MOMENT-DEFORMATION
CHARACTERISTICS
OF PRETENSIONED CONCRETE
BEAMS SUBJECT TO
FLUCTUATING LOADS
D. D. 0. Paranagama Imperial College of Science
and Technology
London, England
A. D. Edwards
Imperial College of Science
and Technology
London, England
The limit state of collapse of noncyclically loaded prestressed and reinforced concrete beams and frames
has been the object of much research
in recent years. Most structures,
however, are subject to large variations in load intensity and load configuration during their useful life.
Current design practice assumes that
any adverse effects due to load history will be offset by the load factor
provided in the design. New design
procedures, which use improved
methods of analysis, aim at more
economical designs and consequently reduce the true load factor. Thus
there is a need for the investigation
of the effects of load history on the
ultimate load carrying capacity of
concrete structures.
Structures, when subjected to repeated load cycles in which the
loads act in one direction only, have
collapsed even when the maximum
load in any cycle is less than that
which would cause failure under
non-cyclical static loading. Under
such loading conditions failure may
62
be by fatigue or due to incremental
deformation.
Fatigue failure is associated with
deterioration in the load carrying
capacity of a material when it is subjected to a large number of cycles of
loading. Fatigue life of prestressed
concrete beams is not considered in
this report.
Incremental collapse, which is a
failure due to the accumulation of
deformation caused by formation of
plastic hinges in the course of a loading cycle, is well established in
structural steel theory. Investigations of this type of collapse in concrete structures must be carried out
at two levels: 1) the effect of repeated loading on the stress-resultant deformation characteristics of
sections; and 2) the effect on the
load-deflection characteristics of
structures. Studies of the behaviour
of reinforced concrete members
which have been subjected to a
small number of overload cycles
have been reported by Aoyama,
Burns, Agrawal, Sinha and RasmusPCI Journal
The results of tests on eight rectangular pretensioned concrete
beams are presented. The beams represent under-reinforced,
over-reinforced and balanced design cases. Five beams were
subjected to cyclic loading; the remainder were subjected to
statically increasing load until failure. An outline of an analytical
procedure to obtain the envelope and unloading and reloading
characteristics is presented.
sen (Refs. 1-6). Corresponding work
on prestressed elements has been reported by Abeles(7, ․).
SCOPE OF INVESTIGATION
The primary objective of the experimental investigation was to determine the effect of a small number
of, repeated loadings on the momentdeformation characteristics of pretensioned rectangular beams.
Tests on eight beams are reported.
Four beams, A l , A2 , A 3 and A4,
were of balanced design; two beams,
B l and B 2 , were under-reinforced;
and two beams, C l and C 2 , were
over-reinforced. The first beam from
each group was tested statically to
collapse. The remainder of the
beams were subjected to four cycles
of loading at each of four levels of
intensity. An ideal program for the
investigation would have involved
testing a large number of beams,
each subjected to the same number
of load cycles but having different
load intensities applied. Such a program was beyond the time and casting facilities available. The compromise program was restricted in scope
to permit any one test to be carried
out in a maximum of ten hours.
A computer program was written
to obtain the theoretical momentcurvature relationship under noncyclic and cyclic loading. The experimental and theoretical results
were compared and the latter adjusted to obtain the best correlation.
TEST SPECIMENS
The eight rectangular preten-
Table 1. Concrete strengths
Beam
Cube strength, psi
Cylinder strength, psi
August 1969
A,
Az
A3
A4
B,
B2
C1
C2
7900
5065
7800
5500
7600
5550
7460
5100
8210.
5430
8390
5500
7130
4010
6580
3850
63
Table 2. Details of beams
Beam
A, A2 A3
A4
B, B 2
C, C2
Overall
depth, D
Width
b
in.
in.
8
6.625
8
6
4
4
4
4
sioned beams were cast in pairs.
Concrete proportions by weight
were 1:1.2:2.8 with a maximum aggregate size of 3/s in. and a watercement ratio by weight of 0.56. Rapid hardening Portland cement and
Thames Valley aggregate and sand
were used. Average values of three
tests of 6-in, cubes and 6 x 12- in.
cylinders, tested on the same day as
the corresponding beam test, are
given in Table 1. Dimensions, details
and reference numbers of the beams
are given in Table 2.
The longitudinal reinforcement
was 0.2 in. diameter indented high
tensile steel wire having an ultimate
strength of 246,000 psi and a 0.1 percent proof stress of 214,000 psi at a
Effec- No. of
0.2-in,
tive
depth, d wires
in.
5.390
5.380
5.440
4.045
Depth Spacing
of
of
stirrups stirrups
in.
in.
3
3
3
4
6.0
5.5
6.0
5.0
4.5
4.5
4.5
4.5
strain of 0.0087. The shear reinforcement consisted of 1/a in. diameter
mild steel bars of 48,700 psi yield
stress bent into a square wave form.
Initial stress conditions, together
with the calculated final prestressing
force, are given in Table 3.
LOADING APPARATUS
AND INSTRUMENTATION
The loading frame, the 20-ton
Amsler jack, the spring dynamometer loading cabinet and the Solatron
data logger are shown in Fig. 1. The
test specimen was supported on concrete pedestals by means of a hinge
and a roller bearing.
Thirty PL 30 electrical resistance
strain gauges and twelve PL 60
Table 3. Details of prestress
Initial
prestress
force
Ib. wire
Final
prestress
force
lb./wire
Initial
strain
in
steel
in./in.
A,
A2
A3
3720
3150
0.00335
A4
3720
3000
0.00318
B
B2
4800
4100
0.00431
4700
3900
0.00411
Beam
C,
C2
64
Initial concrete
stress distribution
(compression is +)
psi
+37
(^
+253
—250
+940
+11
+1015
[N
+6
+1290
PCI Journal
gauges were fixed to each beam, together with a number of clinometers
and dial gauges, as shown in Fig. 2.
In the case of beams B2 and C2, two
additional PL 30 gauges were attached to the top surface in line with
the gauges on either side of the center line.
The load was recorded by means
of a dynamometer placed between
the beam and Amsler jack. The dynamometer and electrical resistance
strain gauges were connected to the
Solatron data logger which was operated at two channels per second.
TEST PROCEDURE
Beams Ai, B l and C1 . These
beams were subjected to statically
increasing load to failure. The applied load was increased by increments of approximately 15 percent
of the ultimate load capacity of the
beam until the first crack appeared;
thereafter, the load increments were
reduced to approximately 5 percent
of the ultimate load of the beam. A
complete set of readings was taken
at each load increment. The duration of the test was about six hours.
Beams A2, A3, A4, B2 and C2.
Each beam was subjected to sixteen
load cycles. Each was first loaded to
60 percent of the ultimate load of
the corresponding comparison beam
and unloaded completely, except for
beam A3 . Four loading cycles were
made at this intensity. A complete
Fig. 1. General view of test rig
August 1969
66
0 . 25L
(0.25L-
5) 5
5` (0 .25L-5')
5_
0.25 L
5~
POD — PL 0 — 19 ?^Q 5—
M
2 z^ 2 z" 2 z" 2 z'
Section
I
2
I
1
q
I
1
3'
I
2
^N
3 i"
I
3
I
4
Fig. 2. Locations of test gauges
set of readings: was: taken on the first
and fourth cycle with readings taken
at about six stages during both the
loading and unloading phases. Readings were also taken at the maximum
and minimum load intensities of
each load cycle. The procedure was
repeated with maximum load intensities of 70, 80 and 90 percent of the
ultimate load and the beam was finally loaded to failure. The minimum load intensity of the cycles for
beam A3 was 40 percent of the ultimate load of the comparison beam.
Each load stage occupied about five
minutes. Thus the first and fourth
cycle lasted about an hour and the
second and third cycles about ten
minutes. The total duration of each
test was about nine hours.
Test on prestressing wire. Three
66
random specimens of the prestressing wire were tested in order to obtain the load-strain curve under
repeated loads. Each specimen was
subjected to three cycles of loading
before loading to failure. The minimum load in each cycle was approximately 4000 lb., the expected effective prestress force in the wires. A
complete load-strain history was obtained by measuring load and strain
at frequent intervals during the loading operation.
THEORETICAL MOMENT -CURVATURE
RELATIONSHIP
The behaviour of a composite section is. dependent upon the behaviour of component materials, their
size and relative positions and strain
distribution across the section.
PCI Journal
The stress-strain relationship for
concrete which is subjected to a noncyclic increasing stress was assumed
to be parabolic-rectangular. For concrete subjected to cyclic stresses the
envelope curve was assumed to be
the same and the unloading and reloading curves were assumed to be
linear and identical (see Fig. 3). The
load-strain curve for the steel was
obtained experimentally (Fig. 5) and
was. represented by the algebraic expressions given.
The normal moment-curvature relationship for the untracked phase
was assumed to be linear. The relationship for the phase after cracking
was established by a trial and error
procedure. In this case a compression strain in the extreme concrete
fiber was assumed and the corresponding moment and curvature derived.
Since the envelope curve for a section subjected to cyclic loading was
assumed to be identical with the
normal static loading curve, the reloading curve traced the normal
curve once the previous maximum
load was exceeded.
The unloading and reloading
curves were obtained using the following assumptions in addition to
those already mentioned:
1. The moment, curvature, j¢,
and neutral axis depth at which
the change of loading takes place
is known.
2. The change of strain in the
concrete fibers has a linear distribution across the section.
3. The change in the steel
strain, e, corresponding to the
incremental change in curvature
is given by
e8C =F(4 d - z)
where F is the strain compatibility
factor, d is the effective depth and
August 1969
Strain
Fig. 3. Stress- strain curve of concrete
z is the change in strain in the extreme concrete fiber.
4. The effective modulus of
elasticity of the concrete during its
loading and reloading cycle is unaffected by its previous cracking
history.
5. The concrete fibers in the top
of the section can sustain a small
tensile stress.
Fig. 4 shows typical changes in
the strain and stress distributions
that are assumed to occur when the
beam is unloaded.
The following trial and error procedure was used to obtain the unloading and reloading curves:
1. A value was assumed for the
incremental change in curvature,
0 1.
2. A value was assumed for the
corresponding change in strain in
the extreme concrete fiber in compression, z.
3. The change in the total force
in the concrete was evaluated.
4. The change in the total force
in the steel was evaluated.
5. If equilibrium was satisfied
the values in Step 6 were calculated; otherwise a new value for z
was assumed and the procedure
repeated.
6. The resultant bending mo67
E CA
6C8
Z
C
YC
AYA
A
\
Ya,
Ys
L
Initial Strain
Distribution
Change in Strain
Distribution
l
_R
// Initial
kYA
YA1
Final Strain
Distribution
y^—
yB^
final
Tc
Initial stress
Distribution
Change in Stress
Distribution
Final Stress
Distribution
Fig. 4. Typical stress and strain distributions
ment, curvature and neutral axis
depth found by this procedure
were used as the initial values for
the next increment.
TEST RESULTS
Prestressing wire. The load-strain
history obtained from the three specimens is given in Fig. 5. The average
modulus of elasticity obtained for
the load history shown was
E 1 = 30.30 x 106 psi
E2 = 29.75 x 106 psi
E 3 = 28.00 x 106 psi
The envelope of the load-strain
curve was represented by
68
P = 947,000 e
for 0:€:0.0056
P = 5300 + 94,700 (e — 0.0056)
—142,500,000 (E — 0.0056)2
for 0.0056 : e : 0.0087
P = 6900 + 60,250 (e — 0.0087)
for e ? 0.0087
Beams. Zero external load condition was taken as the datum for all
results. Beam A4 was accidentally reloaded to near failure after the first
cycle of loading and the correct history of loading was subsequently
applied.
The load vs. deflection and load
vs. total rotation curves have been
PCI Journal
d
.3
0
0
0
J
2000
4000
bow
tuuu
Strain x 106
Fig. 5. Load-strain curve for 0.2-in, diameter wire
given elsewhere (s). These curves are
similar in form to the moment-curvature relationships given in Figs. 6 to
8 where the curvature is based on
the strain readings of the appropriate gauges. The curves shown are
for the central section of the beam
and for sections 1 and 2 in Fig. 2.
The envelope and unloading curves
for the cyclically loaded beams are
solid lines; the reloading curves are
dotted as are the curves for the corresponding statically loaded beams.
The cycle in which the given curve
was obtained is indicated.
Residual deflections for beams A2,
B 2 and C 2 are shown in Fig. 9.
DISCUSSION OF RESULTS
Compared to the statically loaded
BeamA-----___----------'''
10
,
Beam
,^,
/
80
o
x6
1
3
5
^
3
3
Section i
Section at
y I
400
800
9
/
9
1=
0
/
Beam A—
A,
Section 2
1200 1600 2000 2400 2800 3200 3600 4000 4400 4800
Curvature in' x 106
Fig. 6. Moment-curvature diagram for beam A2
August 1969
69
Beam B^^s----_—--------
^O
8
x
c6
13
5
g
16
Beam B^—^
^i
Beam
i //
13 16
5 9
i g 13
5
c 40
0
f y
Section 1
Section at
400
800
1200
1600 2000 2400 2800 3200
Curvature in' x 106
Section 2
3600 4000 4400
Fig. 7. Moment-curvature diagram for beam
specimens, there is no significant reduction in the ultimate load carrying
capacity nor in the ultimate deformation capacity of those beams subjected to the load history specified.
The curvature corresponding to the
maximum moment in any one set of
cycles increases, and the increase
per cycle decreases with the number
of cycles. The envelope curves for
those beams which were subjected
to cyclic loading do not vary appreciably from those of the non-cyclic
companion beams.
The moment-curvature curves for
the beams subjected to cyclic loading show that the unloading and reloading paths differ for any two adjacent cycles. The unloading curve
has three approximately linear
phases: the first phase is relatively
short and the beam exhibits a high
stiffness which decreases as load intensity increases; in the second and
by far the largest phase the beam
exhibits a stiffness which decreases
as the maximum load intensity to
which the beam has been subjected
increases; in the third and final unloading phase the beam exhibits a
70
B2
stiffness slightly less than the initial
uncracked phase. The reloading
path has two approximately linear
phases: in the first reloading phase
the beam has a stiffness which approximates that of the initially uncracked beam; in the second phase
the beam exhibits a stiffness similar
to that exhibited in the second phase
of unloading. The reloading path
crosses the unloading path at moments greater than 90 percent of the
maximum load intensity in the cycle.
The dependence of the stiffness in
the second reloading and unloading
phases on the maximum load to
which the beam has been subjected
is well shown by those curves relating to beam A4 (Fig. 10) which was
accidentally loaded to near failure
after the first cycle. Subsequent load
cycling at 60, 70, 80 and 90 percent
of ultimate load did not substantially
alter the stiffness in this phase.
The above behaviour suggests that
the moment-curvature characteristics
of a prestressed concrete section are
unique. Uniqueness here implies
that at any point in the momentcurvature plane there is a unique
PCI Journal
Im
I
C
E
0
f
400
800
1200 1600 2000 2400 2800 3200
3600 4000 4400 4800 5200
Curvaturerril X 106
Fig. 8. Moment-curvature diagram for beam C2
path which the loading or unloading
curves will follow. Sinha, et al have
shown that this property is reasonably true in the case of the stressstrain curve for concrete. Similar
behaviour has been observed by others for the case of reinforced concrete beams. It is emphasized that
such behaviour does not imply that
geometrically similar sections of a
beam or frame that has been subjected to some loading history will
follow similar moment-curvature
paths in the same cycle of loading.
This is well illustrated in Fig. 7 by
the unloading path in cycle 13 for
the three sections given for beam B2.
The plots of residual deformation
against number of cycles, Fig. 9,
shows that the residual deformations
rapidly tend to a limiting value
when the maximum load intensity of
the cycles is less than 90 percent of
the ultimate load.
COMPARISON OF EXPERIMENTAL
AND THEORETICAL CURVES
Fig. 11 shows a typical theoretical
curve obtained by the procedure
outlined above. In this case, the
August 1969
moduli of elasticity of the steel and
concrete and the compatibility factor were kept constant. One result is
that the unloading and reloading
curves coincide. It is possible to
make the theoretical and experimental moment-curvature curves, which
correspond to statically increasing
load, coincide by varying the value
of the compatibility factor. The variation in F required to achieve this
for beam B 2 is shown in Fig. 11. It
can be seen that F assumes a rather
high undefined value when the beam
first cracks and then decreases rapidly during the initial stage of the
post-cracking phase. Finally F tends
to a value close to unity.
The theoretical stiffnesses of the
beam in the cracked unloading and
reloading phase as given in Fig. 11
do not agree with the corresponding
experimental values obtained. The
parameters which influence theoretical stiffness in the cracked unloading
phase are:
1. The value of the neutral axis
depth from which the unloading
commences.
2. The modulus of elasticity of
71
VIo
,
X
90%3^o
O
p
C)
X
X
CQ
t0
o'
8 0y
0
d
d
90
90./
of
1
80%
70°
1 23 4
No of cycles
1
2 3 4
No of cycles
1
2 3 4
No of cycles
B2
A2
80°/^
o--
Z2'
VIN
C2
Fig. 9. Residual deflection vs. number of cycles
be expected. A significant change in
stiffness in the unloading and reloading cracked phase could only be
brought about by varying the compatibility factor.
The values of F required in beamB 2 to make the theoretical unloading
curves have similar stiffnesses to the
experimental curves are shown in
Fig. 12. These values are of a similar order to those required to make
the theoretical and experimental
the concrete.
3. The modulus of elasticity of
the prestressing steel.
4. The compatibility factor, F.
The influence of the first parameter on the stiffness is negligible as
seen in the unloading curves obtained. It was also found that change
in stiffness was not significantly affected by varying the modulus of
elasticity of both the steel and the
concrete over the range of values to
Beam A„
120
Beam A1
-------- `'— _1—
\yam'
100
mo 80
x
c 60
40 l
ice/
13
5
/i
9
131i
/
/
//
^
1
i
3
1
o
Section at
400
800
1200
q
Section 1.
1600 2000 2400 2800 3200
Curvature
3600 4000 4400 4800 5200
in' x 106
Fig. 10. Moment-curvature diagram for beam A4
72
PCI Journal
non-cyclic static load curves coincide.
CONCLUSIONS
1. The ultimate load carrying capacity and ultimate load deformation capacity of pretensioned rectangular beams subjected to cyclic
loading was not significantly different from their capacity under statically increasing load.
2. Subsequent to loading beyond
the static cracking load, the unloading and the reloading moment-deformation characteristics can be
approximated by a hi-linear relationship; the two stiffnesses are approximately those corresponding to the
untracked and the appropriate
cracked phases.
3. The stiffness of the cracked
phase is governed mainly by the
maximum load intensity to which the
beam has been subjected during its
load history.
4. The decrease in the stiffness in
the cracked phase can be attributed
to a variation in the compatibility
factor since the effect of the other
parameters appears to be small.
5. The envelope curve can be reasonably represented by the curve
corresponding to non-cyclic increasing load.
REFERENCES
1. Aoyama, H., "Moment Curvature Characteristics of Reinforced Concrete Members Subjected to Axial Load and Reversal of Bending Moment," Flexural
Mechanics of Reinforced Concrete, Proceedings of the International Symposium, Miami, Fla., November 10-12,
1964.
2. Burns, N. H. and Siess, C. P., "Repeated and Reverse Loading in Reinforced Concrete," Journal of the Structural Division, Proceedings
of
the
10
LL 5
0
C_
400
800
1200
1600
2000 2400 2800 3200
Curvature my x 106
3600
Fig. 11. Theoretical moment-curvature diagram for beam B and required variation in F
August 1969
73
c
0
Curvature
in-' x 106
Fig. 12. Theoretical unloading momentcurvature diagram with varying F for
beam B
American Society of Civil Engineers,
Vol. 92, No. ST5, October 1966, pp.
65-78.
3. Agrawal, G. L., Tulin, L. G. and
Gerstle, K. H., "Response of Doubly
Reinforced Concrete Beams to Cyclic
Loading," American Concrete Institute
Journal, Proceedings Vol. 62, No. 7,
July 1965, pp. 823-835.
4. Sinha, B. P., Gerstle, K. H. and Tulin,
L. G., "The Response of Singly Reinforced Concrete Beams to Cyclic Loading," American Concrete Institute Journal, Proceedings Vol. 61, No. 8, August
1964, pp. 1021-1038.
5. Sinha, B. P., Gerstle, K. H. and Tulin,
L. G., "Stress-Strain Relations for Concrete under Cyclic Loadings," American Concrete Institute Journal, Proceedings Vol. 61, No. 2, Feb. 1964, pp.
195-212.
6. Rasmussen, B. H., "Incremental CoIlapse of Ordinary Reinforced Concrete
Beams," International Association for
Bridge and Structural Engineering, Vol.
16, 1956, pp. 439-456.
7. Abeles, P. W., "Report on Prestressed
Concrete Sleepers Tested as Simply
Supported Beams," Concrete and Constructional Engineering, Vol. XLII, Nos.
4 and 5, April and May 1947, pp. 123132 and 155-161.
8. Abeles, P. W., "Static and Fatigue
Tests on Partially Prestressed Concrete
Constructions," American Concrete Institute Journal, Proceedings Vol. 51, No.
12, Dec. 1954, pp. 361.
9. Paranagama, D. D. 0., "Pretensioned
Prestressed Beams under Repeated
Overloads," thesis submitted for the degree of M. Phil., University of London,
1967.
Discussion of this paper is invited. Please forward your discussion to PCI Headquarters
by November 1 to permit publication in the February 1970 issue of the PCI JOURNAL.
74
PCI Journal