DELFT UNIVERSITY OF TECHNOLOGY
DEPARTMENT OF CIVIL ENGINEERING
Report 5-84-1
Constant-amplitude tests on plain concrete in
uniaxial tension and tension-compression
Dr.lr. H.A.W. Cornelissen
STEVIN LABORATORY
Delft University of Technology Report 5-84-1 Department of Research No. 7804 Civil Engineering January 1984
Constant-amplitude tests on plain concrete in uniaxial tension and tension-compression
Adron
Dr.Ir. H.A.W. Cornelissen
Tedbnische Hogeschool
Jh»m= Bibliotheek
AideUng: Civiele Techniek
3 r^ó'^- 8£/^ \ Stevinweg 1
^ ' f postbus 5048 2600 GA DeMt
Mailing address: Technische Hoqeschool D e l f t Vakgroep Betonconstructies Stevinlaboratorium 2
Stevinweg 4
2
-Acknowledgements
This investigation has been carried out in the Stevin Laboratory of the Delft University of Technology in close co-operation with CUR-VB (Netherlands Committee for Research, Codes and Specifications for Concrete). The financial support of CUR-VB and MaTS (Marine Techno-logical Research) and the discussions of the members of CUR-VB Committee C-33 are gratefully acknowledged.
The author wishes to thank Ing. G. Timmers for the computer calculations and the development of the test equipment, W.J. van Veen for perform-ing the tests and Ir. H. Voorsluis of the Measurperform-ing and Instrumenta-tion Group for designing the microcomputer for signal generating and data acquisition.
• fi,
i
No part of this report may be published without the written permission of the author.
CONTENTS page Notation 5 Summary 6 1 INTRODUCTION 7 2 LITERATURE SURVEY 9 3 OBJECTIVES 11
4 CONCRETE COMPOSITION AND PREPARATION OF THE SPECIMENS 12
4.1 Concrete composition 12
4.2 Specimens 13
5 TESTING EQUIPMENT 14
6 CENTRICITY CONTROL OF THE LOADING EQUIPMENT 15
6.1 General 15
6.2 Starting points 15
6.3 Measuring device 16
6.4 Eccentricity test of loading machine 17
6.5 Static eccentricity tests 17
6.6 Dynamic eccentricity tests 18
7 PERFORMANCE OF THE EXPERIMENTS 20
8 RESULTS OF STATIC TESTS 21
9 RESULTS OF DYNAMIC TESTS 23
9.1 General 23
9.2 S-N diagrams 23 9.3 Goodman diagram 32 9.4 Cyclic strain 33
page
9.5 Reloading strength of run-outs 34
9.6 Fracture locations 35
9.7 Statistical distribution of number of cycles to 36 failure
10 EVALUATION OF SECONDARY CREEP VELOCITY IN RELATION 40 TO FATIGUE LIFE OF CONCRETE
11 APPLICATION OF A FRACTURE MODEL BASED ON THE MAGNITUDE 47 OF CYCLIC CREEP
12 CONCLUSIONS 50
REFERENCES ' 52
APPENDICES:
A, Static test results for B 45 concrete 55
B, Static test results for low quality concrete 56
C, Dynamic tensile test results for B 45 concrete 57 at 6 Hz
D, Dynamic tensile test results for B 45 concrete 60 at 0.06 Hz
E, Dynamic tensile test results for low quality concrete 61 at 6 HZ
F, Cyclic strain results 62
G, Specimen code 74
H, Probability of failure 75
I, Secondary creep velocity 76
J, Reloading strength (m(t)) 77
Notation
D(n) - measure of destruction at n cycles
E - Young's modulus N/mm^
f' - static compressive strength N/mm^
f ^ - Static tensile strength N/mm^
f 1 - static splitting strength N/mm^
- indicates average value
— m ^
m(t) - factor representing preloading
N - number of cycles to failure
n - number of cycles
r - eccentricity in static tests mm
r , r . - average eccentricity at a , a . in dynamic tests mm
max min ^ •' max min "^
t^ - time to failure seconds
V - coefficient of variation
%
ei - strain at maximum stress in static tests
£. - instantaneous strain
é - secondary creep velocity per second
e^.^4. - total strain
tot
a„,>,, ö„- - maximum, minimum stress level N/mm^
max mm
(|)^ - creep coefficient
6
-Summary
This research report is the continuation of Stevin Laboratory Report No. 5-81-7 "Fatigue of plain concrete in uniaxial tension and in alter-nating tension-compression" [1], in which test set-up, loading equip-ment and preliminary results have been described.
The present report deals with the total set of constant-amplitude tests on concrete loaded in direct tension and in tension-compression. This set includes mainly fatigue tests at 6 Hz on specimens having one con-crete composition (quality B 4 5 ) . Tests were performed on "drying" as well as on "sealed specimens". Besides these tests in an additional program, the effect of concrete quality and test frequency was also studied. The results are presented in S-N relations and a Goodman diagram. It can be concluded from this diagram that stress-reversals cause extra damage, which significantly reduces fatigue life.
Based on cyclic deformation, which was recorded continuously during the fatigue tests, relations were established between fatigue life on the one hand and secondary creep velocity and ultimate cyclic strain on the other hand. Particularly the secondary creep velocity proved to be a good predictor for life of concrete subjected to fatigue loading
1 INTRODUCTION
In view of the increasing tendency to design slender structures with dead load forming only a small part of the total load capacity, and the use of new types of structures such as marine structures subjected to wind and wave loading, more attention is being focused on the fa-tigue behaviour of concrete. The interest concerns concrete structures as a whole, but also the constituent materials. This means that in-formation is needed with respect to the fatigue properties of con-crete. This information should include experimental results as well as theoretical modelling.
Because of the interest in the Netherlands in structures subjected to fatigue loadings, an extensive national research program has been executed. The experimental work has been carried out by the Institute TNO-IBBC for Building Materials and Building Structures (concrete in compression) and by the Delft University of Technology in the Stevin Laboratory of the department of Civil Engineering (concrete in direct tension). Also, the Ghent University in Belgium has cooperated with centre point bending tests.
This report deals with the direct tensile tests as being carried out in the Stevin Laboratory.
Knowledge of the tensile properties of concrete is important for many reasons. For example, in special structures tensile stresses are per-mitted; also, the tensile strength governs the cracking behaviour of concrete which affects the stiffness, damping and durability. Tensile properties are important, too, for the bond of the reinforcement and for shear behaviour.
Moreover, because of the increasing need for safety analyses of struc-tures, information on the tensile properties of concrete is needed.
In this report are presented the results of constant-amplitude tests on plain concrete subjected to repeated tensile and to alternating tensile-compressive loadings. The report is a continuation of the report "Fatigue of plain concrete in uniaxial tension and in alter-nating tension-compression", in which the loading system and the electronic test equipment have been described in detail and test results are given [1]. In this report only some improvements of the
8
-measuring system are indicated. Results are presented which were not published in [1]. The analyses, however, are based on the total set of results.
The experiments are direct tests which have the advantage that the stress in the cross-section is known, contrary to bending or split-ting tests where a stress gradient is present which changes during the course of the experiment because of stress relaxation.
The objectives of the research were to determine S-N relations for concrete in direct tension and in tension-compression. However, attention is also given to internal damage of concrete subject to fatigue, which may be represented by longitudinal cyclic deformations.
LITERATURE SURVEY
Two limited investigations on the fatigue of concrete in direct tension are known [2, 3 ] . In [2] constant-amplitude tests are described for lean concrete mixes (115 kg cement per m ^ ) , one with coarse and the other with fine grading. The maximum aggregate particle size was in both cases 18 mm. The lower stress level in the tests had always the same value. The results are indicated in Fig. 1. As can be seen, finely
Omox/fctm
Fig. 1 S-N relations for fine and coarse graded concrete and for mortar loaded in pure tension [2, 3]
graded concrete has a shorter life, possibly because of its more brittle nature. This tendency can also be observed in the diagram for mortar as indicated by the tests from [3]. A more extensive investigation has been reported in [4]. With aid of friction grips direct tensile fatigue tests have been performed on plain concrete with a water-cement ratio of 0.54, a cement content of 350 kg/m^ (ordinary portland cement), and crushed stone aggregate with a maximum size of 20 mm. In all the
tests the minimum stress level was about 8% of the static tensile strength, Paraffin wax on the surfaces of the specimens prevented them from drying out. The resulting S-N diagram is given in Fig. 2. The formula for the average S-N line proved to be:
log N = 23.96 - 24.27 j
max
ctm
(1)
Stress-strain curves were also determined. An example is shown in Fig. 3, where the maximum stress is 3 N/mm^, i.e., 87.5% of the static tensile strength. The curves are almost linear and their slopes change little during most of the lifetime. However, in any cycle the strain has
in-10
Q^mox/fctm
Q7d »min/fchn»<108
sPv»
• average values for PsO 5 O arithmetic avetoge values
logN
F i g . 2 S-N diagram f o r concrete subjected to d i r e c t t e n s i l e load-ings [ 4 ]
stress (N/mm^)
100 /i50 strain (xlOn
Fig. 3 Stress-strain curves for concrete under repeated tensile loadings [4]
creased in relation to the preceding cycle. Therefore during tensile fatigue, even at relatively high stress levels, there is only little cracking until the last stage of life has been reached and crack propa-gation increases quickly.
No results of direct tension-compression on concrete are available in the literature. From splitting tests, however, it was concluded that stresses, alternating between tension and compression cause only a slight reduction of life in comparison with the results of tests with the same maximum stress and zero minimum stress [5].
OBJECTIVES
The objectives of this research were to determine S-N lines for con-crete in tension and in tension-compression. Also, the accumulation of damage during fatigue loading was studied by evaluation of longi-tudinal deformation of the specimens.
In general one concrete mix denoted as B 45 was tested at a frequency of 6 Hz. Drying as well as sealed (wet) specimens were used.
In additional tests the frequency was lowered to 0.06 Hz. The effect of concrete quality was moreover investigated by performing tests on concrete specimens of lower-grade concrete. These tests included drying as well as sealed specimens.
For the application of a damage model it was necessary to determine the preloading strength. After various given numbers of cycles the dynamic test was stopped and the static tensile strength was measured in a load controlled test.
An overview of the experimental scheme is presented in Table 1.
Table 1 Experimental scheme (CA = constant amplitude tests)
type of test - CA - CA - CA - reload concrete quality B 45 B 45 lower ing strength test frequency 6 Hz 0.06 Hz 6 Hz (model) test humudity sealed; drying sealed; drying sealed; drying sealed; drying
12
-CONCRETE COMPOSITION AND PREPARATION OF THE SPECIMENS
4.1 Concrete composition
Concrete composed of rapid-hardening portland cement and river gravel with a maximum particle size of 16 mm was mainly used for the tests. The average compressive strength was about 47 N/mm^. For the tests with lower-strength concrete, part of the cement volume was replaced by silica powder. This was preferred to increasing the water-cement ratio, because a high water-cement ratio will cause an inhomogeneous mix. This is unfavourable for tensile tests. The compressive strength of the lower-strength concrete was about 20% lower than that of the basic concrete quality. Both concrete mix compositions are given in Table 2.
Table 2 Concrete mix compositions
component
cement content
quartz powder
water-cement ratio (incl. silica powder)
aggregate (sand + grav(
maximum size type of cement 51)
ba
sic quality 325 kg/m^0
0.5
1942 kg/m^ 16 mmpo
(B 45)rtl
typeand
III
lower quality 250 kg/m^ 63 kg/m^0.5
1942 kg/m^ 16 mm cement(B)
In the course of the research program the grading of the aggregate was improved by adding a fine fraction (0.10-0.25 mm) to achieve better compaction of the specimens. The original and the improved grading curves are presented in Fig. 4. The improved grading curve was applied from casting No. 132 onwards.
proportion retained en sieve (%)
iOSOl 2 V % 16 sier« aperture (mm)
Fig. 4 Original and improved grading curves
Fig. 5 Test specimen
4.2
SpecimensThe tests were carried out on tapered cylindrical specimens (jo 120 X 300 mm^) cast vertically in steel moulds. After casting and demoulding, the specimens were placed in water for 14 days. After this period the specimens were stored in the laboratory
("drying specimens") or wrapped in plastic sheeting ("sealed specimens"); they were tested in the fifth week after casting. For the application of the load, steel platens were glued to the top and bottom of the specimens. For that purpose the cement layers were removed in a grinding machine. In order to provide plane parallel and axial connection of the platens, a special gluing press was designed and applied. A specimen provided with these platens is shown in Fig. 5, More details are given in [11.
14
-TESTING EQUIPMENT
A complete description of the testing equipment and measuring system has been published in [1]. From casting No. 96, however, some improve-ments were introduced:
- The Schenck system was replaced by a newer and electronically more stable type. This change had no effect on the test results. The new unit is shown in Fig. 6.
- Measurement of the longitudinal deformations was improved by using more suitable linear voltage displacement transducers (HBM type W1T3). The attachment and instrumentation was also improved (see Fig. 7 ) .
General
In centric testing, bending moments in the specimen should be avoided in order to obtain uniform stress-distribution in the cross-section. It is then assumed that eccentricity is mainly caused by the appli-cation of the loading and not by inhomogeneities of the material. In the static and dynamic tensile tests the specimen is placed be-tween two swivel heads. However, if the loading axis does not coin-cide with the geometric axis of the specimen, bending will result. Because of dynamic effects, bending moments in static and in dynamic tests may be different. So both situations have to be checked.
Starting points
In the dynamic tests the overall loading accuracy is 2% of the full scale (i.e., 100 k N ) ; therefore it is stated that the stresses due to eccentricities or bending moments should not exceed 2%. So for a loading capacity of 100 kN and a specimen cross-section of 11300 mm^ the accuracy of the stresses should be 0.18 N/mm^. This is about 6% of the tensile strength. The permissible maximum moment M is:
M = a.W -> M = 0.18 X ^ X (120)^ « 30000 Nmm (2)
W = section modulus (mm^)
From this moment, the permissible eccentricity can be calculated, which depends on the magnitude of the loading and varies from less than 1 mm at the ultimate tensile loading (fa 30 kN) to less than 2 mm at 50% of this ultimate loading. In compression higher loadings up to 100 kN will occur during the dynamic tests. For obtaining com-parable relative accuracies as in tension ( ~ 6 % ) , at 100 kN compres-sion an extra stress of about 0.60 N/mm^ is allowed here. This cor-responds to an eccentricity of 1 mm.
15
-Stress inaccuracies result also in strain inaccuracies. Because the average E-modulus is about 35000 N/mm^, a deviation of the stress of 0.18 N/mm^ will cause a strain deviation of about 5 microstrain. By measuring the strain over two opposite sides of the specimen these deviations are compensated in the elastic region. In fact, local deviations from the measured average strain values will occur.
3 Measuring device
For determining the real eccentricity an apparatus was designed, con-sisting of three parallel steel bars provided with strain gauges.
The centre of the force can be calculated from the ratio of the measured strains of these bars. Fig. 8 shows the "eccentricity-tester".
Fig. 8 Measuring device for determining loading eccentricities (a) Dimensions of the tester (b)
Because eccentricity can be influenced by dynamic effects during load-ing and because the concrete specimen is part of the closed-loop hydraulic control system, the instrument stiffness was taken as about equal to the stiffness of the concrete specimen. To facilitate mount-ing the instrument in the loadmount-ing machine, the external dimensions of the instrument and the specimen were equal.
1
*iy
shnin qouqe j• / \
A
• «351 I ^
i 60By the application of error propagation laws and control tests it was found to be possible to obtain a measuring accuracy of about 0.1 mm for the eccentricity.
The instrument can also be used to calibrate the load-cell with an accuracy better than 0.3% of full scale.
Details of the design and application of the eccentricity-tester are described in [6].
Eccentricity test of loading machine
To determine eccentricity the eccentricity-tester was mounted in the loading machine and connected to the electronic instruments. The measuring system is shown schematically in Fig. 9. The signals from the eccentricity-tester and also the signal from the load-cell are
load cell
excentricity tester
muttiptexer
transient-recorder floppy disc
Fig. 9 Test set-up for testing eccentricity of the loading machine (schematic)
transmitted to a multiplexer in which the signals are sampled every 0.8 millisecond and sent to a transient recorder where they are also stored on a floppy disc. Afterwards the signals are analysed.
Static eccentricity tests
Eccentricities were determined for static loadings from 100 kN com-pression to 100 kN tension. The results are presented in Table 3.
18
-The eccentricity (r) is defined as the distance between the centre of the load and the centre-line of the eccentricity-tester. For the interpretation of the results it is noted that the tensile failure loading of the specimen was about 30 kN; the compressive failure load-ing can be calculated as about 525 kN.
Table 3 Eccentricity (r) in static tests
1. loading 2. unloading 3. loading
load
kN
- 100 - 75 - 50 - 2525
50
75
100
r
mm
0.8
0.8
0.8
0.7
0.9
1.2
0.9
0.9
loadkN
100
75
50
25
- 25 - 50 - 75 - 100r
mm
0.8
1.0
1.2
1.4
0.6
0.7
0.7
0.8
loadkN
- 100 - 75 - 50 - 2525
50
-r
mm
0.8
0.7
0.8
0.9
1.0
1.0
-+ = tension; - = compressionIt can be concluded from Table 3 that the eccentricity at the lower loading (i.e., -i- 25 or - 25 kN) is in agreement with the required accuracy. At higher loadings, important for compression, eccentricity remains nearly constant and is also in accordance with the require-ments.
6.6 Dynamic eccentricity tests
Eccentricity during dynamic loading was determined at 6 Hz (sine). This frequency corresponds to the loading frequency most applied to the concrete specimens. The lower and upper limits of the cyclic load-ing were chosen in the same range as in the dynamic experiments on concrete. For each combination of lower and upper limit, five subse-quent cycles were analysed. The average eccentricities as measured
at the lower limits (f^^-j^) and the upper limits (f'™,^^ °^ ^^^ cycles
are given in Table 4. For tensile fatigue r is especially important. II Id A
It can be seen that in general the results satisfy the requirements as previously stated.
Table 4 Eccentricity during dynamic loading at various loading ranges loading range r„. ^ ^ min
kN
mm
max mm loading range r„. ^ ^ minkN
mm
max mm0
0
0
-> -> -> + + +50
30
15
+ 10 ^ + 501.1
0.6
0.7
1.2
0.6
- 100 -V -^ 50 - 75 ^ + 50 - 50 ^ + 50 - 50 -> -^ 150.8
0.7
0.6
0.6
0.8
0.7
0.7
1.2
20
-PERFORMANCE OF THE EXPERIMENTS
At an age of 28 days the average tensile strength of 5 or 6 specimens
was determined in a load-controlled static test (rate of loading
0.1 N/mm^.s). This tensile strength was taken to represent the tensile
strength of the specimens subjected to dynamic loads. The upper and
lower limits of these dynamic loads were referred to the average
ten-sile strength. The compressive strength of the specimens was estimated
using the average value of the static cube strengths measured on
three standard cubes (rate of loading 0.47 N/mm^.s). By determining
the compressive strengths of eight tapered specimens from two batches,
these compressive strengths were found to be about 7% lower than the
cube compressive strength. At an age of 28 days three control cube
splitting tests were also performed. The dynamic tests were carried
out during the fifth week after casting.
8 RESULTS OF STATIC TESTS'
To check concrete quality and to determine the reference values for
dynamic testing, static tests were performed. In Table 5 the average
values are presented for compressive tests, splitting tests and
uni-Table 5 Average values of static tensile test results of B 45 concrete
compression splitting uniaxial tension
f'
f
f F
cm csplm ctm cm ^im^
N/mm2 N/mm2 N/mm^ N/mm^ 10"°
46.8 2.89 "drying" 2.50 36410 95
"sealed" 2.92 34280 129
3.7 6.2 "drying" 8.5 3.1 10.6
"sealed" 3.7 3.0 6.6
32 32 "drying" 15 13 13
"sealed" 18 15 15
For all constant amplitude tests
average 47.3 2.85 "drying" 2.45 36140 96
"sealed" 2.88 34690 127
average 3.6 7.0 "drying" 7.6 5.2 9.6
V % "sealed" 3.7 5.0 7.3
k 82 82 "drying" 54 26 26
"sealed" 27 24 24
k = number of concrete batches, each consisting of about 6 specimens
subjected to static tension, 6 for dynamic testing, 3 cubes for
static compression and 3 for splitting tests.
axial tensile tests on B 45 concrete. In the upper part of the table
results are given which complement the results published in [1]. In
the lower part the average values refer to static tests which had to
be performed for the execution of all our constant-amplitude tests.
An overview of all the results can be found in Appendix A.
The results of uniaxial tensile tests are also shown in the mean
a-e
lines of Fig. 10. It is seen that, on the average, the tensile
strength as well as the ultimate strain of "sealed" specimens are
higher than of "drying" specimens.
average
average
22
Similar properties have been determined for the lower quality concrete. The average values are presented in Table 6. In Appendix B the indi-vidual values are given.
Table 6 Average values of static tensile test results of lower quality concrete compression f' cm N/mm^ 39.5
3.5
4
splitting csplm N/mm^ 2.577.9
4
uniaxial tension "drying" "sealed" "drying" "sealed" "drying" "sealed" ^ctm N/mm2 2.57 2.732.6
2.7
2
2
^cm N/mm^ 32220 337104.7
3.0
2
2
^im--10"^108
122
9.0
2.8
2
2
average average V %k = number of concrete batches, each consisting of about 6 specimens subjected to static tension, 6 for dynamic testing, 3 cubes for static compression and 3 for splitting tests.
Comparing the results of B 45 concrete and the lower quality concrete, it appears that the compressive cube strength is about 2 0 % lower. The differences in tensile properties are much smaller.
If the ratio of compressive to tensile strength is conceived as a measure for the material brittleness, then the lower quality concrete can be regarded as more ductile than the B 45 concrete.
3.0 25 20 1.5 1.0
Qsi
m
Get (N/mm^)A
/ / / ^ .<<«s«ded" / ^ \ "dryinq-" ^ 1Fig. 10 a-e lines for "drying" and "sealed" concrete subjected
to direct tension 25 50 75 100 125 150
RESULTS OF DYNAMIC TESTS
9.1
General
In a constant-amplitude test the load fluctuates with constant
fre-quency between given minimum and maximum levels (see Fig. 11). In the
various tests the maximum level was between 40 and 90 per cent of f^. .
'^
ctm
Repeated tensile tests on drying and sealed specimens were performed
with four fixed minimum levels, namely, 40, 30, 20 and 0 per cent of
tensbn
compression
Fig. 11 Constant amplitude tests
f . . In order to investigate the effect of compressive stresses on
ctm
tensile fatigue, fixed minimum compressive levels were included in
the test program. These levels were 5, 10, 15, 20 and 30 per cent of
f' . Minimum levels of 10, 20 and 30 per cent of f' were applied to
cm ^ cm '^'^
drying specimens, while sealed specimens were tested at 5 and 15 per cent.
By means of a statistical analysis it was possible to combine these
re-sults. In general one concrete mix (B 45) was tested. However, the
effect of lower quality concrete was investigated too. The frequency
during the experiments was mainly 6 Hz. In a few additional tests the
frequency was lowered to 0.06 Hz.
9.2
S-N diagrams
For each minimum level investigated, the test results are presented
in an S-N diagram in which the number of cycles to failure is related
to the maximum stress level. For B 45 concrete tested at 6 Hz these
24
-diagrams are given in Figs. 12 to 15 for repeated tension (including
a . = 0) ai
min
reversals).
a • = 0 ) and in Figs. 16 to 20 for compression-tension tests (stress
min ^
:/'ctm
Fig. 12 S-N diagram for B 45 concrete at 6 Hz; a^^-^ = 0.40 f
ctm
Log N
Fig. 13 S-N diagram for B 45 concrete at 6 Hz; a„- = 0 . 3 0 f
^ ^ min :/*ctm
ctm
0 1 2 3 L 5 6 7 '°LogN F i g . 14 S-N diagram f o r B 45 c o n c r e t e a t 6 Hz; a^.^ = 0.20 f ctm'max /fctm CL. 3 confidence limits A = sealed specimen O s drying specimen —> s run out 5 6 7 '°Log N
Fig. 15 S-N diagram for B 45 concrete at 6 Hz; a . = 0.00
min
''max ^'ctm
Fig. 16 S-N diagram for B 45 concrete at 6 Hz; o = 0.05 f
^ ^ min '
cm
Fig. 17 S-N diagram for B 45 concrete at 6 Hz; a .
26
-°max/'ctm
Fiq. 18 S-N diagram for B 45 concrete at 6 Hz; a„. =
^ ^ min 0-15 f' cm
^max^'ctm
Log N
Fig. 19 S-N diagram for B 45 concrete at 6 Hz; o. = 0.20 f
cm
''max /'ctm
Log N
F i g . 20 S-N diagram f o r B 45 c o n c r e t e a t 6 Hz; o . = 0.30 f
The lines drawn in these diagrams are the result of a multiple linear regression analysis in which repeated tension (including minimum level 0.0) and compression-tension were treated seperately. Also, it was decided to consider the maximum number of cycles of the run-outs as the number of cycles to failure. Results of 189 repeated tension and 144 compression-tension tests on "sealed" and "drying" specimens were comprised in the regression analysis.
The total data set is given in Appendix C.
The following expression was derived for "drying" specimens:
repeated tensile tests (c • /f . > 0 ) :
— — u _ — — — . — . - . — — — . _ . — _ — _ _ r m r i r-rm min' ctm
a a • max o -irs min
log N = 14.81 - 14.52 ^^^ + 2.79 r ^ (3) ctm ctm
It turned out that the humidity ("drying" or "sealed") was significant at the 95% level. The S-N relation for "sealed" specimens was:
repeated tensile tests (a • /f j. :* 0 ) : --e min' ctm
log N = 13.92 - 14.52 j ^ + 2.79 J ^ (4)
ctm ctm
.For drying as well as for sealed conditions the 90% confidence regions were log N ± 1.74.
A similar expression is valid for stress reversals:
compression-tension tests (a • /f' > 0 ) :
log N = 9.36 - 7.93 J ^ - 2.59 j ^ (5)
ctm cm
The 90% confidence regions were estimated as log N ± 1.38. Here the difference between drying and sealed was not significant.
Note: Because more test results were involved in the regression analy-sis the constants in formulas (3), (4) and (5) differ slightly from the constants published in [1].
28
-As appears from the S-H diagrams, the results show considerable scatter. By means of a statistical method as described in detail in
[1], part of this scatter can be explained by the variability of the stress-strength levels in the fatigue tests, because of variation in concrete static strength. In Fig. 21 and 22 it is shown that a con-siderable part of the scatter band has its origin in variability of the strength. The rest of the scatter can be explained by the stochastic nature of fatigue and the manner of performance of the experiments.
1.0 0.8 0.6 0.4 0.2 0.0, 'max /'ct
m
"min/'ctm'0-3 0 : caused by estimation of stress-strength! levels
10 log N
Fig. 21 Scatter due to variability of static strength; a^. = 0 . 3 0 f^,^ min ctm 'max /'ctm
ao
caused by estimolion of stress-strength levels I 1 ' 6 7 ^°log NFig. 22 Scatter due to variability of static strength; o . = 0.30 f'
Some average S-N relations for drying specimens as calculated from formulas (3) and (5) are shown in Fig. 23. It appears that at a given maximum stress-strength level a lowering of the minimum level results in shorter lives. If the minimum level is a compressive stress a marked reduction of life is observed.
-^max/'ctm
1 2 3 4 5 6 7
'°LogN
Fig. 23 Average S-N r e l a t i o n at various lower s t r e s s - s t r e n g t h levels
0.8 Q6 04 02 00 ^max j j ^ ,tctm ^mm^ stwin/ 0 kol OS. w moms.gt saito. imc ^ lUiams 1 irrett 3 ^ 2 6 7 logN
Fig. 24 Comparison of tensile fatigue results [2, 3, 4]
In Fig. 24 the test results have been compared with results reported by other investigators. Only results with a^. « 0 were available. ^ •> m i n It can be concluded that there is good agreement, taking into account the scatter inherent in tensile fatigue tests.
30
-A few preliminary tests were performed at a frequency of 0.06 Hz for B 45 concrete on drying as well as on sealed specimens. Minimum levels were chosen at 0.4 f . , 0.0 and 0.15 f' . The results have been
{_i> L r 1 1 I V ^ l I I
plotted in Fig. 2 5 , 26 and 27. They are also presented in Appendix D. In the diagrams the appropriate average S-N lines as derived for 6 Hz are also represented. As can be seen, the numbers of cycles to failure at 0.06 Hz are in general lower than at 6 H z , especially at high maxi-mum stress-strength levels. At lower maximaxi-mum stress-strength levels, however, this cannot be proved because of the run-outs. A more detailed comparison of 0.06 Hz and 6 Hz results is given in Chapter 10.
1.0 0£ 06 Q4 02 00, 'max /'ctm 00 o 1 "min ' ~ " -/»ctm ' ' O " ^ ).4 1 • ^ - Q l > > O-^ '1 o . 6Hz a06Hz 1 (Irying 6 7 "'LogN
Fig. 25 S-N diagram for B 45 concrete at 0.06 Hz; a . = 0.40 f
^ ^ min
ctm
:/'ctm
Fiq. 26 S-N diagram for B 45 concrete at 0.06 Hz; a • = 0,
1.0 OS Q6 04 02 0.0, 'max /'ctm X -£C) J b ^ Omin/'èm =• 0-15 I I ^ H ^ 0.06 Hz O s drying A : sealed 6Hz 10
LogM-Fig. 27 S-N diagram for B 45 concrete at 0.06 Hz; a • = 0.15 f
^ ^ min
cm
For orientation some experiments with specimens of lower quality con-crete were also executed. Constant-amplitude tests at 6 Hz were
per-formed at a lower stress-strength level in tension (0.40 f^^^) and
a lower stress-strength level in compression (0.15 f'^^)- The results
are given in the S-N diagrams of Fig. 28 and Fig. 29 (see also Appen-dix E ) . For comparison also the appropriate lines of B 45 are indicated in those diagrams. For repeated tensile tests (a^.j^/f^^^ = 0.40), the results are in accordance with the line. For alternating compression-tension (a . /f' = 0.15), however, in the case of lower quality con-crete more cycles to failure were generally found. If the individual results of B 45 concrete are compared with the results of lower strength
1.0 08 06 04 02 QO Omax/fctm 90% a, "q*'—p -^-CL Omin/fctm=0.tO ^90% CL -^2=^ low quality Oïdryino asseaierl 'o->.,.^o^' lines for B45 6 7 LogN
Fig. 28 S-N diagram for lower quality concrete at 6 Hz;
a . = 0.40 f ^
32 -1.0 0.8 06 0.4 02 pmax/fctm \ <3D o-Omin/fctm«0.15 ooi L \ 90% CL • ^ . X V i ; j ^ ^ ffl <^ •"N-\ ^ \ X low quality 0 = dfying A : sealed V,iCr* \ . ^ \ lines forBiS 90v.a 0 1 2 3 4 5 6 7 logN
Fig. 29 S-N diagram f o r lower q u a l i t y concrete a t 6 Hz; 0^. = 0.15 f '
min cm
concrete, the differences can be neglected (see Fig. 12).
In our fatigue tests failure was governed by tensile stress and the
difference in tensile strength between the two types of concrete was
small. Hence, the fatigue behaviour was similar. From these tests it
cannot be concluded if S-N lines for relative stresses are unique and
independent of the absolute strength.
9.3
Goodman diagram
For B 45 concrete tested at 6 Hz on "drying" specimens the results
have been evaluated and presented in a Goodman diagram. For that
pur-pose at a given number of cycles to failure the combinations of minimum
and maximum stress-strength levels have been calculated from formulas
1,0 0,8 Q 6 Q4 0,2 Q 2 0,4 0,6 0,8 \0
Fig. 30 Modified Goodman diagram for "drying" B 45 concrete tested
at 6 Hz
(3) and (5). The resulting modified Goodman diagram is given in Fig. 30. In this diagram again shorter lives of stress reversals in comparison with repeated tensile tests can be observed, especially for tests of long duration (log N i 5) where the maximum tensile stress is rela-tively low and the effect of compression may be more important. This more damaging effect of stress reversals was confirmed in post peak tension-tension and tension-compression tests [7]. This can be explained by interaction of tensile and compressive microcracks and by residual stresses during unloading.
9.4 Cyclic strain
In the dynamic tests the longitudinal strain was measured eight times per cycle. From these measurements the maximum and minimum strains per cycle were calculated with the aid of fast Fourier transformation
(see [1]). In the case of stress reversals extreme values of the strain (at a and at a • ) were referred to the strain at zero stress in
max min
the loading branch of the cycle (see Fig. 3 1 ) .
Fig. 31 Reference value of maximum and minimum strain
Some typical examples of the development of the maximum and minimum strains for a repeated tensile and an alternating compression-tension test are shown in Fig. 32. As can be seen, the maximum strain increases rapidly during the first 10% of the total life; after that there is a gradual increase of strain during about 80% of the life, followed
34
-by a strong increase until failure occurs.
The total set of cyclic strain results has been compiled in Appen-dix F, where cyclic creep curves are presented for various combina-tions of minimum and maximum stress-strength levels. As already indi-cated, these levels show scatter because of the variability of the concrete strength. This consequently results in variability of the cyclic strain, which is represented by scatter in the position of the cyclic creep curves.
ISO 125 100 75 50 25 -150 -175 -200 -225 -250 -275 total strain (10 1
r
r ^ ^ E max 2 , ^ emin I i i 1 1 i _ ^ ^ L Ql Q2 0.3 Q4 0.5 Q5 0.7 08 Q9 1.0 1 , , , , , , cvcle ratio n/N 1k
2 1 emin 1. 1 2. 11
! 1 1 num. level N 0508 Q20-070 10154 DUO -015-0.70 5733Fig. 32 Development of maximum and minimum strain for a repeated tensile and an alternating tensile-compressive test
9.5
Reloading strength of run-outsThe dynamic tests had been continued until failure of the specimen or up to 2 X 10 cycles. In the latter case the test was stopped and the static tensile strength of the run-out was determined and compared with the average tensile strength (I'-.tni^ °^ non-preloaded specimens from the same concrete batch. The results for various minimum and maximum stress levels are presented in Table 7.
In this table only the results of B 45 tested at 6 Hz are given. As can be seen, the tensile strength has slightly increased in general because of the cyclic preloading at relatively low stress levels. This
Note: The specimen code as indicated in the diagrams is explained in Appendix G.
finding is in agreement with the results published in [1]. The reason for this gain in strength may be the stress redistribution resulting
in lower stress concentrations. At higher stress levels preloading did not affect significantly the tensile strength, as is pointed out in Fig. 42 of Chapter 11.
Table 7 Reloading strength of run-outs
No. a • a f ./f . min max et ctm 107-06 108-06 101-13 109-11 111-11 95-15 97-05 110-12 112-12 114-12 97-01 111-12 106-10 0.05 f' cm 0-05 f' cm
0.2
0.2
0.2
0.2
0.2
0.4
0
0
0
0
0
^ctm ^ctm ^ctm ^ctm ^ctm ^ctm 0.40 0.50 0.50 0.60 0.60 0.70 0.70 0.60 0.60 0.60 0.70 0.70 0.65 ^ctm ^ctmctm
ctm
^ctm ^ctm ^ctmctm
^ctm ^ctm ^ctm ^ctm f ^ ctm 1.01 1.09 0.81 1.06 1.06 1.12 1.08 1.16 1.12 1.21 1.19 1.33 1.07 Fracture locationsTapered specimens were chosen for the tensile tests to reduce the in-fluence of the loading platens on the behaviour of the specimen, and to reduce the bond stress of the glue. However, because of this shape of the specimen, stress concentrations are introduced in the zone where the cross-section decreases. These stress concentrations may be not very important because of the inherent inhomogeneity of con-crete, which also gives rise to stress concentrations. Moreover the effect of the stress concentrations on the results of the dynamic tensile tests is reduced by referring the stresses to the static strength which is determined on specimens of the same shape.
36
-A measure for the presence of stress concentration is the fracture location. Normally the distribution of this location should be uni-form over the length with reduced cross-section.
In Fig. 33 the distribution of fracture locations is represented by means of a histogram. This histogram refers to 318 (tapered) speci-mens used for all types of tests as described before. These types were analysed together because in [1] no significant difference was found between the various types (such as static, dynamic, run-outs). As can be seen, the fracture locations are fairly well distributed
relative frequency [%)
Fig. 33 Distribution of fracture locations
over the height of the specimens, except close to the top, where
about 24% of the specimens failed. This can be explained by the inter-action of stress concentration and less compinter-action during vibration after casting of the specimens.
9.7 Statistical distribution of number of cycles to failure
As can be observed in the S-N diagrams, considerable scatter was found in the number of cycles to failure. Although part of this scatter is due to incorrect adjustment of the stress-strength levels, because
of the variability of the static strength values, the remaining scatter is quite substantial. The question arises as to what type of statis-tical distribution this scatter obeys. In the S-N diagrams it is assumed to be a logarithmic normal distribution. To support this assumption the following theoretical considerations are important.
As outlined in detail in [ 8 ] , for the determination of the most appro-priate distribution the relationship between the phenomenon (in our
case fatigue) and its causes has to be considered. The uncertainty
in the number of cycles to failure in fatigue tests may be the result
of many individual causes, each difficult to isolate and observe.
How-ever, in many cases we can determine how these causes affect the
phe-nomenon. With respect to this, three causes can be considered: where
the individual causes are additive, where they are multiplicative and
where their extremes are critical.
^' I!2§_§y'D§_!r9^§li_G9!2i§l_^l§5ri^y5l20
If a phenomenon is the result of the sum of causes, a normal
dis-tribution can be applied. Suppose, as a specific example, that
the total length of several pieces is desired, and each piece has
been measured. The error in the total length depends on the number
of pieces and is the sum of all the individual errors.
In many situations, however, the normal distribution is adopted
even when no or little physical justification for it exists. The
normal distribution is then regarded as a "not-unreasonable" model.
2- T!]§_Br2dy95§_!I'9^§li_l93_D9rü]Ël_^l§tributign
The log normal distribution can be used to consider the
distribu-tion of a phenomenon which arises as the result of a multiplicative
mechanism acting on a number of factors. For instance, Freudenthal
[9] conceives the internal damage during fatigue tests (compression)
as a multiplicative mechanism. It is assumed that internal damage
Y„ after n cycles can be written as:
n -^
\ =
^-1
\
(6)
W represents the internal stress state resulting from the n
loading cycle, which is subject to variation because of internal
differences in material at microscopic level. Starting from the
initial damage before application of the loading, Y can be written
as:
38
-Here Y is expressed as the product of a number of variables, Then, taking the logarithm of the expression:
log Y^ = log Y^ + log W^ + log Wg + + log W (8)
If W. is normally distributed, it can be shown that loq W. is also
1 1
normal and consequently log Y is normally distributed.
C- I^§_§^ïr§1)§§_ir9^§Ii_§ïïr§'!)§_ÏËly§_^l§ïi;;ibutions
Extreme value distributions are of interest when the distributions of the smallest (or largest) of many variables are studied. These types of distribution are often used to describe the strength of brittle materials. The basic argument is that a specimen of a brittle material fails when the weakest of a series of many micro-scopic elementary volumes fails. Depending on assumptions, dif-ferent extreme value distributions can be applied. In the well known Weibull distribution it is assumed that a lower bound of strength exists below which no failure occurs. This lower bound is often taken as corresponding to zero stress.
In our fatigue tests a model consisting of the combination of model B and model C appears to be appropriate, because the multiplicative damage (model B) occurs mainly in the weakest cross-section (model C ) . However, for the sake of simplicity model B was adopted for the anal-yses. Also other investigators have shown that for the analysis of fatigue results in compression as well as in tension the log normal
distribution forms a good approximation (see for instance [4, 10, 11]). For our tests this model was checked for two test series containing sufficient individual results. These series were: repeated tensile tests (0.00 - 0.70 f , : sealed) and alternating compression-tension (0.10 f' - 0.70 f , : drying). The probability of failure was cal-culated with the "rank method", in which the rank of the specimen is divided by n + 1 (n = number of specimens) [12]. The results are pre-sented in Appendix H and have been plotted on log normal probability paper as shown in Fig. 34. Straight lines approximately fit the data, which indicates that the log normal distribution is a reasonable dis-tribution for our tests.
40
-EVALUATION OF SECONDARY CREEP VELOCITY IN RELATION TO FATIGUE LIFE OF CONCRETE
For metals it is well known that a relation exists between the minimum strain rate in the secondary part of the creep curve and the life at-tained. Theoretical backgrounds have been presented in [1], For con-crete this relation exists too. In that case the minimum strain rate is found to be nearly constant in the secondary branch of the creep curve (see Fig. 3 5 ) .
strain
time Fig. 35 Cyclic creep curve
In the literature close relations have been described for gravel as well as for lightweight concrete (Lytag) subjected to repeated com-pressive loadings at two given stress rates, namely 0.5 and 50 N/mm^.s
[131. The formulas derived turned out to be independent of these stress rates, but dependent on the type of aggregate. They are:
f 9r_9!r§yêI_99D9r§5Ê •
log N 2.66 - 0.94 log èsec
(9)
f 9r_LY5§9_92!39!r§^e: log N = - 3.79 - 1.06 log ksec
(10)In these formulas the strain rate è is per cycle. Formulas (9) and (10) can be transformed into a relation between time to failure (t^r)
and t per second. These transformed formulas for frequencies of
f o r gravel concrete_at_6_Hz: log t . = - 2.17 - 0.94 log è , ^ , (11) T 5cC
f9ir_9r§y§I_92D9rÊï§_§5_5-?§_y? •
log t^ = - 2.59 - 0.94 log è^^^ (12)
f2r_!:yï§9_99!29r§ï§_§5_§_t!? •
log t^ = - 3.74 - 1.06 log
k^^^
(13)
f9r_LY5Ë9_92Q9r?5Ë_Ë^_OiO§_y?•
log t. = - 3.86 - 1.06 log è,^^ (14)
In the formulas (11) to (14) ê.^. is the secondary strain rate per
ScCsecond. The relations represented by these formulas are shown in Fig. 36,
As can be seen, the influence of frequency is small: at given è ,
time failure is independent of the test frequency. Gravel concrete,
however, shows longer life than Lytag concrete at similar
t .
For
concrete loaded in pure tension a few results from the literature are
also available. These relate to tensile creep tests. In [14] this type
of test has been described for normalweight concrete with crushed
ag-gregate and for a lightweight concrete. The relations between
t
and t^ are represented by:
f2!r_D2r!I?Ël^§ig!]t_concrete:
log t^ = - 5.99 - 1.18 log
t^^^
(15)
f2r_ll9[)5^ei ght_concrete:
log t^ = - 6.92 - 1.24 log è^^^ (16)
In these formulas t^ is in seconds and è is given per second.
Re-t sec ^ '^
lations (15) and (16) are represented in Fig. 37. Here lightweight
concrete also shows shorter life for given
t
42
-'"logêsec (persecond)
4 5 6 7 ^oiogtf (sec)
Fig. 36 Time to f a i l u r e versus secondary creep rate f o r compressive f a t i g u e t e s t s [13]
J°logêsec (per second)
-7 -8 -9 -10 -11 -12 -13
k^
\ 1 liqht ^ .V ^
weiqht ' concrete1
1V
s \ ^ \ 1 1 1 stevinloborotory 1 nishibayashi 2.3 1 1 1 cnjshed stone s,^corx:rete \ ^ gravel N > ^ &(WeIè \ V \ 5 6 7 lOlog t^ (sec)Fig. 37 Relation between time to failure and secondary creep rate for tensile creep tests [14, 15]
The relation determined in our laboratory is also shown in Fig. 36. It relates to tensile creep tests on "sealed specimens" with concrete composition B 45 (see Table 2) [15], The equation is:
log t^ = - 4.37 - 0.96 log e^^^ (17)
It appears that these lines are in reasonably good agreement with each other. For our fatigue tests, secondary creep rate was also evaluated. As shown in [1], no significant difference was found between repeated
tensile test results and the results of compression-to-tension tests, nor between "drying" and "sealed" specimens. This is demonstrated again in Fig. 38 in which the secondary creep velocity is indicated per second and life as the number of cycles to failure. The equation
M o g êsec (^ per second )
-6 -7 -8 -9 -10 -11 -12 "^•^0 1 2 3 4 5 5 7 10 logN
Fig. 38 Number of cycles to failure versus secondary creep rate for tensile fatigue tests at 6 Hz and at 0.06 Hz
of the regression line of the 6 Hz results is:
log N = - 3.25 - 0.89 log è^^^ (18)
Results for 0.06 Hz have also been plotted in Fig. 38. They are well approximated by a line representing 100 times shorter life than the regression line of the 5 Hz results. So it can be concluded that at given ê (per second!) tests at 0.06 Hz result in 100 times shorter
44
-to failure than 6 Hz tests. This was already shown in Figs. 2^-21.
In Appendices C, D, E and I all è values are listed for the tests in this report as well as for the tests from the preceding report [1]
otal strain
time
Fig. 40 Cyclic creep curve for 6 Hz and 0.06 Hz tests
The relation between log t^ and log é of fatigue tests on B 45 con-crete can be compared with pure tensile tests performed on exactly similar specimens and concrete composition. In Fig. 41 the average lines have been drawn. It can be seen that the two relations are approximately equivalent. Obviously, secondary creep rate is a good
- 3 -5 - 7
-a
-9 •10 -11 -12: 13N
V
\ tffl ^ \ sie fatiq tensile creep N,K
s,
^ \\. 1
^ 5,„ 5 7 '°logtf(sec)Fig. 41 Comparison of the relation log è - log t^ for tensile creep and tensile fatigue
life (in cycles) than tests at 6 Hz. It can also be concluded that
if the results are plotted against time to failure, all these results
can be represented by one regression line; for given è the time
to failure is independent of the loading frequency (in the
experi-mental range). This conclusion is in agreement with the compression
fatigue test results as shown before.
log £sec (
t
per second)
i°logtf(sec)
Fiq. 39 Total è results plotted against time to failure
^ sec
'^
In Fig. 39 the total set of results has been plotted as a function
of the logarithm of time to failure in seconds. Included are also the
results which have been published before in [1]. The regression line
in this diagram conforms to the formula:
log t^ = - 4.02 - 0.89 log è^^^ (19)
This formula expresses that the life t^ can be predicted for given
è , independently of the loading frequency. However, it must be
noted that loading frequency does influence è • A given loading
cycle at 6 Hz results in higher strain rate per second than a
load-ing cycle at 0.06 Hz. This is clarified in Fig. 40. To obtain similar
strain rates at both frequencies the stress in the 0.06 Hz fatigue
test must be higher than in the 6 Hz test. In that case equal lives
(in seconds) will be found. In terms of the number of cycles to
fail-ure, 0.06 Hz tests consequently show less than 100 times fewer cycles
46
-predictor for the time to failure, irrespective of whether this rate is caused by a constant or a varying stress.
Information on secondary creep rate can be applied to rearrange test results in S-N diagrams. Only the procedure is described here.
Let c be measured during a fatigue test; then the number of cycles sec 3 3 » J
to failure can be estimated from the known relation between è sec
and N. This relation is not influenced by variation of stress-strength levels. This variation, however, is present in S-N diagrams. But it is assumed that the average S-N line represents N for actual stresses in the specimen. In formulas:
(1) log N = - A - B log è^^^ (20) (2) log N = C - \^j!^ + E ^ (21) ctm ctm Q A \ o •
^ ^
" ^ " ^ l ^
' s e c " ^ f —
-ctm -ctm (22) aFormula (22) shows that for given é^^^, -J^^ can be estimated. This
s^^ 'ctm
is the real stress-strength level of the specimen. In this approach it is assumed that minimum stress-strength level is only slightly affected by variations of f-4.„; this level is correctly adjusted.
The method for correcting S-N diagrams that has been described has
also been applied in [12].
As pointed out before in Chapter 9.2, in S-N diagrams show consider-able scatter due to variation in stress-strength levels caused by the variability of the static strength properties. By means of the
rela-tion between t and N the real stress-strength level can be
esti-sec ^ mated and the individual results can be rearranged in the S-N
dia-grams, resulting in narrower confidence bands around the average S-N lines. These narrow bands include mainly variations due to the stochastic nature of fatigue and measuring errors. So these bands coincide with the inner bands in the S-N diagrams as shown for in-stance in Fig. 21 and 22.
11 APPLICATION OF A FRACTURE MODEL BASED ON THE MAGNITUDE OF CYCLIC CREEP
A fracture mechanics approach was used by Wittmann and Zaitsev [16] to develop a theoretical model for the fracture behaviour of cement-based materials. This model was verified by short- and long-term tests. Now this model will be applied to the cyclic test results.
In a given material unstable crack propagation will occur when a criti-cal crack length is reached. This criticriti-cal crack length is assumed to be independent of the type and duration of the loading [16]. In
[17] it has been deduced that the crack le^^gth S, at time t, can be expressed as:
S(t) = F{C.M(t)} (23)
As a good approximation, C is a material constant, and the function F has to be determined by numerical simulation methods and depends on the material configuration (pores, aggregates). The so-called measure of destruction D(t) can be derived from:
In this formula the creep coefficient 4)(t) also represents creep at the crack tip and m(t) takes into account the effect of preloading (stress-redistribution). For short-term tests to failure (a(t) = f (t)), D(t) will be unity, because (j)(t) = 0 and m(t) as well as E^(t)/E^(t=0) are equal to unity. As already stated, at failure S(t) in formula (23) is independent of the type of test, and so are C and F. So for dif-ferent test conditions D(t) = 1 at failure, and D can be regarded as a failure criterion,
Factor m(t) in the formula represents the influence of preloading on the strength, which affects the true stress-strength level in the test. In a separate test program this effect was studied. Therefore the tests were stopped after 100 x 10"^, 200 x 10^ or 400 x 10^^ load cycles, and afterwards the static tensile strength was determined and compared with the static strength of virgin specimens. The results are shown
in Fig. 42 and in Appendix J. In this diagram appropriate test results of run-outs have also been included. It can be concluded that the change in tensile strength was within the normal scatter band of the static
48
-s t r e n g t h . So no influence of preloading could be proved, and con-se- conse-quently the value of m(t) is taken as 1.
m|t)=fct preioaded)/fctm 13 12 11 10 0.9 Q8 Q7 90% scatter banc; o( fct \ 0.1 02 mm mox 0 -0.1 — O . i „ O.i—0.7 drying ° 0,4 — 0.6 X - 0 0 5 - 0 4 sealed + 04 —0.8 0.3 04
Ar
2.0 xlO^ nFig, 42 Factor m(t) after 100 x 10^, 200 x 10^, 400 x 10^ cycles and for the run-outs
Because only tests with log N «: 6 have been analysed (< 48 hours at 6 H z ) , it is assumed that variation of Young's modulus can be neglected, Equation (24) is thus simplified to:
D(n) -£ '^max
'ctm
'
1 - Vl^"^
(25)
xycl (n) was calculated from the total peak strains in the tensile
parts of the cycles (e.^^.), and the instantaneous strain (e-) at a„,^ th(
(n)
as determined from E of the appropriate casting number cm
({> i ( n )
^cycl
-tot^
(26)
A typical relation between peak strain and n, as well as the defini-tion adopted for the ultimate strain at failure, are shown in Fig, 43,
D at failure was calculated for 122 dynamic tests. The results are presented in the histogram in Fig. 44 and in Appendix K. The average D(n=N) was 1.02 and its coefficient of variation 9.8%. No signifi-cant difference was found between "drying" and "sealed" and between
repeated tensile tests and tension-compression. It can be concluded
that with D as a failure criterion the magnitude of ^ •, orovides
^ eye 1
an i n d i c a t i o n o f t h e accumulated damage i n the m a t e r i a l . I t can a l s o be i n t e r p r e t e d as a f a i l u r e c r i t e r i o n based on maximum s t r a i n . Cyclic deformation ju strain 100 (^.0.2.10-8) K n ° _ _ ^ _ _ _ _ _ _ _ ^ y ^ ^ " ^ Omax = 0-7 fctm Omin = 0.0 - J \ t : I ultimate stram 200 400 600 800 number of cycles F i g . 43 U l t i m a t e s t r a i n i n c y c l i c t e s t s .frequency (%) 0.70 080 0.90 1.00 1.10 1.20 1.30 1.40 D(n=N) F i g . 44 D i s t r i b u t i o n o f the measure o f d e s t r u c t i o n as d e f i n e d i n (25)
50
-12
CONCLUSIONSThe conclusions are based on the experiments as described in our first report on fatigue [1] and in the present report.
1. Based on 189 repeated tensile tests and 144 compression-tension tests S-N relations were derived:
!r§BËÊ5§Ë_!^§D§il§_裏5§_2D_"^rYi!]9"_§9§2i']]§0§-log N = 14.81 - 14.52max
ctm
+ 2.79min
ctm
(3)
rÊB§Êϧ^_ tens il_e_tests_gn_"seal^ed"_sgeci mens:
log N = 13.92 - 14.52
max
''''ctm + 2.79min
ctm
(4)
§5!r§§§_!r§Y§EHl§_2D_"^rylD9"_50^_"§§Êl§dü_§2§9iü}§!]§'
log N = 9.36 - 7.93max
ctm
2.59 ''min 7 ^ cm(5)
2. For repeated tensile tests "drying" specimens showed more cycles to failure than "sealed" specimens. For stress reversals no signifi-cant difference was found.
3. Stress reversals showed relatively much damage, resulting in shorter lives than tests with zero lower level.
4. At 0.06 Hz fewer cycles to failure were found, as compared with tests at 6 H z , at the same stress-strength levels.
The difference, however, was less than a factor 100.
5. No difference in life was found between tests on high and on low quality concrete at equal relative stress levels. The difference between the strength properties of the two concrete m i x e s , how-ever, was siight.
6. At a given stress-strength level the number of cycles to failure conforms to a log normal distribution.
7. A marked relation exists between secondary creep rate and life. This secondary creep rate proves to be a good predictor for the number of cylces to failure.
8. The relation between secondary creep rate (per second) and life (in seconds) is independent of loading frequency and is even approximately the same for cyclic and for creep tests. The rela-tion, however, depends on the composition of the concrete. The relation is as follows:
log t^ = - 4.02 - 0.89 log h^^^ (19)
9. It proved possible to apply a failure model based on ultimate strain for fatigue life prediction (D-model).
10. No significant difference was found between the strength of
non-preloaded specimens and specimens non-preloaded at relatively high stress-strength levels. Preloading at low stress-stress-strength levels seemed to be slightly beneficial for tensile strength.
52
-REFERENCES
1. Cornelissen, H.A.W., Timmers, G.
Fatigue of plain concrete in uniaxial tension and alternating tension-compression
Report No. 5-81-7, Stevin Laboratory, University of Technology, Delft, 1981.
2. Kolias, S., Williams, R.I.T.
Cement-bound road materials: strength and elastic properties measured in the laboratory
TRRL report No. 344, 1978.
3. Morris, A.D., Garret, G.G.
A comparative study of the static and fatigue behaviour of plain and steel fibre reinforced mortar in compression and direct tension Int. Journal of cement composites and lightweight concrete, Vol. 3, No. 2, May 1981 , pp. 73-91.
4. Saito, M., Imai, S.
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