are defined by the following equations:
Region AB: eP < ePb
(23)
fP = e P6P
steel strain
in which
=Pb
steel strain at Point B (proportional limit)
steel stress .
modulus of elasticity o f pr$stressing steel
RegionBC:
epb < ep < epc
^pc ep c
^pb e pb
e Pc r e Pb
e pb epc ^ p b
e P (^pc
^p
(24)
ep b )
f= steel strain at Point C
in which
f,Pb =■ stpel stress at Point B
Lpc = steel stress at Point C
Region CD: e pc < ep ^ e pu
(fpu - W
(25)
LePu ePc J
in which
epu = ultimate steel strain
f
= ultimate steel stress
Numerical values for the stresses and strains at Points B, C, and D should be obtained
from experimentally measured stress-strain curves for the prestressing steel used.
The parameters used for Grad 270 in this study follows those determined experimen­
tally by Thompson and Park [58]. They are;
i
D
OO
Figure 10. Stress-strain relations for prestressing steel [58].
fpb = 0.007
fp c = 0 .0 1 0
e PU
= 0.050
Yb =r 189 ksi
= 240 ksi
= 281 ksi
■= 27,000 ksi
30
CHAPTER II
ANALYTICAL STUDY
Introduction
For this analytical study, square reinforced and prestressecf concrete columns were
analyzed. Figure 11 shows the columns properties.
An important consideration in the analytical study is the determination of the maxi­
mum usable concrete strain. Since for the development of moment-curvature curves the
concrete strain is incremented by 0.001, its choice would dictate somewhat the final values
o f curvature ductility. Limited experimental results have suggested that the maximum con­
crete strain is unlimited. However, this assumption is bnsed on a “confined” concrete. If
the hoop steel fractures, the core is no longer confined. Therefore, the question of permis­
sible maximum concrete strain is complicated by the characteristics of the hoop steel,
For this study, the maximum concrete strain is assumed to be 0.052 in/in. The basis
for this choice was (a) limited full scale experimental results measured concrete strains
approaching 0,05 in/in and (b) the strain was chosen large enough to indicate any variation
o f the moment-curvature curves at values of high concrete strains.
To follow the concrete strains on all moment-curvature plots, a symbol was placed %t
every 5th increment of strain beginning at ec = 0.003. Therefore, a symbol will appear on
the plots for concrete strains o f 0.003, 0.008, 0.013, 0.018, 0.023, etc. This should prove
useful in the interpretation o f the curves.
Values of 3000 psi and 5000 psi are used for the strength o f the concrete in rein­
forced and prestressed columns, respectively. These values were used to approximate usual
Pre stressed
fp u
= 270 ksi
fyh =
40 ksi
f'c =
16" X
5 ksi
16"
3 layers of steel
Figure 11. Column properties.
Reinforced
fy = 60 ksi
fyh
f'c
= 40 ksi
=
16" X
3 ksi
16"
2 layers of steel
32
building practices. For the reinforcing steel, the assumptions are as follows: Grade 60 for
non-prestressed longitudinal steel, Grade 40 for ndn-prestressed hoop steel and Grade 270
for prestressing strand. These also were considered practical choices.
The moment-curvature curves are plotted in non-dimensional form. The column size
for both reinforced and prestressed columns is 16 in. X 16 in. This is a common building
size. The choice o f the amount o f steel, 3% for the reinforced concrete column, was chosen
for two reasons. First, it satisfied ACI 318-77 and ACI 318-83 steel requirements and
second, it provided a strength capacity approximately equal to a prestressed concrete col­
umn with S-1A "0 Grade 270 strands. This should enable certain comparisons to be made
between the two columns.
The balanced load for each columns is approximately 0.35 f'c Ag. The axial loads
considered in the analyses are 0.10 f'c Ag, 0.20 f 'c Ag, and 0.30 f'c Ag. This represents the
tension control region of the moment axial load interaction diagram. The value of 0.10 f 'c
Ag corresponds to 0.4 0 Pb, the value ACI 318-77 and ACI 318-83 indicates as the sepa­
ration between beam and column behavior.
The following variables were considered for the reinforced concrete column analysis:
confinement, strain hardening of non-prestressed longitudinal steel, longitudinal steel con­
tent, and concrete cover. For prestressed concrete columns variable investigated were: con­
finement, axial load, bond compatibility factor, concrete cover, and percentage of nonprestressed steel.
The analysis for the columns is presented in terms of curvature ductilities. The selec­
tion o f 0y , the curvature when the steel yields, is dependent on the choice of ey for the
prestressing steel. Unlike non-prestressed steel, prestressing steel has no well defined yield
point. ASTM suggests a yield point for prestressing steel. In this study, the ASTM recom­
mendation was used, i.e., 0y = 0 .0 1 , but in no case was the yield point o f the steel con­
sidered to have occurred at a concrete strain greater than 0.003. This was done to-conform
33
to recommended practice [39] and also to facilitate comparison o f reinforced and pre­
stressed moment-curvature curves.
For the prestressed concrete columns, the initial strain in the prestfessing strand is
considered to be the result of an initial prestress force of 0.70 f
X Apg. The concrete
strain, ece, is considered to be the resulting prestress force divided by the gross concrete
column area. However, after the concrete cover spalls, the concrete strain due to prestress
is increased due to the resulting smaller concrete core.
For each figure showing the analysis results, all parameters are indicated in the cap­
tions. These computer generated plots (Figures 14 through 32) are located in Appendix A.
Confinement
The effect of confining hoop steel on the moment and curvature ductility is shown in
Figure 14. The confinement, Z, is shown in Table I below with the hoop steel correspond­
ing to each Z.
Table I. Relationship Between Z and Hoop Steel.
Z
6
12
18
24
30
36
Hoop Steel
1
#
#
#
#
#
#
4
4
4
3
3
@ 5 % " ,# 3 @ 4"
@ 9 " , # 3 @ 6"
@ 1 2 " , # 3 @ 8%"
@ 10%"
@ 13"
3 @ 15"
The axial load considered is 0.20 f 'c Ag and the longitudinal steel content is 3% (dis­
tributed equally to the top and bottom). The concrete strength is 3000 psi.
At a concrete strain of 0.052, the curvature ductility for Z = 6 is 32.6 and for Z f 36
is 25.7. These curvature ductilities woud usually be considered adequate for seismic load­
ing, according to Park and Paulay [37] and Park and Sampson [39].
34
The loss of moment capacity is disturbing. Just after the concrete spalls, the section
loses approximately 13% o f its moment capacity, regardless o f the confining steel. Investi­
gators [35,37,39] have proposed that a useful curvature ductility criteria be defined at a
moment capacity of not less than 0.85 o f the maximum moment capacity. If this criteria
were adopted, an examination of Figure 14 indicates that the confinement represented by
a Z o f 6 and 12 would be acceptable. Other levels o f confinement do not adequately main­
tain moment capacity at high ratios o f curvature ductility.
ACI 318-77, Appendix A, requires columns to be confined at a Z level equal to 12,
Figure 14 indicates this to be a prudent requirement for the proposed moment capacity
reduction criteria.
Figure 15 represents 3 M - <t> curves of Figure 14. It shows moment-curvature ductil­
ity curves for Z values of 6, 12, and 18. Figures 16 and 17 show curves for Z values of 6 ,
12, and 18 for axial loads of 0.10 f'c Ag and 0.30 f ^ Ag, respectively.
The effect of steel strain hardening is shown in Figures 18, 19, and 20, A point C
represents the start o f strain hardening of the compression steel and a point T represents
the start of the tension steel strain hardening.
Figure 16 indicates the inadequacy of the ACI 318-77 provisions. For low axial load
levels, the required confinement could be reduced. The curves of Figure 16 incJiQate a Z
o f 6, 12, and 18 to be satisfactory in maintaining moment capacity, The initial reduction
in moment, due to concrete spalling, is approximately 8%. The moment capacity continues
to increase after the first initial moment reduction. Moment-curvature ductility curves
shown in Figure 17 represent an axial load o f 0.30 f'c Ag and Z levels o f 6, 12 and 18.
After concrete spalling occurs, there is approximately a 23% reduction in moment capacity.
Regardless of Z, the moment reduction continues until thb onset o f steel strain hardening.
35
While the curvature ductility appears to be adequate, the large reduction in moment capa­
city due to concrete spalling could be disastrous during an earthquake, It indipates a
anomaly with the ACI 318-77 provisions. At large axial loads, the confinement leyel has
no effect on satisfying the minimum moment capacity criteria.
For the column section analyzed, an axial load o f 0.30 f'c Ag represents approxi­
mately the balanced load (Pb = 0.35 f'c Ag) and an axial load of 0.10 f'e Ag represents
approximately 0.4 0 Pb. ACI uses 0.4 ^Pfe as the dividing line between beam and column
behavior, The results shown by Figures 14, 15, 16, and 17 indicate the ACI provisions to
be adequate for columns with axial loads below 0.4 0Pb. However, above this level of axial
load, the ACI provisions would be considered inadequate based on current criteria. The
results indicated by this study wpuld require the level of confinement to be increased as
the axial load increases. The large reduction in moment capacity due to concrete spalling,
at axial loads Approaching Pb, is of great ppncem. The effect o f confinement is of no value
in counteracting this reduction. The model used for the concrete o f course assumes al} tjie
concrete to spall when the extreme concrete fiber strain reaches 0.p04. A limited amount
of work [38,43,58] indicates that the initial reduction in moment capacity will occur
more gradually than indicated by the curves. However, the final moment reduction will be
close to the predicted moment capacity, regardless of the gradual spalling.
Non-Prestressed Steel Strain Hardening
From previous results o f moment vs. curvature ductility curves, it is obvious that steel
strain hardening increases the moment capacity of the column sections. It was felt neces­
sary to investigate the beneficial effect of steel strain hardening since ^.STM standards have
no requirements for the initiation of steel strain hardening.
Strain hardening o f reinforcing steel may begin at any plastic strain, e§h. For this
study, strain hardening was assumed to begin at esh f Ip ey . Figures 18, 19, and 20
36
represent an egh o f °° (no strain hardening), 16 ey , and 8 ey for axial load levels of 0,10 f'c
Ag, 0.20 f 'c Ag, and 0.30 f 'c Ag. The concrete strength is 3000 psi apd Z = 12.
Figure 18 indicates that early onset o f strain hardening (egh = 8 ey) may help in
, reducing the initial moment capacity drop. It also indicates the largest available moment
capacity of the section. An esh = 16 ey represents the middle curve on all three figures. For
an egh = °°, it provides the lowest possible moment capacity. Figure 18 indicates that at
low axial load levels (0.10 f'c Ag) the effect o f steel strain hardening is not required in
order to meet proposed moment capacity criteria.
However, Figure 19 shows that if steel strain hardening is nqt included in the analysis,
the moment capacity will continue to decrease after concrete spalling due to the continued
yielding of the steel. At higher axial !pads, the occurrence of early steel strain hardening is
shown to produce a rapid moment increase. Hoxyever, if strain hardening is neglectpd, the
moment capupity rapidly (lecreases.
Bepause ASTM does not standardize e?h or the required shape of the strain bardening
region, it would appear wise and also conservative to neglect strain hardening when devein
oping general design requirements. Neglecting strain hardening will place a greater emphasis
on confining steel to achieve stable moment capacity at high curvature ductility ratios.
Longitudinal Steel Content
The steel content considered is 1%, 3%, and 5%. The values o f 1% and 5% represent
the minimum and maximum steel ratios permitted for earthquake design by the ACI Code.
The axial load levels are 0.1 Q f fc Ag, 0.20 f ^ Ag and 0.30 f'c Ag, Thp cpncrpte strength is
3000 psi and Z - 12.
Figure 2 1 in creates that the effect of an axial load decreases the curvature ductility
as the longitudinal steel content decreases. Table 2 gives the results o f maximum moment
reduction and curvature ductility obtained for different axial load and steel content.
37
Table 2. Moment Reduction and Curvature Ductility.
Steel Content
'
3%
1%
5%
Axial Load
Max
Moment
Reduction
Curva­
ture
Ductility
Max
Moment
Reduction
0.10 f'c Ag
19
59
7
43
4
36
0.20 f'c Ag
36
32
15
30
10
29
0.30 f'c Ag
76
24
31
24
20
24
Curva­
Max
ture
Moment
Ductility Reduction
Curva­
ture
Ductility
The results o f Table 2 indicate the following:
1. As axial load increases, curvature ductility decreases. However, the higher the
longitudinal steel content, the smaller the ductility decrease.
2. As axial load increases, the maximum reduction in moment capacity increases.
The increase in moment reduction is less for higher steel ratios.
3. For a given axial load, the lower the steel ratio, the higher the curvature ductility.
However, this behavior is only really evident for columns with an axial load less
than 0.4 0Pb (beam behavior).
Confinement of longitudinal reinforcing for a constant axial load has the same
approximate beneficial result regardless o f longitudinal steel content. Figure 22 shows this
to be true. Some investigators [52,54] have found that higher steel contents tend to assist .
the hoop steel in confining the inner concrete core. Due to the limited tests on the variety
o f positions the reinforcing may be placed, this refinement appears to be unwarranted.
Concrete Cover
The effect pf concrete cover (cc = 1.5, 2.5, 3.5) is considered for axial load levels of
0.10 f'c Ag, 0.20 f'c Ag, 0.30 f'c Ag. It was no surprise that as the concrete cover increased,
the amount of initial moment reduction increased. It is interesting to note that Figures 23,
38
24, and 25 show that as the axial load increases, the moment reduction increases for a con­
stant value o f concrete cover. This would indicate that statistical variations in concrete
cover (workmanship) would be more important for heavily loaded columns.
Prestressed Concrete
•
Moment-curvature curves were developed for prestressed concrete columns for the
following variables: confinement, axial load, bond compatibility factor, concrete cover,
and non-prestressed reinforcing.
Confinement
For prestressed concrete columns, the effect o f confinement is illustrated in Figure
26. The column was analyzed for an axial load of 0.20 f' Ag, and f' = 5000 psi and a
concrete cover of 2.5 inches. It is obvious that an increasing confinement (increasing Z)
provides a more stable M - $ curve at higher curvatures. For an ACI specified confinement
Z = 12, the moment capacity remains relatively constant from curvature ductilities of 8 to
20.
In all curves, the moment capacity is increasing after the concrete cover spalls. This is
due to the shape of the stress strain curve for the prestressing steel. For the prestressing
steel used, the stress continues to increase for any increasing values o f strain.
At an extreme compression fiber strain o f approximately 0.018, for all curves, the
moment capacity begins to decrease. After investigating the changing steel strains, neutral
axis locations and compression zone characteristics (a and 7 ), the cause o f the decrease is
due basically to the compression prestressing strand reaching its yield point. When it
reaches its yield point, it increases in stress, but at a very slow rate. The total compressive
force (Cc + Cs) drops in magnitude due to the decreasing mean stress factor (a) and little
increasing compression steel stress. In order for the total tension force (T1 + T2) to be
39
equal to the total compression force, the distance from the extreme fiber in compression
to the neutral axis must increase. This process effectively diminishes the internal level arm,
and therefore decreases the moment capacity since Cs and T remain relatively unchanged.
Perhaps the most obvious and disturbing feature o f Figure 26 is the large drop in
moment capacity after the concrete spalls (at ec = 0.004). Generally speaking, for any level
o f confinement, the reduction in moment capacity after spalling is approximately 50%.
This must be considered very poor behavior o f a column subjected to earthquake loading.
The confinement is of little concern, since after spalling just about any level of confine­
ment produces an increasing moment capacity.
' Axial Load
The effect of axial load on the M - 0 curve is interesting. As expected', since the axial
loads considered are less than the balanced load, an increase in axial load should increase
the moment capacity. This is indeed true as shown in Figure 27. Table 3 shows some
important information obtained from Figure 27 and additional computer printouts.
Table 3. Moment Reduction and Curvature Ductility.
Axial Load
Reduction in Moment
Capacity, %
Curvature
Ductility
0.10 f'c Ag
39
12
0.20 f'c Ag
51
22
0.30 f'c Ag
65
23
For the lowest axial load, 0.10 f'c Ag, the prestressing strand fractured at a concrete
strain of 0.023. The curvature ductility was therefore very low. It is interesting to note
that an increase in axial load increases the curvature ductility. This is somewhat different
than the results of curvature ductility o f reinforced concrete columns. The increase in axial
40
load permits a lower tension strain at each concrete strain considered. This is due to the
lowering o f the neutral axis to achieve internal force equilibrium.
The percent of initial moment capacity reduction increases as axia] load increases. In
all three cases, the moment reduction is Still greater than current acceptable criteria.
Bond Compatibility Factor
It is well known that the outer shell of concrete columns will spall at a concrete strain
of approximately 0.004. For a prestressed member, local spalling may leave the prestress-ing strand exposed. The bond or lack of bond between the strand and the surrounding con­
crete will invalidate the use o f a strain diagram. If the strand is considered to have no bond
at various spots or locations along the column, it can be treated in the same manner as an
unbonded prestressed member. A bond compatibility factor (F) was, used to relate con­
crete strains to steel strains, eg = Fe,c. The bond compatibility factor was varied to indicate
any problems associated with loss of bond due to concrete shell spalling.
Figure 28 indicates M - 0 curves for bond compatibility factors of 1.0, 0.9, 0.8, 0.7,
0.$, and 0.5. The maximum curvature ductility was approximately the same for all curves,
The only major difference was for “perfect bond” where the strand fractured.
Between a curvature ductility of 8 to 16, the moment capacity begins to decrease.
This again, as explained in the previous section, is due to the prestressing steel reaching its
compression yield strain. As the bond compatibility factor decreases, the steel strain
increases but at a decreasing rate.
The bond compatibility factor was introduced analytically info the problem as soon
as the concrete spalled. Tne major difficulty is still the 50% reduption in moment capapity
upon onset o f concrete spalling.
41
Concrete Cover
The effect of the amount of concrete cover specified for a column was studied. This
effect was primarily studied after other figures indicated the large reduction in moment
capacity after the outer concrete shell has spalled.
The following amount of concrete cover was considered: 0.5, 1,5, 2.0, and 2.5 inches.
This is shown in Figure 29. It is easily observed that a decrease in concrete cover, i.e., a
decrease in the amount o f spalled concrete area, will decrease the reduction in moment
capacity at ec = 0.004. Table 4 indicates the decrease in morpent capacity at 0.004 for
each value of concrete cover.
Table 4. Moment Reduction.
Concrete Cover
inches
Reduction in Moment
Capacity, %
0.5
1.5
2.0
2.5
19
36
43
51
ACI 3 1S--VV and 318-83 recommend a clear cover value for precast prestressed con­
crete of 1.5 inches. However, even if a cpver of 1.5 inches is used, the moment capacity
reduction would be more than considered acceptable.
Note that for the concrete cover cases studied, as the cover increased, the curvature
ductility increased.
Improving Curvature Ductility of Prestressed Columns
It has always been interesting to encounter technical literature arguing the prqs and
cons of improving the curvature ductility of a member by either adding non-prestressed
steel or confining hoop steel. Both definitely would improve the ductility but to what
degree was uncertain. Figure 30 shows M - $ curves in which hoop steel has been increased
or non-prestressed reinforcing was added.
I
42
The basic case for this example was considered to be a prestressed concrete column
with light hoop reinforcing, Le., that satisfying Chapter '7 of the ACI Code (Z = 76). The
decision is made to upgrade the ductility of the column for earthquake loading. The two
choices are to
I
(a) increase the hoop steel to conform to Appendix A of the ACI Code [10] (i.e.,
Z = 12) or
(b) to add non-prestressed steel to improve the ductility.
Figure 3 1 indicates that the obvious choice is to add more confining steel rather than
non-prestressed steel. This is chosen assuming column strength to be satisfactory. The
moment capacity continues to decrease for all columns that had nominal hoop reinforcing
and bar reinforcing. The moment capacity remains fairly constant for all columns in which
extra hoop steel was added. In addition, the curvature ductility is greater for the column
with the extra hoop steel.
An increase of both extra hoops and bar reinforcing will improve the amount of
moment capacity reduction after concrete spalls. Table 5 shows the possible beneficial
effort o f increasing both the hoop steel and bar reinforcing.
Adding non-prestresSed steel provides no benefit in improving curvature ductility and
very little in improving the initial reduction in moment capacity.
Table 5. Column Characteristics.
Confinement
(Z)
Non-prestressed
■ Steel
Curvature
Ductility
Reduction in Moment
Capacity, %
12
12
12
_
2 -# 4
4 -# 4
20
20
20
44
42
40
43
CHAPTER III
ENGINEERING EXAMPLES
To indicate the usefulness o f the computer program developed, two practical design
examples were investigated. The first is a reinforced concrete column in the Life Sciences
Building (Johnson Hall) on the campus o f Montana State University. The second, a standard
PCI prestressed concrete pile is investigated using AASHTO guidelines for confining steel.
The Life Sciences Building is a ten story reinforced concrete structure designed for
Seismic Zone 3. The column chosen for investigation, D-6, is a 20" X 20" interior column.
The column was symmetrically reinforced with 16 -# 11 bars and confined with 3 overlap­
ping # 4 bars at 4" o.c. for the first 28" from the beam and then thereon with 3 overlap­
ping #3 bars at 18" o.c. The engineer has provided greater confinement at expected loca­
tions of plastic hinging. The longitudinal reinforcing is Grade 60 and confining steel Grade
40- The required concrete strength, f'c, is 3750 psi. The dead and live load levels from each
floor are available from the drawings. The column dead load (first floor) is estimated to be
400 kips. A portion of the live load should be considered for the column. It was deemed
appropriate to consider 20% o f the live load. The column load considering total dead load
and 20% live load is 500 kips. Using the above mentioned information, a plot of moment
versus curvature ductility was generated. Figure 3 1 shows the curve.
The results o f the analysis indicate that the difference in axial loads, 400 kips versus
500 kips, only slightly affects the M - 0 curve fqr the confinement o f 3 - # 4 ’s at 4" o,c.
For a light confined column, the axial load affects the M - 0 curve to a more significant
extent.
44
The M - 0 for the column section confined with 3 - # 4 ’s at 4" o.c. (Z = 24) exhibits
acceptable behavior. That is, low initial moment reduction and stable or increasing
moment capacity at high levels o f curvatures ductility. For the column section confined
with 3 - # 3 ’s at 18" o.c. (Z - 130), the initial moment reduction would be considered
unacceptable.
The design of column D- 6 appears to be appropriate. The key to the design is the
assumption o f plastic hinge length, As long as plastic hinging occurs in the heavily con­
fined region, the column behaves satisfactorily. If, however, plastic hinging can occur in
the lightly confined region, the column would be considered inadequate. This example
illustrates the necessity o f an accurate concrete section will add little tp the overall dis­
placement ductility o f the structure if hinging occurs at other locations.
The PCI Design Handbook [40] was used as a reference for a typical prestressed pile.
A 12" X 12" prestressed concrete pile with 8 - 1A"] Grade 270 strands is considered. The
required concrete strength is 5000 psi. The pile is considered to be subjected to axial load
levels of 0.10 f'c Ag and 0.30 f'c Ag. Hoop reinforcement was that suggested by AASHTO
[20,40]. It was assumed that for a bridge foundation, the standard PCI piles are used, but
would be required to satisfy the confining hoop requirements of AASHTO. Two levels of
confinement are considered. A Z = 44 satisfying AASHTO, Standard Specifications for
Highway Bridges, 12th Ed., 1977 and a Z - 3 satisfying Guide Specifications for Seismic
Design o f Highway Bridges, 1983 (AASHTO), are considered for the M - 0 analysis of the
Pile.
The M - 0 curves (Figure 32) for the pile indicate typical results. A high level of con­
finement provides for a smaller decrease in initial moment capacity and a greater moment
at high curvature ductilities. The most evident response showed by the M - 0 curves is the
45
almost complete loss of moment capacity o f the lightly confined pile. This moment loss
has to be considered disastrous. A pile will be very difficult to repair or replace. Prestressed
concrete piles should be considered with extreme caution, if they are likely to be subjected
to dynamic loads during their service life.
46
CHAPTER IV
CONCLUSIONS AND RECOMMENDATIONS
From the results of the analysis, certain general and specific conclusions can be drawn
regarding thp curvature ductility o f reinforced and prestressed concrete columns.
In general, certain shape characteristics o f the moment-curvature curves were evident.
Figure 12 represents the general shape o f a moment curvature curve for reinforced con­
crete column and Figure 13 for a prestressed concrete column.
The most obvious characteristics of the M - 4> curve for reinforced concrete are the
initial reduction in moment capacity at the onset o f concrete spalling, the first increase in
moment capacity due to strain hardening of the tension steel, and then later from the com­
pression steel, and th e;final curvature ductility values controlled by the choice of maptiI
mum concrete strain. \
For the M - 0 curve illustrated in Figure 13, the observable characteristics include a
large initial reduction in moment capacity, the increase in moment capacity because of the
steel performing in the elastic range, the decreasing moment capacity when the steel strains
reach yield, and the final curvature ductility value being dependent on the fracture strain
o f the prestressing steel,
The confinement level tends to affect the increase or decrease in moment capacity
while the choice of maximum concrete strain or steel fracture strain control the final
value of curvature ductility.
Confining steel proves to be more beneficial for reinforced concrete columns than pre­
stressed concrete columns. This benefit was because of the much greater reduction in
moment capacity due to spalling of prestressed concrete columns. The confinement level
47
MOMENT
0 .0 0 3
concrete
SPALLING
STRAIN
CONTROLS-
I
CURVATU RE
12. General M-0 curve for reinforced concrete columns.
MOMENT
0.003
CURVATURE
Figure 13. General M
curve for prestressed concrete columns.
48
specified by ACI 318-77 [1 0 ], Z = 12, is adequate for low levels o f axial load but not for
higher levels o f axial load. This conclusion applies to reinforced concrete columns.
The beginning of steel strain hardening is beneficial in increasing the moment capacity
o f reinforced concrete columns. The beginning o f strand yielding is the start o f a decreas­
ing moment capacity for prestressed concrete columns.
The effect of axial load in changing the initial moment capacity is the same for both
reinforced and prestressed concrete columns. However, its effect on curvature ductility is
different. For reinforced concrete columns, a lower level of axial load increases curvature
ductility, while a low level o f axial load decreases the curvature ductility (strand fractures)
for prestressed concrete columns. This effect on prestressed concrete columns could be
because o f the assumed value of strand fracture.
Concrete cover affects the amount of initial moment capacity reduction in both rein­
forced and prestressed Concrete columns.
Local loss of bond—concrete to strand—appeared to have little effect on performance
o f a prestressed concrete column. A complete bond loss, especially near end anchorages,
will cause a severe reduction in capacity and most likely column failure.
The most effective way to improve stable moment capacity at high curvatures and
greater final curvature ductilities for prestressed concrete columns is by increasing the
level o f confinement.
For the proposed criteria o f acceptable curvature ductility, i.e., permissible moment
capacity reduction o f approximately 15% and a curvature ductility greater than 16, a rein­
forced concrete column could meet the criteria, a prestressed concrete column could not.
Generally speaking, the prestressed concrete columns could not satisfy the level of permis­
sible moment capacity reduction. Also, depending on the stress-strain curve of the pre­
stressing strand, curvature ductilities greater than 16 may not be reached.
Based on this analytical study, the following recommendations are presented.
49
Reinforced Concrete Columns
1. The required level o f confinement varies with axial load.
2. Strain. hardening not be considered in analysis which is used to develop criteria
for building codes. This recommendation could change if ASTM were to consider
manufacturing requirements which include the beginning of strain hardening.
3. Proposed criteria for confinement include statistically permitted variations in
concrete cover. ,
.For Prestressed Concrete Columns
1. The use o f prestressed concrete columns be considered unacceptable for regions
associated with earthquakes, until experimental tests verify otherwise.
2. That engineers (or A STM) consider requiring a specified value of steel strain frac■ture for prestressing steel used in prestressed concrete buildings subjected to earth­
quakes.
3. For ductility, the level of hoop steel should be increased rather than adding nonprestressed steel.
50
REFERENCES
51
REFERENCES
1. Agrawal, G. L., Tulin, L. G., and Gerstle, K. H., “Response o f Doubly Reinforced
Concrete Beams to Cyclic Loading,” Journal o f the American Concrete Institute, Vol1
62, No. 7, July 1965, pp. 823-826.
2. Aoyama, H., “Moment-Curvature Characteristics of Reinforced Concrete Members
Subjected to Axial Load and Reversal of Bending,” Proceedings o f International Sym­
posium on the Flexural Mechanics of Reinforced Concrete ASCE-ACI, Miami, Novem­
ber 1964, pp. 183-212.
3. Baker, A. L. L. and Amarakone, A. M. N., “Inelastic Hyperstatic Frame Analysis,”
Proceedings o f the International Symposium on the Flexural Mechanics of Reinforced
Concrete, ASCE-ACI, Miami, November 1964, pp. 85-142.
4. Barnard, P. R., “Researches into the Complete Stress-Strain Curve for Concrete,”
Magazine o f Concrete Research, VoL 16, No. 49, December 1964, pp. 203-210.
5. Ben-Zui, E., Muller, G., and Rosenthal, J., “Effects o f Active Triaxial Stress on the
Strength o f Concrete Elements,” Proceedings o f the International Symposium on
Flexural Mechanics o f Reinforced Concrete, ASCE-ACI, Miami, November 1964, pp.
193-234.
6. Bertero, V. V. and Vallenas, J., “Confined Conrete: Research and Development
Needs,” Workshop on Earthquake-Resistant Reinforced Concrete Building Construc­
tion (ERCBC), University o f California, Berkeley, July 11-15, 1977, pp. 594-610.
7. Bertero, V. V. and Felippa, C., Discussion o f “Ductility of Concrete,” by Roy, H. E.
H. and Sozen, M. A., Proceedings o f the International Symposium on Flexural Me­
chanics of Reinforced Concrete, ASCE-ACI, Miami, November 1964, pp. 227-234.
8. Blakeley, R. W. G. and Park, R., “Prestressed Concrete Sections with Cyclic Flexure,”
Journal of. the Structural Division, ST8, August 1973, pp. 1717-1742.
9. Blume, J. A., Newmark, N. M., and Corning, L. H., “Design o f Multistory Reinforced
Concrete Buildings for Earthquake Motions,” Portland Cement Association, Chicago,
1961,318pp .
10. “Building Code Requirements for Reinforced Concrete (ACI 318-77),” American
Concrete Institute, Detroit, Michigan, 1977.
11. “Building Code Requirements for Reinforced Concrete (ACI 318-83),” American
Concrete Institute, Detroit, Michigan, 1977.
12. Burden, R. L., Faires, J. D., and Reynolds, A. C., Numerical Analysis, Prindle, Weber
& Schmidt, Boston, Massachusetts, 1978.
52
13. Burdette, E. G. and Hilsdorf, H. K., “Behavior o f Laterally Reinforced Concrete Col­
umns,” Journal of the Structural Division, ASCE, February 1971, ST2, pp. 587-602.
14. Burns, N. H., “Moment Curvature Relationships for Partially Prestressed Concrete
Beams,” Prestressed Concrete Institute Journal, February 1964, pp. 52-63.
15. Burns, N. H. and Seiss, C. P., “Load-Deformation Characteristics o f Beam-Column
Connections in Reinforced Concrete,” Civil Engineering Studies, Structural Research
Series, No. 234, University o f Illinois, January 1962, 261 pp.
16. Chan, W. L., “The Ultimate Strength and Deformation o f Plastic Hinges in Reinforcpd
Concrete Frameworks,” Magazine o f Concrete Research, Vol. 7, No. 21, November
1955, pp. 121-132.
17. Corley, W. G., “Rotational Capacity of Reinforced Concrete Beams,” Journal of the
Structural Division, ASCE, Vol. 92, ST5, October 1966, pp. 121-146.
18. Despeyroux, J., “On Use of Prestressed Concrete in Earthquake Resistant Design,”
Proceedings o f the 3rd World Conference on Earthquake Engineering, Volume III,
1965. [Discussion by Candy.]
19. Fintel, M. and Khan, F. R., “Shock-Absorbing Soft Story Concept for Multistory
Earthquake Structures,” Journal o f the American Concrete Institute, No, 5, May
1969, pp. 381-390.
20.
“Guide Specifications for Seismic Design o f Highway Bridges,” 1983, American Asso­
ciation of State Highway and Transportation Officials, Washington, D.C.
21. Hsu, T. T. C., Slate, F. 0 . Sturm an, G. M., and Winter, G., “Microcracking of Plain
Concrete and the Shape of the Stress-Strain Curve,” Journal o f the American Con­
crete Institute, No. 2, February 1963.
22. Hudson, F. M., “Reinforced Concrete Columns: Effects of Lateral Tie Spacing on
Ultimate Strength,”
23. Iyengar, K. T. R. J., Desay, P., and Reddy, K. N., “Stress-Strain Characteristics of
Concrete Confined in Steel Binders,” Magazine of Concrete Research, Vol. 22, No.
72, September 1970, pp. 173-184.
24. Jennings, P. C. (Editor), Engineering Features o f the San Fernando Earthquake o f
February 9, 1971, Earthquake Engineering Research Laboratory, California Institute
o f Technology, Pasadena, 1971.
25. Karson, I. D. and Jirsa, J. O., “Behavior o f Concrete Under Compressive Loadings,”
Journal o f the Structural Division, ASCE, Vol. 94, ST12, December 1969, pp. 25432563.
26. Kent, D. C. and Park, R., “Flexural Members with Confined Concrete,” Journal qf
the Structural Division, ASCE, ST7, July 1971, pp. 1969-1990.
53
27. Koar, P. H., Fiorato, A. E., Carpenter, J. E., and Corley, W. G., Limiting Strains o f
Concrete Confined by Rectangular Hoops, Portland Cement Association Research and
Development Bulletin R D053.0HD, 1978.
28. Krishnaswamy, K. T., “Strength and Micrpcracking of Plain Concrete Under Triaxial
Compression,” Journal of the American Concrete Institute, No. 10, Octpber 1968.
29. Kupfer, H., Hilsdorf, H. K., and Rusch, H., “Behavior o f Concrete Under Biaxial
Stresses,” Journal o f the American Concrete Institute, No. 8, August 1969, pp. 656666 .
30. Lin, T. Y., “Design of Prestressed Concrete Buildings for Earthquake Resistance,”
Journal o f the Prestressed Concrete Institute, Vol. 9, No. 6, December 1964, pp. 1531.
31. Martin, C. W., “Spirally Prestressed Concrete Cylinder,” Journal o f the American
Concrete Institute, No. 10, October 1968, pp. 837-845.
32. Mirza, S. A. and MacGregor, J. G., “Variability o f Mechanical Properties of Reinforc­
ing Bars,” Journal o f the Structural Division, A$CE, ST5, May 1979, pp. 921-937.
33. Nawy, E. G., Denesi, R. F., and Grosko, J. J., “Rectangular Spiral Binders Effect on
Plastic Hinge Rotation Capacity in Reinforced Concrete Beams,” Journal of the
American Concrete Institute, No. 12, December 1968, pp. 1001-1010.
34. Park, R., “Accomplishments and Research and Development Needs in New Zealand,”
Workshop on Earthquake-Resistant Reinforced Concrete Building Construction,
Berkeley, July 11-15, 1977, pp. 255-295.
35. Park, R., “Design o f Prestressed Concrete Structures,” Workshop on Earthquake^
Resistant Reinforced Concrete Building Construction (ERCBC), University of Cali­
fornia, Berkeley, July 11-15, 1977, pp. 1722-1752.
36. Park, R., Kent, D. C., and Sampson, R. A., “Reinforced Concrete Members with
Cyclic Loading,” Journal of the Structural Division, ASCEj Vol. 98, ST7, July 1972,
pp. 1341-1360.
37. Park, R. and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, Inc.,
New York, NY, 1975.
38. Park, R., Priestley, M. J. N., and Gill, W. D., “Ductility o f Square-Confine^ Concrete
Columns,” Journal o f the Structural Division, ASCE, ST4, April 1982, pp. 929-951.
39. Park, R. and Sampson, R. A,, “Ductility o f Reinforced Concrete Column Sections in
Seismic Design,” Journal o f the American Concrete Institute, No. 9, September 1972.
40. PCI Design Handbook, Prestressed Concrete Institute, Chicago, Illinois, 1976.
54
41. Pfrang, E. O., Siess, C. P., and Sozen, M. A., “Load-Moment-Curvature Characteristics
o f Reinforced Concrete Cross Sections,” Journal o f the American Concrete Institute,
VoL 61, No. 7, July 1964, pp. 763-778.
42. Popovics, S., “A Review o f Stress-Strain Relationships for Concrete,” Journal of the
American Concrete Institute, No. 3, March 1970, pp. 243-248.
43. Priestley, M. J. N., Park, R., and Potangara, R. T., “Ductility of Spirally-Confined
Concrete Columns,” Journal o f the Structural Division, ASCE, S T l, January 1981,
pp. 181-202.
44.
“Recommended Comprehensive Seismic Design Provisions for Buildings,” Working
Draft of Applied Technology Council, Palo Alto, California, 1976.
45.
“Recommendations for the Design o f Aseismic Prestressed Concrete Structures,” 3rd
Draft of Commission on. Seismic Structures, Federation Internationale de la PreContrainte, London,-England, July 1976-
46.
“Recommended Lateral Force Requirements and Commentary,” Seismology Com­
mittee, Structural Engineers’ Association o f California, San Francisco, 1973.
47. Richart, F. E., Brandtzaeg, A., and Brown, R. L., “The Failure o f Plain and Spirally
Reinforced Concrete in Compression,” Bulletin No. 190, Engineering Experiment
Station, University of Illinois, April 1929.
48. Roy, H. E. H. and Sozen, M- A., “Ductility o f Concrete,” Proceedings o f the Inter­
national Symposium on the Flexural Mechanics of Reinforced Concrete,” ASCEACI, Miami, November 1964, pp. 213-244.
49.
Sargin, M., Ghosh, S. K., and Handa, V. K., “Effects of Lateral Reinforcement upon
the Strength and Deformation Properties o f Concrete,” Magazine o f Concrete Re­
search, Vol. 23, No. 75-76, June-September 1971, pp. 99-110.
50. Scott, B. D., Park, R., and Priestley, M. J. N., “Stress-Strain Behavior of Concrete
Confined by Overlapping Hoops at Low and High Strain Rates,” American Concrete
Institute Journal, January-February 1982, pp. 13-27.
51. Sheikh, S. A., “A Comparative Study of Confinement Models,” American Concrete
Institute Journal, July-August 1982, pp. 296-306.
52. Sheikh, S. A. and Uzumeri, S. M-, “Strength and Ductility o f Tied Concrete Columns,
Journal of the Structural Division, ASCE, ST5, May 1980, PP- 1079, 1 102.
53. Sinha, B. P., Gerstle, K. H., and Tulin, L. G,, “Stress-Strain Relationships for Com
crete Under Cyclic Loading,” Journal o f the American Concrete Institute, Vol. 61,
No. 2, February 1964, pp. 195-211.
54.
Solimani M. T. M. and Yu, C. W., “The Flexural Stress-Strain Relationship of Con­
crete Confined by Rectangular Transverse Reinforcement,” Magazine o f Concrete
Research, Vol. 19, No. 61, December 1967, pp. 223-238.
55
55. Somes, N. F., “Compression Tests on Hoop-Reinforced Concrete,” Journal of the
Structural Division, ASCE, ST7, July 1970, pp. 1495-1509.
56. Stunnan, G., Shall, S. P., and Winter, G,, “Microcracking and Inelastic Behavior of
Concrete,” Proceedings o f the International Symposinm on Flexural Mechanics of
Reinforced Concrete, ASCE-ACI, Miami, NovemberT 964, pp. 473-499.
57. Taylor, M- A-, “Constitutive Relations for Concretes Under Seismic Conditions,”
Workshop on Earthquake-Resistant Reinforced Concrete Building Construction
(ERCBC), University o f California, Berkeley, July 11-15, 1977.
58. Thompson,. K. S. and Park R., “Ductility o f Prestressed and Partially Prestress^d Con­
crete,” Prestressed Concrete Journal, July 1980, pp. 23-37.
59.
“Uniform Building Code,” VoL I, International Conference of Building Officials,
Pasadena, California, 1970.
60. Uzumeri, S: M. and Sheikh, S. A., “Strength and Ductility of Reinforced Concrete
Columns with Rectangular Ties,” Workshop on Earthquake-Resistqnt Reinforced
Concrete Building Construction (ERCBC), University o f California, Berkeley, Iuly
1 1-15, 1977, pp. 611-623.
61. Warwaruk, J., “Deformation Analysis for Prestressed Concrete Beams,” Prestressed
Concrete Institute Journal, October 1965, pp. 67-80.
62. Wiegel, Robert L., Earthquake Engineering, PrenticerHall, Inp,, Englewppd Cliffs,
NJ, 1970.
63. Yamashiro, R. and Siess, C. P., “Moment-Rotation Characteristics of Reinforced
Concrete Members Subjected to Bending, Shear, and Axial Load,” Civil Engjnpering
Studies, Structural Research Series No. 260, University of Illinois, Dpcpmber 1962,
64. Zia, P. and Guillermo, E. C., “Combined Bending and Axial Load in Prestresspd Con­
crete Columns,” Prestressed Concrete Institute Journal, June 1967, pp. 52-59.
56
APPENDICES
57
APPENDIX A
COMPUTER GENERATED M- 0 CURVES
CD
ro
S j Too
^Too
ToToo
TsToo
2'o. oo
2's.oo
CURVATURE D U C T IL IT Y
Figure 14. Effect o f confining hoop steel.
SrOToo
S1S-OO
0.32
0.30
0.28
0.26
Z= 18
•22
MOMENT -
Z= 12
0.24
M/FOB*H*H
Z=6
cOToo
3 . 00
ToToo
TsToo
ToToo
TsToo
CURVATURE D U C T I L I T Y
Figure 15. Effect of confining hoop steel; P = 0 .2 0 T'c Ag.
ToToo
T s.oo
0.33
0.31
0.29
0.27
0.25
M/FOB*H*H
Z= 18
• 23
MOMENT -
Z= 12
cS- OO
SrToo
16.00
2'4 . OO
3I2 . 0 0
4'o . 0 0
CURVATURE D U C T I L I T Y
Figure 16. Effect of confining hoop steel; P = 0.10 f'c Ag.
48.00
56.00
0 ^3 2
0, - 30
Q. 2 8
M/FC*B*H*H
-
0.26
0-24
Z= 18
22
MOMENT
Z= 12
AO'. OO
4'. OO
FT oo
1'2 • OO T e . do 2'o. o o
CURVATURE DUC TI LI T Y
Figure I 7. Effect o f confining hoop steel; P = 0.30 f'c Ag.
2'4 . OO
28.00
O
1 0 . 0 0 2 0 . 0 0 3 0 7 oo
ToToo
S1O-OO
CURVATURE D U C T I L I T Y
Figure 18. Effect o f steel strain hardening; P = 0 .10 f'c Ag, Z - 12.
o'o.oo
0.33
0.31
0.29
0.27
.23
0.25
M/FOB*H*H
MOMENT -
o>
U)
c Ol-OO
5.00
T o.oo
TsToo
ToToo
TsToo
CURVATURE DUC TI LI T Y
Figure 19. Effect o f steel strain hardening; P = 0.20 f'c Ag, Z - 1 2 .
30.00
T1S-OO
hO
4.00
2.00
16-00
W
CURVATURE DUCTILITY
Figure 20. Effect o f steel strain hardening; P = 0.30 f 'c Ag, Z = 12
28.00
O. 5 0
AOD 0.10 f'c Ag
,0 = 5%
O x + 0.20 f'c Ag
0.40
0-30
0.20
41
P = 1%
.00
MOMENT -
p = 3%
0.10
M/FOB*H*H
X * -V 0.30 f'c Ag
Xl
c U-OO
To. o o
Z o .o o 3 o . o o T oT oo 3 o . o o
CURVATURE D U C T I L I T Y
Figure 21. Effect o f longitudinal steel content; Z - 12.
e 'o .o o
To.oo
O
to
f + O z =6
* x o Z= 12
P =3%
/3 = 1%
CURVATURE D U C T I L I T Y
Figure 22. Effect of confinement and varying steel percentages; P = 0.20 f 'c Ag.
CM
ro
cc = 2.5
cc = 1.5
* O'
O
•
CURVATURE
Figure 23. Effect o f concrete cover; P = 0.10 f'c Ag.
DUCTILITY
cc = 3.5
M/FOB*H*H
CvJ
PO
MOMENT -
CC
CURVATURE D U C TI LI T Y
Figure 24. Effect o f concrete cover; P = 0.20 f'c Ag.
= 2.5
cc = 3.5
0.32
0.30
0.28
0.26
.22
0.24
M /F O B * H*H
MOMENT -
12.00
CURVATURE DUCTILITY
Figure 25. Effect o f concrete cover; P = 0.30 f'c Ag.
0.12
0.10
0.08
.06
MOMENT -
o.
M/FOB*H*H
UD
CURVATURE
Figure 26. Effect o f confinement; P = 0.20 f'c Ag.
20.00
DUCTILITY
28.00
0. i
O1. 10
0.08
0.20 f'c Ag
0.30 f'c Ag
.04
MOMENT -
0.10 f'c Ag
0.06
M/FOB*H*H
o
S d1. o o
41.
oo
SrToo
TTToo
Te. oo
TTToo
CURVATURE D U C T I L I T Y
Figure 27. Effect of axial load; Z - 12.
TrTToo
T s.oo
28.00
Figure 28. Effect o f bond compatibility factor; P = 0.20 f ' Ag, Z = 12.
0.12
0.10
0.08
.06
MO ME N T
-
M / F C * B* H
CD
4.00
16.00
CURVATURE
2'O • 00
DUCTILITY
Figure 29. Effect o f concrete cover; P = 0.20 f'c Ag, Z =12.
28.00
0.20
0-16
0.12
M/FOB*H*H
-
0.08
0.04
Z= 12
6 - #4
4 - #4
2 - #4
Z = 76
.00
MOMENT
6 -# 4
4 -# 4
2 -# 4
93.00
4'. OO
FT oo
T 2 . 0 0 T e . o o 2'o. o o
CURVATURE D U C T I L I T Y
Figure 30. Effect of non-prestressed steel and hoop steel; P - 0.20 f'c Ag.
2 '4 .0 0
^ s . 00
0.48
0.44
0.40
0.36
M /FOB*H*H
3 _#3(&' 18
O + P = 400 kips
a a p = 500 kips
0.32
-
3 - #4 (« 4
.28
MO ME N T
t
SrToo
12 . o o TgT oo T o
CURVATURE D UCTILITY
Figure 31. Ductility o f concrete column Life Sciences Building, Bozeman, MT.
2 8 . OO
0, - 20
O1. I 6
0.12
0.10 f'c Ag, Z = 44
0-08
_
0.30 f'c Ag, Z = 44
0.04
M/FOB*H*H
0.30 f' Agj Z = 3
.00
MOMENT -
0.10 f'c Agj Z = 3
CURVATURE D U C T I L I T Y
Figure 32. Ductility of prestressed bridge piles.
77
APPENDIX B
PROGRAM DOCUMENTATION AND LISTING
78
APPENDIX B
PROGRAM DOCUMENTATION AND LISTING
Documentation for Program Column
A.
Problem Identification Cards:
The.information input on the first ten cards is printed. There must be ten cards, either
blank or containing relevant information concerning the problem. The information on the
first card is printed as a heading on a moment-curvature plot, if a plot is requested.
10 Cards:
Columns
Format
Entry
1-80
A
Problem information to be
output with result?.
B.
Problem Iteration Control Card:
This card identifies the number of problems to be run at one time.
I Card:
Columns
Format
Entry
1-2
12
IK
If IK > I, then the input cards for Sections C through H are required for each IK case.
C.
Plotting and Printing Control Card:
This card provides the user with the opportunity to select the appropriate plots and
output for a problem. It is possible to “Print All” which includes the neutral axis, concrete
compression force, moment, and curvature for each increment o f concrete strain and steel
areas, locations, strains, stresses, and forces in the steel fpr each increment of concrete
79
strain. A summary o f concrete, strains, moments, and curvatures is then printed. A “Print
Summary” option provides a means of reducing the amount of output.
The plotting option provides for a plot of moment versus curvature (ft. - kips vs.
I/in.), moment versus curvature (nondimensional), and to calculate nondimensional coordi­
nates, but to save them for the combined plotting option. If a plot o f all the moment cur­
vature curves is required, the combined plotting option is used. Morp than one moment
curvature curve is generated by using IK on the Problem Control Iteration Card.
II
Print Option
=
O Print All
=
I Print Summary
KK
Plotting Option
=
O No Plot
=
I Plot - Moment vs, Curvature
=
2 Plot - Moment vs. Curvature (Non-dimensional)
=
3 No Plot - Calculates Non-dimensional Coordinates for Combined Plotting
=
4 Plot —Non-dimensional Mpmpnt vs. Curvaturp Duptility
=
5 No Plot - Calculates Non-dimensional Moment vs. Curvature Ductility for Com­
bined Plotting
KKK Combined Plotting Option
=
O No Plot
=
I Plot - Combined Non-dimensional Moment vs. Curvature Curves
=
2 Plot —Combined Non-dimensional Moment vs. Curvature Ductility
80
KI
Plotting Symbol Option
=
I Symbol Every Data Point
=
2 Symbol Every Other Data Point
O
O
=
5 Symbol Every Fifth Data Point
etc.
I Card:
Columns
' 1-2
Entry
Format
II
12 ■
3-4
12
KK
5-6
12
KKK
7-8
12
KI
D.
Cqlumn Information:
The program uses the secant method to determine the neutral axis locatipn for a con­
crete section by iterating the force equilibrium equation. The secant method requires two
initial starting values, CO and C l. The width B, depth, H, and axial force level, AXIAL, are
also inputted on this card.
I Card:
1-10
F1Q.1
Entry
AXIAL
1 1-20
F 10.3
B
21-30
F10.3
H
31-40
F10.3
CO
41-50
F10,3
Cl
Columns
Format
81
E.
Reinforcement Properties:
E l. Prestressed Steel
Properties o f prestressed steel and the number o f layers of prestressed steel, NP, are
inputted on this card. The stress-strain curve for the steel consists of three segments. The
modulus of elasticity, EP, is also inputted. This card is required to be blank if tl>e column
is not prestressed.
I Card:
Columns
1-10
Format
F 10.6
EPB
11-20
F10.6
EPC
21-30
F10.6
EPU
31-40
F10.3
FPB .
41-50
F10.3
FPC
51-60
F10.3
FPU
61-70
F10.1
EP
71-72
12
NP
Entry
E2. Nomprestressed Reinforcement
Properties of non-presfressed steel and the number of layers o f non-prestressed steel,
NN, are inputted on the card. The stress-strain curve for non-prestressed steel is assumed to
be elastic-plastic with a strain-hardening region. The modulus o f elasticity, MOES, is also
inputted. This card requires to be blank if column is only prestressed.
I Card:
/
82
Columns
Format
Entry
MO
F10.6
EY
11-20
F10.6
ESH
21-30
F10.6
ESU
31-40
F10.3
FY
41-50
F10.3
FSU
51-70
F20.3
MOES
71-72
12
NN
F.
Prestressing Information:
The information input on this card is the effective strain in the prestressed steel due
to prestressing, EPE, the effective strain in the concrete due to prestress, ECE, and a bond
compatibility factpr (CF), between the concrete and steel. (ECE, can also be modified to
account fqr long term effects, i.e., creep.) If NP ?= 0, see section El. , this card is not required. EPY, is the yield point o f the prestressing steel.
I Card (optional):
Columns
1-10
Format
F 10.6
Entry
EPE
1 1-20
F10.6
ECE
21-30
F10.6
CF
31-40
F10.6
EPY
G.
Reinforcing Areas and Effective Depths:
G l. Prestressed Reinforcement
The program is currently dimensioned to handle 10 layers or 10 different locations of
prestressed reinforcement. The area of each layer PAS, or each reinforcing bar and the dis­
tance from the extreme fiber in compression to the centroid of the area, PD, is input on
this card.
83
The maximum number o f cards which can be read is 10. The number o f cards required
is equal to NP, input in section E l . If NP = 0, this card must not be input.
0 - 1 0 Cards (optional):
Columns
1-10
Format
F10.3
Entry
PAS
11-20
F10.3
PD
G2. Non-prestressed Reinforcement
The program is currently dimensioned to handle 10 layers pr 10 different locations of
non-prestressed reinforcement. The area of each layer, AS, or each reinforcing bar and the
distance from the extreme fiber in compression to the centroid, D, is input on this card.
The maximum number of cards which can be read is 10. The number o f cards required is
equal to NN, input in section E2. If NN = 0, no cards are required.
0 - 1 0 Cards:
Columns
1-10
Format
F10.3
11-20
F10.3
H.
Entry
AS
D
Column Confinement Information:
Information required to calculate the confined stress-strain curve for concrete is
input on this card. The concrete strength, F C l, volume of hoop reinforcement, PS, width
o f the confined core, B2, transverse spacing, o f hoops, SH, and concrete cover, CC, are all
input via this card. A Z value may be input, bypassing the calculation of Z. The yield
strength of the transverse reinforcement is inputted as FYH.
The user can choose from three different stress-strain curves. The original stressstrain curve for concrete proposed by Park and Paulary, a second which is a modified
version o f the original to account for strength increase due to confining steel, and a third
for a high strain rate version of the modified curve.
84
NM
Stress-Strain Curve Option
=- I
Original Curve
=
0
Modified for Strength Increase
=
I
High Strain Rate Including Strength Increasp
I QW :
Entry
Columns
1-10
Format
F10.3
11-20
F10.5
PS
’ 21-30
F10.5
B2
31-40
F 10.5
SH
41-51
F10.3
CC
51-60
F10.3
Z
61-70
F10.3
FYH
71-72
12
NM
FCl
85
Program Listing
09:11
C
C
C
C
C
C
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C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
cC
C
C
C
C
C
C
C
C
C
C1
C
C
C
C
C
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C
C
C
C
C
C
MAY
15
'84
C O L U M N . UEMBS
***************************************A*A***A**********************
*****
*****
*****
PROGRAM
NAME:
COLUMN
DATE
PROGRAMMED:
A P R I L , 82
*****
*****
*****
*****
*****
*****
*****
*****
COLUMN C A L C U L A T E S THE MOMENTS AND CURVATURES OF. AN
A X I A L L Y LOADED C O N F I N E D P R E S T R E S S E D , R E I N F O R C E D ,
OR P A R T I A L L Y PREST RES SED CONCRETE COLUMN S E C T I O N .
*****
THE C O N F I N E D S T R E S S - S T R A I N CURVE FOR CONCRETE CAN
BE THE O R I G I N A L , M O D I F I E D , OR H I G H S T R A I N CURVES
PROPOSED BY PARK 8 KENT OR PARK 8 OTHERS.
*****
*****
*****
*****
******
*****
*****
THE PROGRAM
THE NEUT RAL
EQUILIBRIUM
*****
USES. THE SECANT METHOD TO DE T ERMI NE
AX I S' L O C A T I O N BY I T E R A T I N G THE FORCE
EQUATI ON.
*****
*****
** ** *
*****
*****
*****
*****
*****
ASSUMPTI ONS:
*****
** ** *
** ** *
BERNOULL' S P R I N C I P L E .
T ENS I L E STRENGTH OF CONCRETE
i
?
5
*****
4
*****
*****
*****
*****
*****
*****
*****
** ***
IS
*****
I GNORED.
B U C K L I N G OF L O N G I T U D I N A L ST EE E JfJ COMPRES SI ON I S
PRE VENTED BY TRANSVERSE R E I N F O R C E M E N T .
THE S T R E S S - S T R A I N CURVE FOR U N CONF I NE D CONCRETE
( CONCRE T E COVER) FOLLOWS THE S T R E S S - S T R A I N CURVE
FOR C O NF I NE D CONCRETE AT S T R A I N S EESS THAN 0 . 0 0 4
FOR S T R A I N S GREATER THAN 0 . 0 0 4 , THE CONCRETE
COVER I S ASSUMED TO HAVE S P A L L E D AND T HE R E F O R E ,
HAS ZERO ST RE NGT H.
S HORT - T E RM L O A D I N G I S C O N S I D E R E D ,
PERFECT BOf f D E X I S T S BETWEEN THE CONCRETE AND
N O N P E E f T R E S S E D OR P RE ST RES SED R E I N F O R C E M E N T .
*****
5
6
*****
*****
*****
*****
*****
*****
*****
*****
*****
**(***
*****
*****
*****
*****
*****
** ** *
*****
PROGRAM
*****
** ** *
*****
* ****
*****
D I M E N S I O N E D TO HANDLE
REI NF ORCEMENT AND 10
R E I N F O R C E ME N T .
THE. PROGRAM C A L CU L A T E S THE MOMENT AfJD CURVATURE
FOR CONCRETE S T R A I N S . 0 0 3 THRU . 0 5 2 , I NCRE ME NT E D
*****
*****
*****
*****
*****
*****
.001
.
ALL
UNITS
ARE
A X I A L LOAD
I F TENSI ON
*****
*****
*****
*****
*****
*****
*****
*****
RESTRICTIONS:
THE PROGR:AM I S CURRENTL Y
1 0 L AY ERS OF P RE ST RES SED
L A Y E RS OF NONPRESTRESSED
*****
IF
AS
ASSUMED
TO
BE
COMPRESSI ON
NEGATIVE.
IN
MUST
I NCHE S
BE
AND
I NP UT
AS
BY.
POUNDS.
POSITIVE
*****
* * ***
*****
** ***
*****
*****
*****
*****
*****
*****
*****
*****
*****
*****
*****
*****
*****
*****
A* * * *
PROGRAM W I L L HANDLE ONLY UNI F ORM CONCRETE S T R A I N
DUE TO PREST RESS I NG I . E t COL UMNS, S YMMMf T R I C A L
RE I N F O R C E ME N T WI T H SAME P RE ST RES S FORCE.
, '
■
*****
*****
*****
*****
********************************************************************
T HJ S
IS
THE
MAI N
PROGRAM
S T EE L S T R A I N S , S T R E S S E S ,
WHEN THEY ARE N E G A T I V E .
THE
AXIAL
LOAD
IS
IN
COLUMN.
ANp
COMPRESSI ON
FORCES
WHEN
ARE
IN
POSITIVE
COMPRESSI ON
AND T E N S I O N
86
C
C
C
SUBROUTI NE Y I E L D CALCULATES THE Y I E L D CURVATURE AND
MOMENT FOR, A PRESTRES$ED, NONPRESTRESSED, OR PARTI ALLY
PRESTRESSED COLUMN,
c
CALL Y I E L D ( E P B » E P C v E P U » F P B p F P C » F P U > E P » E C E # E P E z E C / C P » C 1 ' P 9 ' P AS/ TPE
$ S , P S S f P F S f N P f C F f N M , E Y , E S H f E S U , F S U , F Y , M O E S , A S , E S , S S , E S , NN f AL P HAf F Cl
$, AKfBfAXIALfCfDfGAMMAfCURVYfTOTMYfHfCCfIIfFYHfPSf82fSHfZfEPY)
CO 150 1 = 1 , 5 0
EC = 0 . 0 0 1 * 1 + . 0 0 2
C
C
C
C
C
THE NEXT L I N E DENOTES WHEN TH^ CONCRETE ST RAI N I S
GREATER THAN 0 . 0 0 4 THE COLUMN DI MENSI ONS ARE TO
BE CHANGED.
IF
(EC-0.004)
525,525,220
220
B=B2
919
51 5
H= HI
ECE=ECEI
I F (NP . E Q . 0 . 0 )
DO 91 9 J = I , NP
PD( J) = P D I ( J )
CONTI NUE
I F (NN . E S . 0 . 0 )
DO 929 J = I f N N
D ( J ) = Dl ( J )
CONTI NUE
929
525
GO TO 515
GO TO 525
C
C
C
SUBROUTI NE CONFI NED CALCULATES THE CENTROI p AND MEAN
STRESS FACjTOR FOR THRI-E DJ FF^RENT TYPES QF STRESSSTRAI N CURVES.
C
C
C
CALL CONFI NED ( E C , F C l , FYH, P S , B 2 , S H , Z , NM, A L P HA , GAMMAf I I ' AK)
SUBROUTI NE SOLVE USES THE SECANT METHOD TO CALCULATE
THE NEUTRAL AXI S LOCATI ON.
C
CALL SOLVE ( E P B , EPC , EPU, F P B , FPC, FPU, EP, ECE, EPE, EC, C O , CI , P D , PAS, TPE
SSfPSSfPFSfNPfCFfNMfEYfESHfESUfFSUfFYfMOESfASfESfSSfFSfNNfALPHAfFCI
$ , A K,BfAXIALfCf D f * 1 5 5 , *932)
COMP = - ( A L P H A * F C 1 * A K * B * C )
I F ( I I . E S . I ) GO TO 6 1 7
C
C
SUBROUTI NE OUTPUT WRI TES THE S T R A I N S , Ji TRESSESf
FOR EACH I NCREMENT OF CONCRETE ST RAI N.
C
CALL OUTPUT
617
C
C
( NP, TP E S , PSS, PF S, C , COMP, AX I AL , PAS, P D , ES, S S , F S, AS, D, NN
MOMENT
CALL MOMENT
150
932
FORCES
$)
CONTI NUE
SUBROUTI NE
C
AND
CALCULATES
THE MOMENT QF THE COLUMN.
( N P , P F S , PD, ALPHA , GAMMA , B , C , H , EC , I , T P T M , CURV , FCI , CC, FS
S , D , N N , I I , AK )
CONTI NUE
CONTI NUE
SUBROUTI NE
SUMMARY PRI NTS
A SUMMARY OF THE CONCRETE
STRAI N,
87 •
C
C
C
WHEN
NEGATIVE.
IMPLICIT
REAL
DI MENSI ON
( M)
PSS(IO),PFS(IO),PAS(IO),PD(IO)
D I M E N S I O N T P E S ( I O ) , TOTM ( 5 0 ) , CU R V.( 50 )
DI MENSI ON A S d 0 ) , E S d 0 > , S S ( I O ) , F S d 0 ) , Dl ( 1 0 ) , D ( I O )
DI MENSI ON P O l ( I O) , PXCURV( 5 2 ) , PYTOTM( 5 2 )
DI MENSI ON XARRAY( 5 0 2 ) , YARRA V ( 5 0 2 )
D I M E N S I O N L d O ) , L I ( 0 : 1 0 ) , PXYCUR V d O , SO) , P Y X T O T M d O , 5 Q )
CHARAC T E R * 8 0 H E A D I N G ( I O )
C
C
C
C
THESE L I N E S READ AND WRI T E ANY I N F O R M A T I O N OR H E A D I N G
THAT THE USER WANTS OUT PUT T ED WI T H H I S R E S U L T S .
222
444
226
921
923
C
READ ( 1 0 5 , 2 2 2 )
(HEADING(I), I = I ,10)
FORMAT ( A 8 0 / A 8 0 Z A 8 0 / A 8 0 / A 8 0 / A 8 0 / A S O / A S O / A 8 0 / AS O )
WRI TE ( 1 0 8 , 4 4 4 )
HEADING(I), 1=1,10
FORMAT C l ' , A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 / A 8 0 )
READ ( 1 0 5 , 2 2 6 ) . I K
FORMAT ( 1 2 )
DO 921 N L = I , 1 0
L(NL)=O.O
DO 9 2 3 NI = I , 1 I
NII=NI-I
LI(N I I ) =0.0
DO 9 3 9 J K = I , I K
S UB ROUT I NE
I NPUT
READS
IN
TH^
VARIABLES
TO T HE
MAI N
PROGRAM.
C
CAL L I N P U T ( A X I A L , B , H , C O , C l , 5 P B , E P C , E P U ) F P B , F P C , F P U , E P , N P , FC I , P S , 8
$2,SH,CC,PAS,PD,EPE,5CE,Z,FYH,:NM,EY,ESH,ESU,FY,FSU,MPES,'NN,AS,D,I.I ,
$KK,CF,KKK,EPY,KI)
C
C
C
C
THESE NEXT F I V E L I N E S CHANGE THE
ACCOUNT FOR OUTER CORE S P A L L I N G .
C
C
C
COLUMN
HQ=H
BG=B
AG = B* H
H I = H - 2 * CC
AN=B2 * H I
I F ( NP . E Q . 0 ) C F = I . O
I F ( NP . EQ. O . Q ) E P E = O . O
I F ( NP . E Q . 0 . 0 ) E C E = O . O
THE NEXT L I N E C A L C U L A T E S THE I N C R E A S E I N
DUE TO P R E S T R E S S I N G AF TER THE OUTER CORE
979
41 4
121
230
ECE1=ECE*(AG/AN)
I F ( NP . E Q . 0 . 0 )
DO 9 7 9 J = I , NP
PDI(J)=PD(J)-CC
CONT I NUE
I F ( NN . EQ. 0 . 0 )
DO I 21 J = I , N N
Dl ( J ) = D ( J ) - C C
CO N T I N U E
GO TO
414
GO TO 2 3 0
DI MENSI ONS
TO
CONCRETE S T R A I N
HAS S P A L L E D .
88
C ■ MOMENT,
CURVATURE,
CALL SUMMARY
L(JK)=I-I
C
C
C
C
CURVATURE
DUCTILITY..
( T O T M , CURV , E P E , E C E / N M , C F , F C l , B G z H G , CURV Y)
S UB ROUT I NE CALCOMP
ONLY ONE A N A L Y S I S .
CAL L CALCOMP
SJ K / C U R V Y / K I )
GO TO 9 0 9
AND
PL OTS
MOMENT
VERUS
CURVATURE
FOR
( T O T M / C URV / H E A D I N G ^ K K / H G , B G / F C I » P X q URV z P y T Q T M / L / N P T S ,
155
WRI TE
505
FORMAT ( / / / , T S , ' CONVERGENCE
GO TO 9 4 9
CONT I NUE
(108,505)
IS
NOT
ACHIEVED')
909
C
C
THE F OL L OWI NG L I N E S REAARRANGE I N F O R M A T I O N SO THAT
C
MORE THAN ONE PLOT CAN BE PUT ON A P A G E , I . E . , I T
C
I S P L A C I N G A L L THE I N F O R M A T I O N I N ONE F I L E SO THAT
C
I T CAN SCAL E T HA T F I L E FOR THE PL OT P A R A ME T E RS .
C
DO 35
35
939
Kil =1 , L ( J K )
PXYCURV( J K , K J ) = P X C U R V ( K J )
P Y X TOT M ( J K z K J ) = P Y T o T M ( K J )
CONTI NUE
I F ( KKK . E Q . 0 ) GO TO 9 4 9
DO 3 4 3
JK=IzIK
DO 4 3 4
KJ=IzL(JK)
L I ( O) =O
LI ( J K ) = L ( J K ) + ! , ! ( J K - I )
I=LI(JK)-L(JK)+KJ
NCPTS=I
N C P T S I =1+1
NCPTSP= 1 + 2
XARRAY( i ) = p x y c u r v ( j k , k j )
434
Y A RR A Y ( I ) = P Y X T O T M ( J K , K J )
343
CONT I NUE
C
SUB ROUT I NE CALSUM PL OTS AL L COMB I N E D I N F O R M A T I O N ON ONE
C
GRAPH.
C
CAL L CAL SUM ( X A R R A Y , Y A R R A Y z I K z H E A D I N G z L z p X Y C UR V z P Y X T O T Mz N C P t S z NCPT
$S1/NCPTS2/KKK/KI)
949
CONTI NUE
END
S U B ROUT I NE SOLVE ( E P B z E P C z E P U z F P B z F P C z F P U , EP, E C E z E P E z E C z C O z C I z P D z P
$ A S , T PES z P S S z P F S z NPzCF , NMzEYzE S Hz ES Uz FSUz F Yz MOE Sz A S, E S , SSzFSzNNzALP
$HA,FC1/AK,B,AXIAL/C/Dz*/*>
C
C
C
C
T HI S SUBROUTI NE I TERATES THE
BY USI NG T HE SECANT METHOD.
I S DETERMI NED.
FORCE E Q U I L I B R I U M EQUATI ON
THE EXACT NEUTRAL AXI S LOCATI ON
C
I M P L I C I T REAL ( M)
DI MENSI ON S U M P F 0 ( 0 : 1 0 ) z S U M P F i ( O : 1 0 ) , S U M F S 0 ( O : 1 0 ) z S U M F S 1 ( 0 : 1 0 )
DI MENSI ON P S S ( 1 0 ) z P F S ( 1 0 ) z PAS( 1 0 ) / P D ( I Q ) , TPES( 1 0 )
D I M E N S I O N E S( I 0 ) / S S ( I O ) / F S ( I O ) . / AS ( I 0 ) / D ( I 0 )
JI
* 0.0
I
I
89
260
CONT I NUE
I F ( NP . E Q „
0.0)
GO TO 8 7 8
CAL L P RE S T E E L ( E P B , E P C , E P U , F P 8 , F P C , F_PU, E R , E C E , E P E , E C , C O , P D , P A S , TPE
$S/PSS#PFS,SUMPF0,NP,CF,NM,*301)
CAL L P R E S T E E L ( E PB , E PC , E PU> F |=B » FPC , F P U , EP » ECE , E P ^ , E Cz C I , P p , P A S , Tf> E
$ S , P S S , P F S , S U MP F 1 , N P , C F , N H , * 3 0 1 )
CONT I NUE
I F < NN . EO. 0 . 0 )
GO TO 7 8 7
CAL L S T E E L < E Y , E S H # E S U , F S U , F Y , M O E S , E C # C O , D , A S , E S , S S , F S , S U M F S O , N N , N
S M , *401 )
CAI t L S T E E L ( E Y z E S H z E S U , F S U z F Y z M OE S z E C , C l , D , A S , E S , S S , F S , SU M F S I z N N z N
S M , *401 )
878
CONTI NUE
I F (NP . EQ.
I F (N= . E O .
I F (NN . EQ.
I F ( NN , E Q .
.
0.0)
0.0)
0.0)
0.0)
SUMPFO( O) = O. O
SUMPFI ( O) =O. O
SUMFSO( O) = O. O
SUMFSI ( O ) = O - O
E R R O R O = - ( A L P H A * F C 1 * A K * B * C O > + S UMPF O ( N P ) + A X I A L
ERRORI = - ( A L PHA* F C 1 * A K * 9 * C 1 ) ♦ S UMP F ^ ( NP ) + A X I A L
C2 = C l - ( ( ER R ORI * - ( C 1 - C O ) ) / ( E R R O R I - E R R O f i O ) )
I F ( A 3 S ( C 2 - C 1 ) . L T . 0 . 0 1 ) GO TO 1 6 0
CO = Cl
Cl = C2
JI
* 1.0
+ SUMFSO( NN)
* SUMFSI(NN)
+ JI
F (JI
. G T . 1 5 0 0 ) RETURN I
GO TO 2 6 0
CONT I NUE
C = Q2
GO TO I 71
WRI TE ( 1 0 8 , 4 0 4 )
FORMAT ( / / / , T l 5 z ' STRAND F R A C T U R E S ' )
RETURN 2
WRI TE ( 1 0 8 , 4 0 5 )
FORMAT ( / / / , T l 5 , ' B A R F R A C T U R E S ' )
RETURN 2
CONT I NUE
RETURN
END
SUB ROUT I NE Y I E L D ( E P B , E P C , E P U , F P B z F R C z f P U , ^ P z E C E , E P E z E p , C O , C l , P D , f»
$ A S z T P E S z P S S , P F S , N P , C F z N M , E Y , E SH, E S U , F S U , F Y z M Q E S z A S z E £ , S S , F S z N N , AL P
SHA z F C I zAKz B z AX I A L z C z D z G A M M A , CURV Y T O T M Y z H , CCz I I , F YH P S z B 2 z SHz Z z EPY
$)
I
160
301
404
401
405
171
,
C
C
C
C
C
C
C
,
T H I S S UB ROUT I NE C A L C U L A T E S THE NEUT RAL A X I S ( . OCAT I ON WHEN THE
THE BOTTOM S T EE L ( P RE S T R E ^ S E D OR NONPRESTR ES SED) I S AT I T S
YIELD POI NT.
I T THEN CA L CUL A T E S THE Y I E L D MOMENT AND Y I E L D
CURVATURE AND P R I N T S B OT H.
THE S T R A I N I N THE BOTTOM ST EEL
I S P R I N T E D OUT FOR EACH I T E R A T I O N ( AS A C H E C K ) .
I M P L I C I T REAL ( M)
DI MENS I ON E S ( 1 0 ) z T P E S ( 1 0 ) , T O T M ( 5 0 ) z C U R V ( 5 0 )
DO 2 0 I =1 , 4 0
EC=O- OOI O + 0 . 0 0 0 0 5 * 1
CAL L CO N F I N E D ( E C z FCI z F Y H z P S z B 2 z SH, Z z N Mz A L P H A , GAMMA , I I z A K )
CAL L SOIt VE (E P B , EPCz E P U , F P B z F P C z F P U z E P z EC E z EP E z EC z CO , C l z P D , P A S z TPE
SSzPSSzPpSzNPzCFzNMzEYzESHzESUzFSUzFYzMOESzASzESzSSzFSzNNzAEPHAzFCI
SzAKzBzAXIALzCzD,*155z*932)
90
345
18
456
20
155
43
932
44
21
I =I
678
969
696
C
C
C
C
C
I F ( NP . G1 . 0 . 0 )
GO TO 1 8
WRI TE ( 1 0 8 , 3 4 5 ) E S ( N N )
FORMAT ( / / , T I O , '
ES( NN) I S • , 3 X, F l 0 . 7 )
I F ( A B S ( E S ( N N ) - E Y ) ' - t - T . . 0 0 0 0 5 ) GO TO 21
GO TO 2 0
WRI TE ( 1 0 8 , 4 5 6 ) T P t S ( N P )
FORMAT ( / / , T I O , '
TRES(NP) I S ' , 3 X , F 1 0 . 7 )
I F ( A B S ( T P E S ( N P ) - E P Y ) . L T . . 0 0 0 5 ) GO TO 21
CONT I NUE
GO TO 21
WRI TE ( 1 0 8 , 4 3 )
FORMAT ( / / , T 5 , 1 CONVERGENCE I S NOT A C H I E V E D ’ )
WRI TE ( 1 0 8 , 4 4 )
FORMAT ( / / , T S , ' ST EEL F R A C T U R E S ' )
CAL L MOMENT ( N P , P F S , P D , A L P H A , G A M M A , B , C, H , E C , I , T O T M , C U R V , F CI , CC, F S ,
$D , NN, I I , A K >
TOTMr=TOTM(I)
CURVY = C U R V ( I )
WRI TE ( 1 0 8 , 6 7 8 )
EC
FORMAT ( / / , T I O , ' CONCRETE S T R A I N AT S T E E L Y I E L D I S ' , 3 X , F I 0 . 6 )
WRI TE ( 1 0 8 , 9 6 9 )
TOTMY
FORMAT ( / / , T 1 Q , ' M O M E N T AT S T E E L Y I E L D I S ' , 3 X , F 1 5 . 3 )
WRI TE ( I 0 8 , 6 9 6 ) CURVY
FORMAT ( / / , T l O , ' Y I ELD CURVATURE I S ' , 3 X ^ F l 0 . 7 )
RETURN
END
S UB ROUT I NE C O N F I N E D ( E C , F C I , F YH , PS , B 2 , S Hi. Z , N M , A L P H A , G A M M A , I I , At O
T H I S S U B R O U T I N E D E T E R MI N E S THE MEAN STRESS AND C E N T ROI D
A C ONF I NE D S T R E S S - S T R A I N CURVE.
N M = - I ; S T R E S S - S T R A I N (JURVE OF PARK AND KENT
NM = O I M O D I F I E D S T R E S S - S T R A I N CURVE OF PARK AND KENT
NM = I I M O D I F I E D S T R E S S - S T R A I N CURVE AT H I G H S T R A I N RATES
Cf
LU
5
10
20
30
I
I F (NN
I F ( NM . E O . - I )
I F (NM . E Q . Q)
I F ( N M . E Q . 0)
I F (NM . E Q . 1)
I F ( NM . E Q . I )
I F ( Z . GT. 0 . 0 )
Of
RATE = 1 . 0
AK = 1 . 0
RATE = 1 . 0
AK = 1 . 0 + ( P S * F Y H ) / F C 1
RATE = 1 . 2 5
AK = I . 2 5 * ( 1 . 0 + ( P S A F Y H ) / F C I )
GO TO 5
E 5 0 U = ( 3 . O + 0 . 0 0 2 * F C I ) / ( F Cl - I 0 0 0 . 0 )
ESOH= 0 . 7 5 * P S * ( ( B 2 / S H ; > * * 0 . 5 )
Z » ( 0 . § / ( E . 5 0 U , + E 5 OH - 0 . 0 0 2 * A K ) ) * R A T E
CONT I NUE
E2 0 C = Q . 8 / Z + 0 . 0 0 2 * A K
I F ( EC - 0 . 0 0 2 * A K )
10,10,20
AREAAB = ( ( F C I * E C * * 2 . 0 ) / 0 . OO2 ) * ( I . O r E C / ( 0 . 0 0 6 * A K ) )
ALPHA = A R E A A B / ( EC * FC1 * A K )
,,
FMAAB = ( ( 2 . 0 * F C 1 * E C * * 3 . 0 ) / 0 . 0 0 6 ) * ( 1 . 0 - 3 . 0 * E C / ( 0 . 0 1 6 * A K ) )
GAMMA = 1 . 0 - F MA A . B / ( E C » A R E A A B )
GO TO SQ
I F (EC - E 2 OC) 3 0 , 3 0 , 4 0
AREABC * A K * F C 1 * 2 . 0 * . 0 0 2 / 3 . 0 * A K + A K * F C I * ( E C - . 0 0 2 * A K ) - A K * F C 1 * Z / 2
$ . 0 * ( E C * * 2 j0 - (.002*AK;)**2.:0) ♦ .002*AK*AK*FC1 *Z*(EC - .002*AK)
ALPHA = ARE A B C / ( E C * A K * F C 1 )
91
FMABC = 5 . 0 / 1 2 „ 0 * F C 1 * . 0 0 2 * . 0 0 2 * A K * * 3 . 0 + AK*FC I * 0 . 5 * ( 5 C* * 2 . 0 - ( . 0
$02*AK) * * 2 . 0 ) - AK*FC1 6 Z / 3 . 0 * ( E C * * 3 . 0 - < . 0 0 2 * A K ) * * 3 . 0 ) + A| ( **2. 0*FC
$ 1 * Z * . 0 0 2 * 0 . 5 * ( E C * * 2 . 0 - ( . 0 0 2 *AK) * * 2 . 0 )
GAMMA = 1 . 0 - FMABC/(ECAAREABC)
GO TO 50
AO
AREACD = A K * * 2 . O * FC I * 2 . 0 * . 0 0 2 / 3 . O + A K * F C 1 * ( E 2 0 C
. 0 0 2 * A K ) - AK*F
$C1*Z*0. 5 * ( E 2 0 C * * 2 . 0 ( . 0 0 g * A K ) * * 2 . 0 )
+ A K * * 2 . 5 * FCI * Z * . 0 0 2 * CE 2 OC $ . 0 0 2 * A K ) + 0 . 2 * A K * F C 1 * ( EC
E20C)
ALPHA = AREACD/ (EC*AK*FC1 )
FMACD = 5 . 0 / 1 2 . 0 * F C 1 * ( . 0 0 2 * * 2 . 0 ) * A K * * 3 . 0 ♦ AK * FC I * 0 . 5 * < E 2 0 C * * 2 . 0 $ ( i 0 0 2 * A K ) * * 2 . 0 ) - A K * F C 1 * Z / 3 . 0 * ( E 2 0 C * * 3 . 0 - ( . 0 0 2 * A K ) * * 3 . O) ♦ AK**
$ 2 . 0 * F C I A Z * . 0 0 2 * 0 . 5 * ( E 2 0 C * * 2 . 0 - < . Q 0 2 * A K ) * * 2 . O ) + 0 . 2 * A K * F CI * 0 . 5 * (
$ E C t*2 .0 - E20C**2.0)
GAMMA = 1 . 0 50
FMA CD / ( E C* AREA CD)
continue
IF ( I I . E Q . I ) GO TO 7 7 7
WRITE ( 1 0 8 , 6 0 ) EC,ALPHA,GAMMA
60
FORMAT ( * 1 ' , 5 X , ' C O N C R E T E STRAIN I S * , 1 X , F 8 . 6 , 5 X , ' M E A N STREf S FACTOR
$ I S ' , 1 X , F 8 . 6 , 3X, "CENTROID I S ' , T X , F 8 . 6 >
WRITE ( 1 0 8 , 7 0 ) F CI , Z
70
FORMAT ( / / , I O X , ' 28 DAY COMP. STRENGTH I S ' , I X , F l 0 . 3 , S X , ' PARAMETER
$Z I S ' , 1 X , F 1 0 . 6 )
IF ( E C - . 0 0 4 ) 7 7 7 , 7 7 7 , 6 6 6
666
WRITE ( 1 0 8 , 8 8 8 )
888
FORMAT ( / / , 2 5 X , ' C O N C R E T E COVER HAS SPALLED' )
777
CONTINUE
RETURN
END
SUBROUTINE PRESTEEL ( EP B# E P C# E P p , F P B, F P C# F P U, E P » E CE # E P E » EO, C, P D, P A
SS, TPES, Pf S, PFS, SUMPFS, NP, CF, NM, *>
C
C THIS SUBROUTINE CALCULATES THE STRAIN; STRESS, AND FORCE IN EACH
C LAYER OF PRESTRESSING STEEL.
TENSION IS P OSI TI VE.
C COMPRESSION I S NEGATIVE.
C
DIMENSION P E S ( i p ) , P S S M O ) , P F S ( I O ) , S UMP F S ( 0 : 1 0 > , P AS ( 1 0 ) , PD( 1Q>, TPES
$ ( 1 0 ) , T P E Z ( I O)
I F (N M . E Q . I ) F P B = I . 25*FPB
IF ( NM . E Q . I ) F P C = I . 25*FPC
IF (NM . E Q . I ) F P U = I . 25*FPU
DO 171 J = I , NP
PES( J) = 0 . 0
PSS(J) = 0 . 0
PFS(J) = 0 . 0
TPES(J)=O.O
T PEZ( ^ ) = O- O
171
SUMPFS( J) = 0 . 0
SUMPFS(O) = 0 . 0
DO 101 J = I ,NP
PES(J) = ( ( E C « ( P D ( J ) - C ) ) / C ) * C F
T P E S ( J ) = EPE + P E S ( J ) + ECE
TPEZ(J)=ABS(TPES( J ) )
IF ( T P E Z ( J ) . GT. E PU) RETURN I
I F ( T P E Z ( J ) . L T. EPB) GO TO 410
IF ( T P E Z ( J ) . L T, EPC ) GO.TO 420
P S S ( J ) = FPC + ( ( T P E Z ( J ) “ E P C ) / ( E P U ~ E P C ) ) A ( FPU-FPC)
GO TO 320
92
410
420
320
101
P S S ( J ) = TPEZ( J ) *EP
GO TO 320
PARTI P (<; FPC* EPC- FPB* EPB>/< EPC-EPB ) >
PART2 = ( E P B * E P C * ( F P B- F P C ) ) / ( TPEZ<J ) * ( E P C - ERB))
P S S ( J ) = PARTI + P ART 2
CONTINUE
IF ( T P E S ( J ) . L T. 0 . 0 ) CORR = - I . O
IF ( T P E S ( J ) „ GT. 0 . 0 ) CORR=I. O
P F S ( J ) = P S S ( J ) *PAS( J ) *CORR
SUMPFS( J) = P F S ( J ) ♦ S U H P F S ( J - I )
CONTINUE
RETURN
^ND
SUBROUTINE STEEL ( EY,/E SH, ESUz F S Uz FY, MOE S# E C# C, O, AS, E S , S S, F S, SUMF S »
SNN z NMz * )
C
C
C
C
C
THIS SUBROUTINE CALCULATES THE STRAINz STRESS, AND FORCE
IN THE REINFORCING BARS.
TENSION I S P OS I T I VE .
COMPRESSION IS NEGATIVE.
717
200
210
230
202
C
I MPLI CI T REAL (M)
DIMENSION E S ( 1 0 ) z S S ( 1 0 ) z F S ( 1 0 ) z S U M F S ( 0 : 1 0 ) z A S ( 1 0 ) z D ( 1 0 ) * E Z ( 1 0 )
I f (NM . E Q . I ) FY=1. 25*FY
I F (NM . 5 0 . I ) F SU“ 1 . 2 5 * F SU
DO 717 J=I zNN
ES ( J ) = O . O
S S ( J ) =0.0
FS ( J ) = O . O
SUMFS( J ) =O. O
EZ(J)=O. O
R=ESU-ESH
M=((FSU/FY)*((30.0*R+1. 0 ) * * 2 . 0 ) - ( 6 0 . 0 * R ) - 1 . 0 ) / ( I 5 . 0 A ( R * * 2 . 0 ) )
DO 202 J =I zNN
ES(J)=(EC*(D (J)-O )Z C
IF (A3S (ES ( J ) ) . LE. . E Y) GO TO 2 00
IF ( A B S ( E S U ) ) . LE. E SH) GO TO 210
I F (A8S ( E S ( J ) ) . GT. ESU)
RETURN I
IF ( E S ( J ) . LT. 0 . 0 ) CORR=- I . O
IF ( E S ( J ) . GT. 0 . 0 ) CORR=I, O
EZ(J)=ABS(ES(J))
PARTI = ( (M* ( EZ ( J ) - ESH) + 2 .,0) / ( 6 0 . 0* ( E Z ( J ) - ESH) + 2 . 0 ) )
P ART2 = ( (EZ ( J ) - E SH) * ( 6 0 . O-M) / ( 2. . 0* ( ( 3 0 . 0 *R + 1 . 0 ) * * 2 . 0 ) ) )
S S ( J ) = F Y* ( P ART1 + PART2)*CORR
• GO TO 230
S S ( J ) = E S ( J ) *M0 E S
GO TO 230
IF ( E S ( J ) . LT. 0 . 0 ) S S ( J ) = - F Y
IF ( E S ( J ) . GT. 0 . 0 ) S S ( J ) = F Y
SUMFS( Q) =O. 0
F S ( J ) = S S ( J ) *AS( J )
S U M f S ( J ) = F S ( J ) + S UMF S ( J - I )
CONTINUE
RETURN
END
SUBROUTINE MOMENT ( NPzPFSzPDzALPHAzGAMNAzBzCzHzECzIzTOTMzCURVzFCI z
$CCzFSzDzNN, I Iz4K)
93
C
C
C
THIS SUBROUTINE CALCULATES AND PRINTS THE MOMENT AND CURVATURE
FOR A PRE STRESSED CONCRETE COLUMN WITH A GIVEN AXML LOAQ.
I MPLI CI T REAL (M)
dimension
161
163
4pI
413
IF
41 I
409
( NN
IF U I
WRI TE
510
599
(10>/P
d
< 1 0 ) , m p s <10>/Summp? < 0 : 1 0 ) #
t
0Tm < 5 0 > / C u r v (50>
.EQ.
0.0)
GO TO
409
DO 41 I J = I , N N
MS( J) = F S U > * ( D ( J ) - H / 2 . 0 >
SUMMS( J) =MS( J) + § UMMS( J - I )
MC = ( ALPHA*FC1*AK*B*C>* ( H / 2 . O-GAMMA=O
TOTM ( I ) = MC + SUMMPS( NP) + SUMMS(NN)
CURV(I)
500
p fs
DIMENSION F S ( 1 0 ) , D ( I O ) , MS ( I O ) , S U MMS ( 0 : 1 0 )
I F ( I . GT. I.) GO TO 163
DO 161 L = 1 , 5 0
TOTM(L) = 0 . 0
CURV(L) = 0 . 0
SUMMPS(O)=O. O
SUMMS(O)=O.O
IF (NP . E 0 . 0 . 0 ) GO TO 413
DO 401 J = I , NP
MPS( J) = P F S ( J ) * ( P D ( J ) - H / 2 . ) .
SUMMPS(J) = MPS( J ) + SUMMPS( J - I )
CONTINUE
=
EC/ C
. EQ.
1)
(108,500)
GO TO 599
TOTM(I)
FORMAT ( / / , 5X, 9HMOMENT I S , 3 X , F 1 5 . 3 )
WRITE ( 1 0 8 , 5 1 0 ) CURV(I)
FORMAT ( / / , 5 X, 1 2 HCURVATURE I S # 3 X , F 1 0 . 8 )
CONT I NUE
RETURN
END
SUBROUTINE INPUT ( A X I A L , B # H # C 0 , C 1 , E P B , E P C , E P U , F P p , F P C , F P U , E P , N P , F C
$ 1 , P S , B 2 , S H , C C , P A S , P D , E P E , E C E , Z , F Y H , N M, E Y , E S H , E S U # F Y , F S U , M0 E 5 , N N , A S
$, D,II,KK,CF,KKK,EPY,KI)
DIMENSION P A S ( I O ) , PD( I 0 ) , AS( I O) ,.D( I O)
C
C
C
THIS
111
100
1 01
107
104
102
802
2 01
S UB R O U T I N E
READS
THE
INPUTTED
VARIABLES
I M P L I C I T REAL ( M)
I I , K K , KKkOKI
READ ( 1 0 5 , 1 1 1 )
FORMAT ( 4 1 2 )
READ ( 1 0 5 , 1 0 0 )
AXI A L , B , H , CU, Cl
FORMAT ( F I 0 . 1 , 4 F l 0 . 3 )
READ ( 1 0 5 , 1 0 1 )
E P 8 , E P C , E P U , F P B , F P C , F P U , ER
FORMAT ( 3 F I 0 . 6 , 3 F 1 0 . 3 , F l 0 . 1 , 1 2 )
E Y , E S H , E S U , F Y , F S U , MOE S r NN
READ ( 1 0 5 , 1 0 7 )
FORMAT ( 3 F I 0 . 6 , 2 F 1 0 . 3 , F 2 0 . 3 , I 2 ) .
I F ( NP . EQ. 0 ) GO TO 2 0 1
READ ( 1 0 5 , 1 0 4 ) E P E , E C E , C F , E P Y
FORMAT ( 4 F l 0 . 6 )
DO 8 0 2 1 = 1 , NP
PAS( I ) , R D ( I )
READ ( 1 0 5 , 1 0 2 )
FORMAT ( 2 F I 0 . 3 )
CONT I NUE
CONT I NUE
I F ( NN . EQ. 0 ) GO TO 2 0 2 '
94
109
809
202
103
C
C
C
DO 809 I = I , NN
READ ( 1 0 5 , 1 0 9 ) A S ( I ) , D ( I )
FORMAT (2 FI Ot 3)
CONTINUE
CONTINUE
READ ( 1 0 5 , 1 0 3 ) F C 1 , P S , B 2 , S H , C C , Z , F Y H , N M
FORMAT ( F 1 0 . 3 , 3 F 1 0 . 5 , 3 . F 1 0 . 5 , I 2 )
RETURN
END
SUBROUTINE OUTPUT ( N P , T P E S , P S S , P F S # C , C O M P , A X I A L , P A S , P D , ^ S , S S , F S # A
$S, D, NN)
DIMENSION T P E S ( I O ) , P S S ( I O ) , P F S ( I O ) , P A S ( I O ) , PD( I O)
DIMENSION E S ( I O ) , S S ( 1 0 ) , F S ( 1 0 ) # A S ( I 0 ) , D ( 1 0 )
THIS SUBROUTINE PRI NTS OUT INFORMATION
601
600
610
620
622
602
603
625
626
630
640
650
C
C
C
C
I F (NP „EQ. 0 . 0 ) GO TO 622
WRITE ( 1 0 8 , 6 0 1 )
FORMAT ( / / / , T3 O, ' P RE STRES SE D S TEEL' )
WRITE ( 1 0 8 , 6 0 0 )
FORMAT ( / / / , T l O , ' STEEL L O C A T I O N ' , T 3 0 , ' S T R A I N ' , T 4 5 , ' S T R E S S ' , T 6 0 , ' F 0
$RCE'#T75#'AfiEA',T85,'0*)
DO 620 J = I , NP
WRITE ( 1 0 8 , 6 1 0 ) J , T P E S ( J ) , P $ S ( J ) , P F S ( J ) , P A S ( J ) , P O ( J )
FORMAT ( 1 0 X , I 5 , 1 0 X , F 1 0 . 5 , 5 X , E i p . 3 , 3 X , F 1 5 . 3 , 1 X , F 1 0 . 3 , 1 ) f , f i p . 3 )
CONTINUE
CONTINUE
IF (NN . EQ« 0 . 0 ) GO TO 626
WRITE ( 1 0 8 , 6 0 2 )
FORMAT ( / / / , T3 0 , ' NONPREST RE S SE D S TEEL' )
WRITE ( 1 0 8 , 6 0 0 )
DO 625 J = 1 ,NN
WRITE ( 1 0 8 , 6 0 3 ) J , E S ( J ) , S S ( J ) , F S ( J ) , A S ( J ) , D ( J )
FORMAT ( 1 0 X , l 5 , i p x , F 1 0 . 5 , 5 X , F 1 0 . 3 , 3 X , F 1 5 . 3 , 1 X , F 1 0 t 3 , 1 X , F 1 0 - 3 )
CONTINUE
CONTINUE
WRITE ( 1 0 8 , 6 3 0 ) C
FORMAT ( / / , T20 , ' NE UT RAL AXIS I S ' , 3X , F7.-.4)
WRITE ( 1 0 8 , 6 4 0 ) COMP
FORMAT ( / / , T 2 0 , ' C O N C R E T E COMPRESSION FORCE I S' , 3Xf F I 2 . 3 )
WRITE ( 1 0 8 , 6 5 0 ) AXIAL
FORMAT ( / / , T 2 0 , ' AX I AL FORCE IS * , 3 X , f 1 5 . 3 )
RETURN
END
SUBROUTINE SUMMARY ( TOTM, CURV , EPE, ECE, NM, CF , FC I , BG, HG, CURVY)
DIMENSION TOTM( SO) , CURV( SO) , CTOTM( 5 0 ) ,CURVZ( SOJ
DIMENSION CNTOTM(SO), CNCURV(SQ)
THIS SUBROUTINE PRINTS A SUMMARY OF STRAI NS,
AND CURVATURES
700
393
933
MOMENTS,
WRITE ( 1 0 8 , 7 0 0 )
FORMAT C l ' , T 3 5 , ' S U MMA R Y ' )
IF ( NM) ' 3 9 3 , 3 9 4 , 3 9 5
WRITE ( 1 0 8 , 9 3 3 )
„
FORMAT ( / / , T 1 0 , ' O R I G I N A L STRESS-STRAIN CURVE GO TO 945
KENT 8 PARK' )
■
95
394
943
395
953
945
705
706
704
701
710
WRITE ( 1 0 8 / 9 4 3 )
FORMAT ( / / / T I O / ' STRESS-STRAIN CURVE WITH STRENGTH INCREASE ' )
GO TO 945
WRITE ( 1 0 8 / 9 5 3 ) '
FORMAT ( / / / T l 0 / ' HIGH STRAIN RATE STRESS ■!•ST RA IN CURVE WITH STRENGT
$H INCREASE' )
CONTINUE
IF (ERE .ECU 0 : 0 ) GO TO 701
WRITE ( 1 0 8 / 7 0 5 ) ERE
FORMAT ( / / , T S z ' EFFECTIVE STEEL STRAIN I S ' / 3 X , F I 0 . 6 )
WRITE ( 1 0 8 / 7 0 6 ) ECE
FORMAT { / / , T5/ ' PRESTRESSEP CONCRETE STRAIN I S 1 / 3 X / F 1 0 . 6)
WRITE ( 1 0 8 / 7 0 4 ) CF
FORMAT ( / / , Tl 0 / ' STRAI N COMPATIBILITY FACTOR I S " / 2 X , FI Q . 6)
CONTINUE
WRITE ( 1 0 8 / 7 1 0 )
FORMAT ( / / / T I O / * CONCRETE STRAI N*/ T30 /' MOM ENT' / T 4 2 / " CUR VATU RE' / T 6 ( ) /
$ ' MOMENT*/T7 0 / ' MOMENT' / T80/ ' CURVATURE' / T 9 ? / ' CURVATURE' )
1
WRI TE ( 1 0 8 / 7 1 4 )
.
FORMAT ( T l 5 / ' I N / I N « » T 2 6 / * I N * L 8 « / T 4 5 / ' I / I N « , T 6 1 ' E T - K I P « / T 7 3 , « NO ND I M
'
$ENSI0NAL'/T95/'DUCTILITY'////)
DO 8 0 3 . J = 1 / 5 0
EC = 0 . 0 0 1 * J + . 0 0 2
CTOTM(J) = T OTM( J ) / 1 2 0 0 0 . 0
CNT0TM( J ) =T0TM( J ) / ( FC1*BG»HGAHG>
CNCURV(J ) =CURV(J ) *HG
CUR VZ ( J ) = C URV( J ) /CURVY
WRITE ( 1 0 8 / 7 2 0 ) ECz TOTM( J ) / CU R V( J ) / CTOTM( J J / CNTOTM( J ) / C NCURV( ^ ) /
SCURVZ(J)
720
FORMAT ( 1 0 X / F l 0 . 5 / 5 X / . F 1 5 . 3 / 3 X / F 1 0 . 8 / 2 X # F 1 0 » 3 / 2 X / F - 5 » ? / 5 X / F 1 0 . 8 / 3 X / F
$8.3)
800 CONTINUE
RETURN
END
SUBROUTINE CALCOMP ( TOTM r CU RV/H E AD I NG/ KK / HG/BG / F CI/P)^ CUR V/ PY TO TM/ L
S / NP T S / J K/ CURVY/ KI )
DIMENSION T 0 T M( 5 0 ) / C UR V ( 5 0 ) / P XC U R V ( 5 2 ) / P Y T OTM < 5 2 ) / HE ADI NG( I O)
DIMENSION L ( I O )
c c THIS SUBROUTINE PLOTS THE MOMENT CURVATURE CURVE
C
IF ( KK . E Q . 0) GO TO 464
DO 747 KL = I / 5 2
RXCURV(KL)= 0 . 0
747
PYTQTM( Kl ) = O . 0
KJ = O. 3
DO 474 K J = I / L ( J K )
t
|
;
j
714
pxcurv
(
k j
)=
curv( k j
)
•
474
PYTOTM(KJ)= ABS ( T O T M( KJ ) / 1 2 Q 0 0 . 0 )
I F (KK . EQ. I ) GO TO 512
DO 475 K J = I z L ( J K )
475
PYTOTM ( K J ) = ABS( TOT M( K J ) / ( FC1 *BG* ( HG * * 2 . 0 ) ) )
IF (KK . LE. 2) GO TO 512
IF (KK . E Q . 3) GO TO 464
DQ 213 K J = I z L ( J K )
PXCURV( KJ ) =CURV( KJ ) /CURVY
PXCURV( K J ) = P X C U R V ( K J ) * HG
213
1
96
512
I F ( K K . EQ. 5> GO TO 4 6 4
CONT I NUE
CAL L P L O T S ( 0 / 0 / 8 )
CAL L NEURON ( 2 )
CAL L FACTOR ( 2 . 5 4 )
NPTS=L(JK)
N P T S I = L ( J K> +1
NPTS2 = L ( J K ) + 2
CAL L SCAL E ( P XCURV / 7 . 0 / N P T S / 1 )
CAL L SCAL E ( P Y T O T M O / N P T S / 1 )
CALL^AXIS
( 0 . 0 / 0 . 0 / ' CURVATURE
$ E a L L CA X I S 2 ( 0 . 0 , 0 . 0 / ' MOMENT
-
I / I N ' / - 16 / 7 . 0 / 0 . 0 / f X C U R V ( N P T S I ) / PX
^ ■F T - K I P ' / 1 5 / 5 . 0 / 9 0 . p / P Y T 0 T M ( N P T S 1 ) / P Y T
$0TM(NPTS2))
596
c ALL0 AXI S
$ C all
595
AXI S
( 0 . 0 / 0 . O , ' MOMENT' , 6 / 5 . 0 / 9 0 . 0 / P Y T O T M ( N P T S I ) / P Y T Q T M C N P T S 2 )
$)
C ONT I NUE
,
.
, ..
CAL L F L I N E ( PX CURV / PY TO T M/ - N P T S / I / K I / 4 )
CAL L SYMBOL ( 2 . O / I - 0 / . I 4 > HE A 0 I N G ( I ) / 0 . O / 8 O )
C A|_L 0 LQT
464
( 0 . 0 , 0 . 0 , ' C U R V A T U R E ' , - 9 , 7 . 0 , 0 . 0 , P X C U R V ( N P T < ; n , PXCURV UI P T
(12,0/0.0/999)
CONT I NUE
RETURN
SUB ROUT I NE
CALSUM
( X AR R A Y , Y A R R A Y , I K , HE A D t N G / L z P X Y C U R Y f p v XTOTM> NCPT
$S,NCPTS1/NCPTS2,KKK,KI)
C
C
C
C
THIS SUBROUTINE PLOTS COMBINED NO ND IME NS I QN ALI ZE
MOMENT - CURVATURE CURVES.
XARRAY( 5 0 2 ) , YARRAY( 5 0 2 ) , PXCURVC5 % ) , PYTOTM( Sf )
DIMENSION
DIMENS ION P X Y C U RV( 1 0 , 5 0 ) , PYXTOTMC1 0 , 5 0 ) , L ( 1 0 ) , I NTE( I O)
DIMENSION HEAD I NG( I Q)
I NTE( I ) =O
I NT E C2 ) =1
I NTE ( 3 ) = 2
I NTE( 4 ) = 3
I N T E ( S ) =4
I NTE( 6 ) = 5
I NTE( 7 ) = 6
I NT E ( S ) = I I
I NT E ( 9 ) = 1 2
INTE(IO)=I3
CA|_L PLOTS ( 0 / 0 / 8 )
CALL FACTOR ( 2 . 5 4 )
CALL SCALE ( XARRAY , 7 . 0 / NCPTS/ I )
CALL SCALE ( YARRAY/ 5 . O/ NC PT S / I )
IF (KKK . E U . 2) GO TO 313
- 6 * H ' , - 1 5 , 7 . 0 , 0 . 0 , XARRAYtNC PT S I ) , XA
CALL AXIS ( 0 . 0 / 0 . 0 , 'CURVATURE
SRRAYCNCPTS2))
GO TO 314
DUCT I L I T Y* , - 1 9 , 7 . b/ 0. 0, XARRAY( NCPTS1
CALL AXIS ( 0 . 0 , 0 . 0 , 'CURVATURE
313
$) , XARRAY( NCPTSf ) )
CONTINUE
31 4
97
46
43
CALL AXIS ( O . O / O . O z * MOMENT - M/ FC* B* H* Hl » 1 9 # 5 . 0 z 9 0 . 0 z YARRAY( NCP TS 1
$ ) zYARRA Y( NCPTS2) )
DO 43 J = 1 , ns
OO 46 K J = I z L ( J )
NTI =L ( J ) -M
NT2=L( J) +2
PXCURV( KJ) =PXYCURV( JzKJ)
PXCURV(NT1)=XARRAY(NCPTS1)
PXCURV(NT2) =X ARRAY( NCPTS2)
PYTOTM( KJ) =PYXTOTM(JzKJ)
PYTOTM(NTI)=YARRAY(NCPTSI)
PYTOTM(NT2)=YARRAY(NCPTS2)
CALL FLINE ( PXCURV , PYTOTMz- L( J >z 1 z KI z I NTE ( J ) )
CONTINUE
CALL SYMBOL ( 2 . Oz 4 . 5 z . 1 4 z HEADI NG( I ) z O . Oz 8 O)
CALL “ LOT ( 1 2 . Oz0 . Oz9 9 9 )
RETURN
END
MONTANA STATE UNIVERSITY LIBRARIES
3
762 100 1356 O
D378
SuT6 S u p r e n a n t , B . A.
c o p .2
C u r v a tu r e d u c t i l i t y o f
r e i n f o r c e d an d p r e s t r e s s e d . .,
DATE
IS S U E D TO
D378
Su 76
cop. 2