DELFT UNIVERSITY OF TECHNOLOGY
DEPARTMENT OF CIVIL ENGINEERING
Report 5-81-7
Fatigue of plain concrete in uniaxial tension and in alternating tension-compression
experiment and results
H.A.W. Cornelissen G. Timmers
STEVIN LABORATORY
CONCRETE STRUCTURES
Technische Hogeschool
BIBLIOTHEEK Civiele Techniek Stevinweg 1, 2628 CN Delft Tel. 015 - 785472. Retoar uiterlijk vóór; T T ,
D e l f t University of Technology Department of C i v i l Engineering Report 5-81-7 Research No. 7804 A p r i l 1981 Technische Hogeschool
Afdeling: Civiele Techniek Stevjn.vcg 1
Srt r - ^ - r postbus 5048
2600 G A DelftacxVer ü
^^
Fatigue of plain concrete in uniaxial tension and in alternating tension-compression
experiment and results
H.A.W. Cornelissen G. Timmers
Mailing address: Technische Hogeschool Delft Vakgroep Betonconstructies Stevinlaboratori urn 2
Stevinweg 4
2628 CN Delft, The Netherlands
SI
,0-^
-2-Acknowledgement
This investigation has been carried out in the Stevin Laboratory of the Delft University of Technology in close co-operation with CUR-VB (Nether-lands Committee for Research, Codes and Specifications for Concrete). The financial support of CUR-VB and the discussions of the members of CUR-VB committee C-33 are greatfully acknowledged. The authors wish to thank W.J. van Veen for performing the tests, C M . P . van Hoek and staff for preparing the specimens and Ir. H. Voorsluis of the measurement and instrumentation group for designing the data-acquisition system.
No part of this report may be published without written permission of the authors.
-3-CONTENTS • P^ae 5 7 9 12 13 Notation Summary 1 INTRODUCTION 2 OBJECT
3 SET-UP OF THE EXPERIMENTS
4 CONCRETE COMPOSITION AND PREPARATION OF THE SPECIMENS 15
5 TESTING EQUIPMENT 19
6 PERFORMANCE OF THE EXPERIMENTS 27
7 RESULTS OF STATIC TESTS 27
7.1 Compressive strength and splitting tensile strength 27
7.2 Uniaxial tensile strength 28
8 RESULTS OF DYNAMIC TESTS 31
8.1 Number of cycles to failure 31
8.2 Cyclic deformations 33
8.3 Shrinkage 33
8.4 Static tensile test results for specimens subjected to dynamic 36 loadings
9 ANALYSIS OF DYNAMIC TEST RESULTS 39
9.1 Wöhler-diagrams 39
9.2 Influence of scatter of stress-strength levels on the 44 variability of the number of cycles to failure
-4-page
9.2.1 Theory 44
9.2.2 Application to the results of this investigation 47
9.3 Cyclic creep velocity versus number of cycles to failure 50
10 PRELIMINARY CONCLUSIONS 53
REFERENCES 55
-5-Notation
f' - static compressive strength N/mm^
f . -
static tensile strength N/mm^
f 1 - static splitting strength N/mm^
- indicates average value
— m ^
E - Young's modulus N/mm^
N - number of cycles to failure
n - number of cycles
a -
maximum stress-level N/mm^
max '
a • -
minimum stress-level N/mm^
m m '
ci -
strain at maximum stress in static test
C2 - maximum strain in static test
v - coefficient of variation
%
-7-Summary
This research program deals with fatigue of plain concrete subjected to
cyclic uniaxial tensile loadings or alternating uniaxial compression-tension. In the main program more than 250 constant amplitude tests at a frequency of 6Hz have been performed. One concrete composition was investigated (average compressive strength: 50 N/mm^ and average tensile strength: 2.6 N/mm^). The dynamic stress alternated between two stress-strength levels, namely, 40% to 90% of the static tensile strength for the maximum level, and 40% to 0 (tension) or 0 to 30% (compression) for the minimum level. The numbers of cylces to failure have been related to these maximum and minimum levels and presented in Wöhler-diagrams (S-N). Drying as well as sealed specimens were tested. By means of statistical methods the effect of the variability of the static strength on the scatter of the dynamic results was determined. Cyclic deformations were also recorded and given as a function of the cycle ratio (n/N). The secondary creep velocity was compared with the numbers of cycles to failure.
The aim of this report is to describe the testing equipment and to present the results. The theoretical approach will be published later.
i c c h n i s c h e Ilojjeschool
- 9 - Ei';i;::hce': Afdeling: Civiele 'i :;-::niek
1 INTRODUCTION Stcv;rr.vcg l posihus 5ü '3 2600 G A Delft
The behaviour of plain concrete under cyclic axial tensile and tensile-compressive loading is the subject of this investigation, which has been carried out in cooperation with CUR-VB (Netherlands Committee for Research, Codes and Specifications for Concrete). Also represented on CUR-VB Committee C-33 "Dynamic loads" are the Institute TNO-IBBC for Building Materials and Building Structures and the Magnel Laboratory of Ghent University (in Belgium).
The part played by TNO-IBBC consists in experimental research concerning the fatigue behaviour of plain concrete under repeated compressive loads, with the emphasis on checking Miner's rule for different kinds of loads including random loads |_ 1 J .
In the Magnel Laboratory the behaviour of concrete under repeated tensile and tensile-compressive loads is studied by means of centre point bending tests.
Because of the application of concrete mainly in compression, hardly any-thing is known about the dynamic behaviour of concrete loaded in tension or tension-compression. With respect to this behaviour, however, knowledge is important because tensile stresses are permitted to occur in special structures (i.e., partially prestressed concrete) and because failure can be regarded as caused by local exceeding of the tensile strength (or ex-ceeding of the ultimate tensile strain). Investigations in the tensile region will achieve a better understanding of the limit state of the ma-terial and structures.
The dynamic tensile tests as mentioned in the literature are splitting tests or bending tests. Splitting tests were carried out by Linger et al. in 1965 [ 2 ] and by Tepfers in 1979 [ 3 "| .
Linger's tests were performed on cylindrical specimens {0 75 x 150 m m ^ ) ;
those of Tepfers were on cubes (150 x 150 x 150 m m ^ ) . The stresses applied during the dynamic tests were related to the static strength as determined on splitting test specimens from the same concrete. Both investigators compared their results from tension tests with results from compression tests. Fig. 1.1 shows the relation between the relative stress and the number of cycles to failure (Wöhler-diagram), as measured by Linger et al. (details in Appendix A ) .
-10-No difference was found between tension and compression, provided that the stresses were related to the static tensile and compressive strength respectively. This indicated that in both cases the dynamic behaviour of concrete is determined by the same material property.
peak stress/strength
3 /. 5 6 7 ^°log (cycles to failure)
Fig. 1.1 Fatigue life of concrete in tension and compression [_ 2 J
Also by Tepfers, no difference was found between tension and compression. This is demonstrated in his formula:
max
1 - e(i -mm
10
log N (1.1)max
In this formula a and a • are the maximum and the minimum stresses max m m
during a constant amplitude test, f^ is the static strength and N is the number of cycles to failure. The average value of the parameter B proves to be equal for both tension and compression; namely 3 = 0.0685. However, the coefficient of variation of B was about 17% for tension (normal con-crete) and about 20% for compression (normal and lightweight concon-crete)
provided that {o^^Jf) < 0.8 (details in Appendix A ) .
The behaviour of concrete subjected to repeated tensile stresses can also be studied by means of bending tests. An example is the experimental work of Raithby f 4"|. Third-point bending tests with beams (102 x 102 x 510 mm^) were performed. Three kinds of concrete having different ages at loading were used. If the stresses were related to the static modulus of rupture, it turned out that the test results could be represented by one line (see Fig. 1.2) (details in Appendix A ) .
•11-In this diagram the relation obtained with formula (1.1) is also shown. The difference between splitting and bending test results is not expected to be significant, because of scatter.
1.0 0.8 0.6 O.A 0.2 0
Max cyclic stress/modulus of rupture
= Concrete PQ1. U weeks to 5 years = Concrete PO 2, 13 weeks to 2 years = Concrete L C I . 13 weeks to 2 years = Indicates some
spe-clmens did not fail
modulus of rupture determined from flexural tests at
17 kN/m^.s
-0 1 2 3 4 5 6 7 8
^°log( number of cycles to failure)
Fig. 1.2 Fatigue performance related to modulus of rupture f o r 3 con-cretes at various ages, a l l cured under water |_ 4 J
The r e s u l t of s p l i t t i n g tests f 3 ~| i s also indicated
In the case of the above-mentioned methods of examining the dynamic ten-s i l e propertieten-s of concrete the ten-streten-sten-s d i ten-s t r i b u t i o n in the ten-specimen i ten-s not s u f f i c i e n t l y known. For fundamental research, however, i t is necessary to know the magnitude of the s t r e s s . That is why pure t e n s i l e tests are preferable.
U n t i l now no r e s u l t s are a v a i l a b l e w i t h respect to uniaxial tests mainly because of experimental d i f f i c u l t i e s i n connection w i t h mounting the specimens i n the t e s t i n g r i g and applying the loading u n i a x i a l l y .
Moreover, i n the case of uniaxial tension-compression tests special j o i n t s are necessary i n order to provide continuous t r a n s i t i o n of the loading. To perform the tests as described in t h i s report the said problems had to be solved.
-12-2 OBJECT
The object of this research program is to investigate for plain concrete the relation between cyclic axial load and the number of cycles to failure (Wöhler-diagrams). The loads are either tensile loads or they alternate between tension and compression. The investigation has been restricted to sinusoidal loads with a fixed frequency of 6Hz, and to one concrete com-position. All boundary conditions are held constant except the upper and lower limits of the loads, which are the only influential factors. In a later stage, also described in this report, the cyclic deformations are recorded as a function of time (cycles).
The deformation measurements serve as basic input for the development of a theoretical model. To avoid the influence of drying shrinkage on these deformation measurements, an extra series of experiments was performed using sealed specimens. The results of drying and sealed specimens are compared.
In this report mainly the testing equipment and the performance of the dynamic tests are described in detail. The results are summarized and analysed.
The study concerning the development of a theoretical model based on the fundamental material properties will be presented in a subsequent report.
-13-SET-UP OF THE EXPERIMENTS
Dynamic tests were performed with stresses alternating between adjusted
maximum and minimum values. The maximum was always a tensile stress; the
minimum was tension, zero or compression (see Fig. 3.1a and b ) .
stress
max
a
o
o
stress
max
Fig. 3.1 The dynamic stress varies between two tensile stresses (a) or
between tension and compression (b)
The dynamic stresses were related to the static tensile or the static
com-pressive strength. These static strength values were determined on
speci-mens from the same concrete batch as the dynamic specispeci-mens.
In Table 3.1 the investigated combinations of maximum and minimum relative
stress levels are given. In tension more combinations were investigated
because preliminary experiments proved the need to obtain accurate Wohler
diagrams.
For every stress combination a minimum of three specimens from different
batches (in principle) was tested.
The experiments were performed with drying specimens. Also a few tests
were carried out with sealed specimens. In that case the combinations as
given in Table 3.1 were investigated only for minimum stress-levels,
namely, 0.0 and 0.3 x f , .
-14-Table 3.1 Experimental scheme for the dynamic tests
minimum -• maximum 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.40 X X X X X X X tension °min''^ctm 0.30 X X X X X X X 0.20 X X X X X X X 0.00 X X X X X X X compression 0.10 X X X X X X 0 . /f' min' cm 0.20 X X X X X X 0.30 X X X X X X
Because the development of a theoretical model started at a later stage of the research program, the cyclic deformations were determined only for part of the investigated specimens.
In principle e\/ery dynamic test was continued till the specimen failed
or till 2 X 10 cycles were reached. The static tensile strength was
measured imrrediately after the dynamic test, for the unfractured specimens, As mentioned before, the dynamic tests were carried out with a sinusoidal load signal having a fixed frequency of 6Hz and using one concrete com-position (B ^ 5 ) .
-15-CONCRETE COMPOSITION AND PREPARATION OF THE SPECIMENS
Concrete composition
A concrete mix attaining an average 28-day compressive cube strength of about 50 N/mm2 (B 45) was used. From other tests with this mix it was known that variations of the compressive strength could be kept quite small. The mix composition is given in Table 4.1.
Because of the size of the specimens, the maximum particle size was 16 mm, The grading curve is shown in Fig. 4.1.
Table 4.1 Concrete mix composition
type of cement : Portland cement type I I I (B) cement content : 325 kg/m^
aggregate (sand + gravel): 1942 kg/m^ water-cement r a t i o : 0.50
proportion retained on sieve (%)
0.25 Q5 1,0 2.0 Ü0 8.0 16.0 sieve aperture (mm)
-16-Specimens
Tapered cylindrical specimens were chosen for the dynamic static tensile tests. Sizes and shape are shown in Fig. 4.2. Standard cubes (150 x 150 x 150 mm3) were used for determining the compressive strength and splitting strength.
É
0120
0U5
CD7-' en ID \
Fig. 4.2 Specimen geometry
and dimensions
Casting and curing
Because in one week a maximum of 6 specimens could be tested dynamically, and another 6 specimens were necessary for determining the static tensile strength, 12 test cylinders and 6 control cubes were cast per week. By means of preliminary tests a casting and compacting procedure was devel-oped with the aim of minimizing the scatter of the tensile strength.
In the beginning of the research program plastic moulds were used. The cylinders were cast and compacted one by one. After that steel moulds were employed, which were filled together. For the compaction a special frame-work was constructed on the vibrating table (see Fig. 4.3 and Fig. 4.4). After casting, the moulds were covered, and 48 hours later the specimens were demoulded and immersed in water of 20 ° C At an age of 14 days, the
•17-Fig. 4.3 View of steel mould
-18-specimens were placed in the laboratory (temp. ^ 20 °C, RH ^ 45%). When an age of 28 days was reached the static and dynamic tests were performed in a 7-day period. The same working method was followed for the sealed specimens, except that the specimens were immersed in water for 3 weeks and then wrapped in plastic sheeting to prevent drying out. The sealed specimens were also tested at an age of 28 days.
Fig. 4.5 Gluing the steel plates to the specimen
For the application of the tensile loads, steel disks were glued to the bottom and the top of the cylinders. For this purpose the cement layer was removed and the concrete surface was roughened before the steel plates were fixed using a two-component epoxy glue (Bolidt, compound E ) . For the sealed specimens (having a higher tensile strength), it was necessary to saw off about 5 mm from the upper and lower part of the specimen in order to prevent failure in those parts which had been affected by mould oil, wall effects, etc. Also, a different type of epoxy glue was used (Ciba Geigy OW 5-15).
To glue the plates in an axial and plane-parallel manner, a special gluing press was designed (Fig. 4.5).
-19-TESTING EQUIPMENT
The static tensile tests and the dynamic tests were performed in a newly developed testing machine. This machine is shown schematically in Fig. 5.1a and is also illustrated in Fig, 5.1b.
The framework of the machine consists of standard steel beams, and in order to provide axial load application the specimen is placed between two swivel heads. Special prestressed swivel heads were designed for obtaining a
continuous loading signal in the specimen during compression-tension tests.
V///A
prestressed
swivel head
actuator
spiral washers
oad cell
steel platen
prestressed
swivel head
^7777777777,
Fig. 5.1 Test set-up for uniaxial dynamic tests
In Fig. 5.2 the principles of these swivel heads are explained. The basic element is a prestressed spherical bearing. By means of conical roller bearings extra rotation in one direction is possible to facilitate
mount-
-20-ing the specimens. For continuous load transfer the connections between actuator, load-cell and specimen were also prestressed with spiral washers. The capacity of the actuator was 100 kN for both compression and tension. The range of the load-cell was the same. The specimens were mounted in the testing machine by means of bolted connections.
• to loading rig bolts to apply prestress i n 0 and inC ® prestressed spherical bearing © c o n i c a l roller, bearing to specimen
Fig. 5.2 Schematic view of prestressed swivel head, designed to apply alternating tension-compression.
It is to be noted that the dynamic experiments with a minimum stress level of 30% of the static compressive strength were carried out on a standard Schenck dynamic testing machine (400 kN) because of insufficient capacity of the machine as described above (see Fig, 5,3).
Loading system
Schenck equipment was used for load control. In a schematic view of the loading system (Fig. 5.4) it can be seen that the electric signal from
•21-Fig. 5.3 Test set-up for dynamic test with a minimum stress level of 30% of f' cm servo valve i control signal hydraulic power supply J_L actuator c >»• load cell
ii
'^/////////77^
T7
control/data unit program input servo controllerfeedback signal load cell
conditioner
Deformations
The longitudinal deformations Vvnre measured with linear vol Lage 'J/^spl.ace-ment transducers (HBM type W<:;K) , on L'VO oppo:^ite sides of tho sni^cimen. These transduce! ^ were atcached hy me;n>5 of qrip'^ gli'^J ir. thr^ srf:'''i"ipn
(glue; T n d o x , r-88). Fici. 5.;- shows the attdi:;hiMer.t oi' cha tr-inj.i;icers.
di:ipla::.em(?nls transduc.T
universal joint magnetic connection
Fig. 5.5 Attachment of transaucer for the deformation measurements
The measuring signal of the strains was amplified with a HBM type KWS/..-; S-b amplifier. The fact that the cross-section was not constant over the total gauge length was taken into account for the calculations of i;train::.. By comparing these strain measurements with results obtained with a gauge length over a constant cross-section, a fictitious gauge length was
deter-mined. ;• ?0Q 180 150 UO 120 100 SO 60 40 20 "' 0 1 2 3 ;. ;, 6 7 8 9 10 11 12 number of cycleu
Fig. 5.6 r.'ie generated loading signal
-23-Control and recording
For this research program an electronic instrument was developed for two purposes. On the one hand, to generate a load signal for the dynamic tests; on the other hand, to record the data during these tests. The instrument consists of a function generator and a data acquisition system. The control signal for the dynamic loadings is shown in Fig. 5.6.
This figure shows that the loading was applied gradually. After about 6 cycles the adjusted maximum and minimum stress levels were reached and stayed constant during the remaining cycles.
The increasing of the loading as a function of time is expressed by the following relation:
F(t) = (A - B cos 2 Tft)(l ^-t/0.25^ (5.1)
f = frequency (= 6Hz) A = offset (kN) B = amplitude (kN).
During the dynamic tests the load and deformation signals were stored in the memory of the data acquisition system. For that purpose 8 pairs (stress-strain) of measurements per cycle were done after equal time steps of 1/48 sec. at 6Hz (see Fig. 5.7). An adjustable synchronization system
sample interval 1/^8 sec
1 2 3 4 5 6 7 8 time
Fig. 5.7 Sampling the loading (and deformation) signal
provided measurements very close to the maxima of the signal.
The data output was stored on paper tape. However, the complete data set was not punched on paper tape, but only the results of two subsequent
cycles after exponentially (power of 2) increasing time intervals (see Fig. 5.8)
15 15 After an interval of 2 the interval remained constant (2 cycles).
-24-scan
num.
phase
num.
®
1
1
(D
l A A
2 1
(D
A A |
2 ^
2'
, 2\.
M» »4« X-/A
12 13 14
!AA
©
'AA !AA
„ 2 ^ , 2^^, 2^^time interval (cycLi)
Fig. 5.8 Organisation of data output
The procedure as indicated in Fig. 5.6 remained unchanged until the in-stant of failure. Then the total content of the memory was punched on
paper tape. This means that the last 397 pairs of results (i.e., 50 cycles) became available. A schematic view of the whole measuring system is shown in Fig. 5.9. load cell 3 *
ÏI
papertape punchJL
0
displacements transducersttf
D -oscilloscope @ recorder- t f l
teleprinter control/data unit temp 1 peak temp 2 ^ , detection "•" (load or strain) amplifier amplifier time base generatorFig. 5.9 Schematic view of the control and measuring system
Temperature control
In tension the ultimate strain is about 100-200 microstrain. The thermal linear coefficient of expansion of concrete is about 10-15 microstrain. So it was necessary to restrict temperature variations in the specimen. For that purpose a temperature unit was placed around the specimen. In
•25-this unit the maximum temperature variation was about + 0.2 °C. Throughout the execution of the test program the average temperature was about 21 C and the RH about 40-45%.
Fig. 5.10 shows the equipment for temperature control. It consists of a thermostat-controlled water bath connected to the temperature unit by a plastic tube. A similar tube has been placed spirally in this unit (see also Fig. 5.11). actuator grip cooling load cell I thermo electric couples environmental chamber prestressed swivel head VTTTTTTTTZ F i g . 5.10 Equipment f o r temperature c o n t r o l Fig. 5.11 Part of environmental
-26-In Fig. 5.10 the grip cooling is also shown. This cooling prevented heat transmission to the specimen. The temperature was measured with thermo-couples.
A general view of the control system is shown in Fig. 5.12 and Fig. 5.13.
hydraulic power supply ^ V777Z777777Z7.
v^y/zz
^777777777T, recorder teleprinter control/data unit s t o p peak displ sync force data temp 1 temp 2 gen time base generator thermal compensator cycle counter time base generator recorder amplifier amplifier servo controllera
Fig. 5.12 Overall schematic diagram of test equipment
Fig. 5.13
Control and
-27-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 tensile strength. The compressive strength of the specimens was estimated using the average value of the
static cube strengths as 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 1% 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.
More details on the execution of these experiments are given in Appendix B.
RESULTS OF STATIC TESTS
Compressive strength and splitting tensile strength
To check the concrete quality, the splitting strength and compressive strength were determined. The compressive cube strength was also used as the reference value for the stress levels in the dynamic tension-compres-sion tests. The average of three test results and the coefficient of variation for each investigated concrete batch are given in Table 7.1. The splitting strength has been calculated from the formula:
F
f T = 0.64 ~ (7.1)
cspl a2 ^ ' F = ultimate force (N)
-28-Table 7,1 Compressive strength and splitting tensile strength
casting no. 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 59 60 61 62 averag f' cm N/mm2 45.07 45.69 46.96 47.24 48.96 50.83 46.86 51.20 48.81 49.18 50.86 48.02 45.60 47.18 46.24 48.82 49.24 46.73 49.08 48.12 47.11 47.41 53.03 49.34 50.72 e V % 3.6 10.9 4.7 5.8 6.1 3.4 8.7 4.1 1.9 1.7 3.4 2.3 20.8 5.9 4.6 4.0 3.4 2.9 1.6 2.2 2.2 8.4 5.3 1.6 1.5 f csplm N/mm^ 2.99 2.89 2.82 3.39 3.32 2.90 2.96 2.98 2.76 2.94 3.00 2.91 2.86 3.25 3.51 3.01 3.00 2.82 2.43 2.53 2.45 3.19 2.94 3.14 2.93 V 6.3 2.4 6.1 9.8 9.7 11.7 8.8 9.8 12.9 1.2 2.3 14.4 12.4 5.1 6.8 4.5 9.6 8.5 9.6 10.1 8.8 7.2 7.4 4.5 3.9 casting no. 64 65 67 68 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 f' cm N/mm2 49.08 51.51 47.18 43.62 47.27 50.28 45.85 42.47 45.07 46.90 45.52 44.29 43.67 44.68 43.49 45.85 43.26 45.54 44.49 47.51 51.73 50.52 47.27 50.34 50.14 47.54 V t 8.1 3.9 3.3 0.6 0.5 2.3 1.9 2.1 2.1 3.1 0.8 1.5 1.9 2.4 3.9 5.4 2.2 7.1 4.8 1.8 3.8 3.6 2.9 0.6 2.2 3.6 csplm N/mm2 3.87 3.22 2.87 3.07 2.59 2.88 2.36 2.40 2.86 2.81 2.84 2.25 2.52 2.62 2.21 2.39 2.30 2.44 2.46 2.71 2.49 2.98 2.78 2.92 3.09 2.83 V % 12.7 5.9 4.5 1.9 12.1 3.2 7.0 11.5 13.2 11.4 6.6 1.0 5.4 9.6 4.6 6.1 7.7 7.0 8.2 8.0 13.7 7.7 0.6 5.3 8.7 7.5
Uniaxial tensile strength
The tensile strength of the dynamic test cylinders was estimated by deter-mining the average static tensile strength of 5 or 6 specimens from the
same concrete batch. To characterize the concrete used, the ultimate tensile strain, the strain at the ultimate load and the Young's modulus were also measured (see Fig. 7.1).
The results are summarized in Table 7.2.
The fracture locations are indicated in a histogram in Fig. 7.2. In Fig. 7,3 a typical fracture surface is shown.
-29-stress ' c l O i f c l -A\Ec strain Fig. 7.1 Load-controlled s t a t i c t e n s i l e t e s t D e f i n i t i o n of f . , E , c-^ and ez
Table 7.2 Results from s t a t i c uniaxial t e n s i l e t e s t s ; average values and c o e f f i c i e n t s of v a r i a t i o n of f . , E , ci and ez c a s t i n g no. 17 18 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 59 60 61 62 64 65 66 67 68 71
K
11^
8 0 ; 81* 82^ 84" 85 86 87 88 89 90 91 number of specimens 6 7 6 6 6 6 4 4 4 7 7 6 4 5 5 6 6 6 5 7 6 5 5 5 5 4 6 5 5 5 5 5 5 3 3 4 5 4 4 4 5 5 5 4 5 5 4 5 ^ctm N/mm2 2.39 2.07 2.35 2.03 2.37 2.09 2.20 2.40 2.12 2.36 2.45 2.38 2.52 2.40 2.48 2.41 2.54 2.42 2.59 2.35 2.57 2.45 2.63 2.47 2.55 2.56 2.57 2.72 2.56 2.52 2.60 2.39 2.75 2.92 2.72 2.73 2.93 2.78 2.80 2.78 2.69 2.55 2.29 2.30 2.49 2.62 2.37 2.64 V % 5.1 7.2 5.9 9.7 6.7 9.9 10.0 6.1 7.8 12.6 7.1 6.9 5.9 7.9 9.8 8.3 9.0 8.8 9.6 7.4 5.3 8.3 3.5 2.8 4.5 9.8 8.1 3.8 5.5 7.3 4 . 8 8.0 1.9 4.6 4.2 5.2 3.6 5.1 3.2 3.5 2.2 6.7 6.6 7.2 5.9 8.5 6.3 6.3 E cm N/mm2 V % not determined 34574 34808 34657 33408 34010 39020 35585 37880 36011 32277 34616 34078 38816 35170 34054 38337 35256 33932 36200 37135 37336 37582 7.9 7.1 6.7 1.9 1.2 21.4 6.0 12.1 3.4 7.4 8.1 6.0 18.2 8.1 5.9 5.5 5.7 3.3 14.0 5.2 14.2 1.5 E l 10-6 V % not determined 105.9 104.4 96.2 105.2 108.6 87.3 116.7 123.9 116.0 129.8 133.0 125.9 117.9 125.5 121.2 90.1 94.6 89.4 91.2 92.6 89.5 97.3 9.1 7.2 7.0 12.4 5.2 10.4 3.0 10.7 6.4 11.2 6.8 12.9 8.9 10.2 5.8 5.5 11.8 8.1 7.9 8.7 9.3 8.2 10-6 V % not determined 107.2 108.2 100.1 109.4 113.7 92.0 121.8 129.7 118.3 136.7 138.3 127.9 123.7 130.0 126.0 95.0 98.7 99.3 97.5 99.9 97.1 100.9 8.2 8.8 8.0 12.2 4 . 1 10.9 4.7 9.6 5.8 10.9 5.3 11.8 6.4 9.7 5.6 3.5 9.5 6.2 7.6 7.7 8.6 4 . 8 sealed specimens
303 0
-relative frequency (%)
^ - ^ ^
• direction of casting
Fig. 7.2 Fracture locations in the static axial tensile test
Fig. 7.3 Typical fracture mode and fracture surfaces of a specimen subjected to uniaxial tensile loading
-31"
RESULTS OF DYNAMIC TESTS
Number of cycles to failure
A complete survey of the results of the 219 dynamic tests is given in
Appendix C, both for drying and sealed specimens. The averages of the
""•^log of the number of cycles to failure are summarized in Table 8.1.
A survey of the fracture locations is given in a histogram (see Fig. 8.1)
Table 8.1 Results of dynamic tests. The averages of the log of the
number of cycles to failure are given for the minimum and
maximum stress-strength levels investigated. The standard
deviations are given in brackets
niinimutn -*-maximum °max' ctm 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 tension °min/'''ctm 0.40 run-out 3.47 run-out 3.21(0.03)** 2.96 2.93(1.10) 0.30 run-out run-out 3.35" run-out 3.24(0.40)*' run-out 2.27(0.51)" 2.84(0.63). 2.42(0.28)** 1.87(0.80) 2.29(0.64)** 0.20 run-out run-out 3.17(1.03) 3.21(0.75)** 2.79(0.91) 1.95(0.84) 2.08(0.72) 0.00 run-out run-out run-out run-out 3.18(0.01)** 2.49(0.33) 3.20(0.94)** 2.58(0.78) 2.49(1.37) 2.28(0.63) 2.94(0.94) 1.79(0.10)** compression a . / f ' min cm 0.10 run-out run-out 4.44(1.12) 3.14(1.06) 4.44(0.81) 2.82(0.25) 2.89(0.66) 1.98(0.37) 0.20 run-out 4.78(1.08) 4.96(0.89) 3.16(0.96) 2.83(0.83) 1.64(0.42) 0.30 run-out**** 3.72(0.31) 3.54(0.66) 2.48(0.49) 2.48(0.13)
XX includes specimens which did not fail before 2 x 10^ cycles X sealed specimens
-32-Table 8.2 The log of the numbers of cycles to failure at the indicated stress-strength levels for drying concrete | 5 ~|
no. 48-8 ** 48-11 48-14 49-14 50-1 50-12 53-11 53-12 53-15 54-12 57-1 57-5 58-11 50-6 50-8 51-2 51-4 51-8 51-12 52-5 56-5 57-8 stress-st ''min' ctm 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 0.44 **48-8 -ength level "max ctm 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.81 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 log cycles to f a i l u r e 3.02 1.85 2.22 3.45 1.93 3.83 2.41 2.89 3.70 4.60 3.15 4.12 3.43 4.50 1.48 4.08 > 5.69 4.86 > 5.58 > 5.71 2.75 > 5.70 no. 49-11 52-15 55-4 56-7 55-12 57-14 58-2 58-15 48-15 49-8 51-11 56-2 57-7 58-6 stress-strE °min' ctm 0.11 0.11 0.11 0.11 0.11 0.11 0.43 0.43 0.08 0.08 0.08 0.08 0.08 0.08 ngth level "max' ctm 0.68 0.68 0.68 0.68 0.68 0.68 0.73 0.73 0.43 0.43 0.43 0.43 0.43 0.43 log cycles to f a i l u r e > 6.00 > 6.15 > 5.95 > 5.97 5.53 > 5.99 > 5.72 > 5.96 > 6.20 > 6.30 > 6.15 > 6.17 > 6.17 > 6.39
casting no. specimen no,
relative frequency (%)
- ^ ^
direction of casting
^ ^
-33-Research for a Master's degree thesis is comprised in this program |_ 5 J. For the purpose of this research, similar specimens were used under the same conditions as described before. The drying specimens were tested at the stress levels given in Table 8.2 in which also the number of cycles to failure are stated.
8.2 Cyclic deformations
Cyclic deformations were measured in the direction of the loading on several specimens as indicated in Appendix C. Because most damage will occur at the relatively high stress-strength level in the tensile part of the loading cycle, the strains at these levels were computed as a function of time.
By means of a method described in Appendix D, the tensile peak strains were derived from the eight discrete measurements carried out in every cycle.
In the case of compression-tension tests the strain at zero stress, in the loading branch of the cycle, was used as a reference value for the calculation of the maximum tensile strain of that particular cycle. Some typical results are shown in Fig. 8.2a and 8.2b for tension-tension tests. Examples of strain measurements results for compression-tension tests are presented in Fig. 8.3a and 8.3b. More results are given in Appendix E.
8.3 Shrinkage
The drying specimens also show shrinkage deformations while under load. As the dynamic tests were performed during 8 days after a curing period of 28 days, shrinkage was measured from t = 28 days to t = 36 days. The results also reported in |_ 5 ~| are presented in Fig. 8.4. In the case of sealed specimens no shrinkage was detected in this period.
It can be concluded that during a testing period of 4 days (= 2 x 10 cycles at 6Hz) the average deformation due to shrinkage will be about 17 microstrain for the drying specimens.
-34-UO 120 100 80 60 /.O 20 microstrain
1*!]
r
^ ^ ^ —""""" -—•—• • _ _ _ . i_ - - — - — — nunn 1. 820A 2 8112 3.7105 i. fi715 -3 level (03-08) (03-08) (0.3-0.8) ( 0 3 - 0 8 ) sealed • ^ l^logN 263 191 293 3 99 0 0.1 0.2 0.3 0.^ 0.5 0.6 07 0.8 0.9 1.0 n/NFig. 8.2a Total peak strain versus cycle ratio for dynamic tension-tension tests on drying and sealed specimens
UO 120 i n n 80 bO AO 20 C microstrain /'^ ^ -[ • ^ — " ^^-^ / ) 0 _ - - : 1 0 '•^ -- - • 2 0 — — -3 0 - -
—-u
0 — — _ 5 0 _1__ — 2 " 3 4 5 6 0 num 1 8009 2.7512 3 6708 4 7104 5 6411 7 0 __ •— level (02-0.75) (0.2-075) (02-075) (O2-0.70) (02-0.70) l e a l e d 8 0d
nJ
l ° l o g N 268 3 73 2 68 593 493 9 1,( n/NFig. 8.2b Total peak strain versus cycle ratio for dynamic tension-tension tests on drying and sealed specimens
3 5
-uo
120 100 80 60 AO 20 microstrain ^ — 2 • ^ -num 1 8905 2. 8812 3 8715 4 9004 _ _ . ^ = = level (-01-0 85) (-01-0 80 (-0.1-080 (-01-0 85 ^ ^ ™log N 2.32 3.02 287 3.6001
0.2 0.3 O.A 0,5 0.6 0.7 0.8 0.9 1.0 n/NFig. 8.3a Total tensile peak strain versus cycle ratio for dynamic compression-tension tests on drying specimens
UO 120 100 80 microstrain 60 ^ i,0 20 ^ 1 3 num 1. 9114 2. 9104 3 9008 level 1-02-080) (-02-0.70) (-0.2-0.80
1
J
y
™log N 3.54 4.82 376 0.1 0.2 0.3 O.A 0.5 0.6 07 0.8 0,9 1.0 n / NFig. 8.3b Total tensile peak strain versus cycle ratio for dynamic compression-tension tests on drying specimens
Bibl'othse!^
-36-shrinkage microstrain
6 8 10 time (days)
Fig. 8.4 Drying shrinkage between 28 and 36 days after casting, measured on 8 test cylinders from different batches but the same concrete mixes
Static tensile test results for specimens subjected to dynamic loadings
The uniaxial static tensile strength of the run-outs (no failure before 2 X 10 cycles) was determined. This strength was compared with the aver-age uniaxial tensile strength of the specimens (from the same concrete batch), which were not dynamically loaded before, according to:
Af Ct f .^(preloaded)
ctm
ctm
(8.1)The results are given in Table 8.3. In this table the modulus of elasticity and the strains at failure are also mentioned and are compared with the results obtained on the non-preloaded specimens. A survey of the differ-ences of tensile strength is presented in Fig. 8.5. The fracture locations are indicated in the histogram of Fig, 8.6,
-37-Table 8.3 Results of static tensile tests on specimens previously subjected to dynamic loads
specimen number 36-10 32-06 33-08 28-04 28-09 29-08 30-12 89-09 88-04 89-09 81-02** 65-05 26-11 23-12 23-10 23-02^ 84-02** 83-11** 68-14 66-11 20-08 72-04 59-02 18-12 18-08 83-06** 62-15 62-07 60-07 38-12 65-06 72-15 67-02 64-15 21-12 21-11 21-10 64-05 61-01 37-10 36-05^ 85-01** 72-09 72-06 22-09
!ii^^-^^ ^
min. comp 0.30 0.20 0.20 0.10 0.10 0.10 0.10 0.10 0.10 0.10 max. 10^ ^ct°f run-outs N/mm2 ^^ct % ression-tension tests 0.70 0.40 0.40 0.40 0.40 0.40 0.50 0.50 0.50 0.50 >3.2 >3.0 >2.1 >2.0 >1.0 >2.4 >0.5 >2.0 >2.0 >2.5 tension-tension 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.20 0.20 0.20 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.55 0.60 0.60 0.60 0.60 0.60 0.65 0.65 0.65 0.65 0.70 0.65 0.70 0.70 0.70 0.65 0.70 0.70 0.70 0.70 0.75 0.75 0.75 0.80 0.80 0.80 0.80 0.70 0.70 0.70 0.70 0.80 0.80 0.80 0.90 X sealed specimens XX see F ig. 7.1 >2.0 >2.5 >2.0 >0.5 >0.5 >1.4 >3.0 >0.5 >5.3 >2.0 >2.0 >2.6 >2.5 >1.0 >2.4 >2.4 >1.5 >2.1 >2.0 >2.6 >2.1 >5.5 >2.5 >0.4 >0.5 >1.9 >0.5 >2.5 >2.1 >3.1 >2.0 >1.0 >2.9 >2.0 >2.4 2.92 2.61 2.47 2.32 2.44 2.19 2.34 2.46 2.30 2.45 tests 2.70 2.70 2.77 2.39 2.28 2.53 2.92 2.77 3.40 2.94 2.49 3.08 2.69 2.42 2.28 2.83 2.52 2.72 2.85 2.62 2.92 2.83 2.79 2.87 2.50 2.47 2.19 3.10 2.51 2.44 2.87 2.70 2.83 2.79 3.12 11.3 5.0 2.4 -5.6 -0.4 -8.7 -7.7 -6.1 -7.6 -6.5 -3.0 4.8 23.5 12.6 8.3 17.4 7.9 -0.4 25.9 7.5 5.6 22.4 8.9 14.5 9.2 1.8 - 1.2 6.3 7.7 1.9 12.0 15.6 8.2 10.8 18.8 17.8 7.3 17.4 1.6 3.7 9.8 5.6 15.6 14.3 24.0 E/** of run-outs N/mm2 27446 27666 33172 36489 40607 36250 35100 38976 34022 35891 34260 40544 34804 34260 ^^c % -24.1 -25.5 - 2.7 5.5 15.5 8.5 0.8 - 0.1 - 3.3 3.8 - 1.1 5.8 -10.8 -12.2 XX j-£l of run-outs 10-^ 96.6 104.5 105.4 100.5 96.8 117.3 100.5 97.6 106.9 94.6 104.5 85.7 95.6 100.5 Aei /o 5.9 12.9 -17.0 - 5.1 -22.9 11.5 - 3.7 11.8 -14.8 -10.7 8.6 - 4.9 9.5 15.1 £2 of run-outs 10-6 106.4 107.4 107.4 100.5 103.8 123.2 100.5 97.6 112.9 94.6 107.4 88.7 99.5 107.4 A E 2 % 9.1 7.5 -16.0 - 6.3 -20.2 12.6 - 7.1 6.1 -13.2 -11.8 7.3 - 6.6 8.2 16.73 8 -% (fct'P''^"'°°'^^^'-fctm )/fctm 120 100 80 60 /.O 20 O
a = before cyclic loadings b = a f t e r cyclic loadings
(average values of t a b l e B.3 I
O.lxfcm 0.0 0.2xfctm 0.3xfctm 0./.xfctm minimum stress - strength level
F i g . 8.5 Comparison of s t a t i c t e n s i l e strength values of specimens subjected to dynamic loadings and non-preloaded specimens
relative frequency (%)
- V
direction of casting
Fig. 8.6 Histogram of f r a c t u r e locations i n s t a t i c t e n s i l e t e s t s on run-outs
-39-9 ANALYSIS OF DYNAMIC TEST RESULTS
9.1 Wöhler-diagrams
The results of the dynamic tests given in Table 8.1 and in Appendix C can be represented in so-called Wöhler-diagrams in which the number of cycles to failure is related to the dynamic stress level parameters. For ewery investigated minimum stress-strength level the test results have been plotted in the Wöhler-diagrams of Fig. 9.1 to Fig. 9.7.
The relation between the logarithm of the number of cycles to failure and the dynamic stress-strength level was investigated by means of multiple linear regression. Distinction was made between tension-tension and compression-tension tests.
To each of the two sets of results a linear model was fitted having the general form:
1^^ M D , D r , D min , n max , „ min max ^ ^ /o i N
log N = B -H Bi .C + 02. J + B3. -7 -I- B(+. r . j + e^ (9.1)
ctm ctm ctm ctm
B . . . Bh = c o e f f i c i e n t s 0
C = experimental c o n d i t i o n ; C = 0 f o r drying specimens and C = 1 f o r sealed specimens
IB = random e r r o r .
Tension-tension tests
The zero-tension results (see Fig. 9.4) were also included in t h i s part of the regression analysis. The t o t a l number of r e s u l t s was 137.
I t was decided to t r e a t the run-outs as normal r e s u l t s . In t h a t case the maximum number of cycles applied to the specimen was regarded as the number of cycles to f a i l u r e . Because the run-outs w i l l in f a c t f a i l (or not) at a higher number of cycles, t h i s approach r e s u l t s in r e l a t i v e l y less steep (safer) Wöhler-lines. When on the other hand the run-outs are ne-glected in the regression a n a l y s i s , the calculated Wöhler-lines are found to underestimate the number of cycles w i t h respect to the run-outs, where-as t h i s part of the Wöhler-diagram is most important in p r a c t i c e .
-40-Qmox/^ctm Log N F i g . 9 . 1 Wöhler-diagram f o r p l a i n c o n c r e t e s u b j e c t e d t o t e n s i o n - t e n s i o n a t a lower s t r e s s l e v e l o f 0.4 x f
ctm
1.0 0,8 0,6 OA 0,2 0.0 '^max /fctm J„~O.A J o cr-A-crSo A-O'-O.c^C ^ - f . , ^ A+0.,A~-•90%C.L. °min/fctm=0-3 I C.L = confidence l i m i t — O = drying specimen - - A = sealed specimen — > = run out I 0 10 Log N F i g . 9.2 Wöhler-diagram f o r p l a i n c o n c r e t e s u b j e c t e d t o t e n s i o n - t e n s i o n a t a lower s t r e s s l e v e l o f 0.3 x f , ctm 1,0 08 0,6 0.^ 0,2 0,0 ' - ' m a x / ^ c t m 90 % C.L -"min' 'ctm :0.2 _ L _ C.L. = confidence limit O = drying specimen A = sealed specimen —» = run out I ' ' ' 6 7 i ° L o g NFig. 9.3 Wöhler-diagram for plain concrete subjected to tension-tension at a lower stress level of 0.2 x f ,
-41-Log N
F i g . 9.4 Wöhler-diagram f o r p l a i n concrete subjected to tension-tension at a lower stress level of 0.0
LogN
F i g . 9.5 Wöhler-diagram f o r p l a i n concrete subjected to tension-compres-sion at a lower stress level of 0.1 x f '
cm °max''^ctm
Log N
Fig. 9.6 Wöhler-diagram for plain concrete subjected to tension-compres-sion at a lower stress level of 0.2 x f'
-42-Log N
Fig. 9.7 Wöhler-diagram f o r p l a i n concrete subjected to tension-compres-sion at a lower stress level of 0.3 x f '
cm
The regression analysis resulted in the following equations:
for_dr,^i^ng_sgecimens log N = 15.02 -f9!2_s§al§d_sgeci_mens
14.90 . 5 ^ + 3.13 . 5 ^
ctm
ctm
(9.2)log N = 14.29 - 14.90 '^max , o 10 ^niin
-E + J.io . -J
ctm
ctm
(9.3)
In (9.2) and (9.3) the standard deviation of the error is 1.11. It turned out that the experimental condition (drying or sealed) was significant (on 95% level) whereas the interaction term a . /f ^ . a /f was
m m ' ctm max' ctm not significant.
The regression lines obtained with (9.2) and (9.3) have been drawn in
Fig. 9.1 to Fig. 9.4. The 90% confidence regions have also been indicated. These regions were calculated from: log N + t . 1.11.
The statistic t is obtained from the Student t-distribution and is 1.66 at a confidence level of 90% and 133 degrees of freedom.
Compression-tension tests
In this case 82 test results for drying specimens were analysed. The run-outs were included in the regression procedure, which resulted in
-43-the following equation:
log N = 9.46 - 7.71 . - ^ - 3.78 . , ^ (9.4)
ctm cm
The standard deviation of the error was 0.87. The interaction term
^min/^cm " "max/^ctm P^°^^^ "°t to be significant.
In Fig. 9.5 to Fig. 9.7 the calculated Wöhler-lines have been indicated as well as 90% confidence regions from: log N + t . 0.87. The value of t is 1.67 (90% and 79 degrees of freedom).
Results of dynamic tests can alternatively be represented in Goodman-dia-grams: for a given number of cycles to failure the relation between the minimum and maximum stress-strength level is given. In Fig. 9.8 a modified Goodman-diagram was constructed using the calculated mean values of (9.2)
Fig. 9.8 Representation of dynamic test results in a modified Goodman diagram. The results in the compression-compression part are taken from f 6 ~| and [ 7 ~]
-44-and (9.4) for log N = 3, 4, 5 -44-and 6. As the compressive strength proved to be about 20 times the static tensile strength, it must be realized that for equal stress-strength ratios the actual stresses are about 20 times higher in the case of compression.
For comparison, the results of |_6, 7 J are depicted in the compression-compression part of the Goodman-diagram.
9.2 Influence of scatter of stress-strength levels on the variability of the number of cycles to failure
9.2.1 Theory
In order to conduct dynamic tests the maximum and minimum stress levels have to be adjusted to target values, these being generally percentages of the static strength of the specimen to be tested. This static strength cannot be accurately determined and is estimated from static test results of other specimens from the same concrete batch, resulting in deviating stress-strength levels and therefore scatter of test results. The better the static strength of dynamic test specimens can be estimated, the less the scatter of the results will be. That is why the variability of the static tensile and compressive strengths have to be minimized. To compare test results of different investigators it is important to know what part of the scatter can be attributed to random errors of the adjusted stress-strength levels and what part is caused by the possible stochastic nature of fatigue. Moreover, in practice the scatter of static strength is often taken into account by means of characteristic strength (lower limit of the strength distribution). When this strength is taken as the reference value, scatter in the unsafe direction of lifetime is mainly caused by the prob-ability aspects of fatigue itself, because strengths below the reference value are unlikely.
In the following, a method for distinguishing between random error of the stress-strength levels to be adjusted and the remaining error (stochastic nature, etc.) will be described. This method is due to Fedorov ] 8 |.
-45-The general linear model describing the dynamic test results as indicated
in equations (9.1) to (9.4) is:
a . 1)
^ogJl-B^,B,.^.Bz.^^e
(9.5)
ctm cm
The underlining denotes stochastic quantities. The dependent variable
log N is stochastic because of the random error £. It is assumed that the
distribution of e is normal, with mean zero and variance
a^,
and that e
— 0 —
is not correlated with log N.
In the dynamic tests maximum and minimum stress levels are adjusted.
These levels are intended to be ratios of the real static strength of the
specimen. Let the target values of the maximum and minimum stress-strength
levels be c and d respectively. Because the strength of the dynamic
speci-men is estimated, the maximum and minimum stress levels actually realized
are, however:
a
= c . f ,
max ct
a .
= d . f'
m m c
(9.6)
indicates estimated value.
With (9.6) model (9.5) will be as follows:
B . ^ B i ^•^ct y +B2 ctm d . f ' c ^cm
log N = Bg + Bi .-^
+B2.Y'
— + i (9-7)
ctm cm
Since the estimated values of the strengths can be written as the
ex-pected values plus an error term, (9.7) can be expressed more generally
as:
log N = B ^ - H B I . (c+ hi ) + B ? . (d-h h?) + e (9.8)
jii and J22 are the error terms of the maximum and minimum stress-strength
levels respectively. It is assumed that the expectations (the means) of
hj and h2 are zero and that their distributions are normal.
Equation (9.8) can be rewritten as:
-46-log N = B^ -I- B i . c + B2.d + Bi.Jii + B2.JI2 + £ (9.9)
Except f o r the two e r r o r terms with h^^ and h2, equation (9.9) i s i d e n t i c a l with equation ( 9 . 5 ) . The expectations of (9.9) and (9.3) are equal:
L log N = B^ + Bi.c + B2.d . (9.10)
I = expectation
The consequence of (9.10) is that the mean value of log N, or the position of the regression lines in the Wöhler-diagrams, is not affected by random errors of the stress-strength levels. However, the scatter of log N wil 1 be influenced. In equation (9.9) the variance of the total error is:
var(B^.hi -1- B2.h2 + e) =
Bf var _hi + B | var h^2 + 2Bi B2 cov(h^i, h^2) + ^ Q ( ^ - H )
Assuming that for the experiments the relation between tensile and com-pressive strength can be described by a linear relation, the terms of (9,11) can be calculated (see Appendix F ) :
var jij = v^ . c^
var ho = v^ . d ' — c
cov(fii, h2) = v,^ . v^ . c . d (9.12)
V, and V are c o e f f i c i e n t s of v a r i a t i o n ( r a t i o of standard deviation and t c ^ mean) of s t a t i c t e n s i l e and s t a t i c compressive strength r e s p e c t i v e l y . In the case of dynamic tension-tension t e s t s tiie expressions f o r these variances are s l i g h t l y d i f f e r e n t :
u 2 2
var Jli = V. . c
var J22 = v^ . d^
4 7
-In equation (9.11) the variance due to the error of stress-strength levels is separated from the remaining variance (a ) . So it is possible to deter-mine which part of the total scatter as measured in the dynamic tests is caused by erroneous adjustments of the stress-strength levels.
9.2.2 Application to the results of this investigation
In order to separate the scatter due to error of stress-strength levels from the remaining variance ( a ^ ) , equation (9.11) can be applied to the results obtained from the Wöhler-tests as described by model (9.2) for tension-tension and model (9.4) for tension-compression tests. Only the results of drying specimens are analysed here.
For V and v, the mean values of the coefficients as given in Tables
c t ^ 7.1 and 7.2 are used; namely v = 3.56% and v = 6.62%. The result of
the statistical analysis is shown in Fig. 9.9 to Fig. 9.15.
A considerable part of the scatter proves to have its origin in the
variability of the adjustments of the stress-strength levels. This effect increases at higher maximum levels. Because of a better estimation of the static compressive strength, the scatter caused by the levels in tension-compression tests is less than in tension-tension tests. On the other hand, with respect to o!^ (stochastic nature and also random errors caused by the execution of the e x p e r i m e n t s ) , more variation was found in tension-compression tests. 1.0 0.8 05 0^ 02 00 /fctr 'max' 'ctm '^min''ctm=°-^
0 1
Q : caused by estimation of stress-strength levels _!_ 6 7 i°log NFig. 9.9 A part of the 90% confidence interval in the Wöhler-diagram ^, is caused by scatter of static strength
-48-1,0 0,8 0,6 0.^ 0,2 0,0, 'max /fct m . ^ 1 ° m i n / ' c t m = 0 ' 3 1
. ^ 5 ^
^ ^ . . ^ ^ = ^ ^ 1 1 1 • : caused by estimation of stress-strength levels 10 F i g . 9.10 log N F i g . 9.11 1.0 0.8 0.6 OA 0.2 0.0 <^max O m i n = /^ctm "^"^5 D.O^ ^
^ ^ ^>^^^ ^
n:t;°u str« sed by ' s s - s t r e ^ estimat ngth lev : ^ on of els 0 1 6 7 ^°log N F i g . 9.12-49-1.0 0,8 0,6 OA 0.2 0.0 'max /fct m N O m i n ' ^ >
vj
im-O' . \ : N N . . ^ S ^k^
^k>
k
[ 1 1 T [ ] ] ; caused by estimation of stress-strength levels • 0 1 6 7 ^°log N F i g . 9.13 1,0 0,8 0,6 OA 0,2 n n Omaxx
°min' / f c t m:-^-^K
' S . ^ cm I_,xNlX
N ^ n : c a i sir \ ^N
jscd by ess-str( \ , estimat >ng!h le ^ ion of /els 5 6 7 ^°log N F i g . 9.14 F i g . 9.15F i g . 9.10 - 9.15 The i n d i c a t e d p a r t s o f c o n f i d e n c e r e g i o n s are caused
by s c a t t e r o f the s t a t i c s t r e n g t h , r e s u l t i n g i n v a r i a t i o n s o f the s t r e s s - s t r e n g t h l e v e l s i n the dynamic t e s t s .
-50-As can be seen here, the scatter in the Wöhler-tests cannot be entirely explained by the variability of the static strength values as found by others |_1, 7 |. Different lifetimes must be expected even when the static strengths of concrete specimens are equal. These differences are the result of the stochastic nature of fatigue and/or the random error due to the experimental procedure.
9.3 Cyclic creep velocity versus number of cycles to failure
Based on reaction kinetics, the following expression for creep velocity (ê) can be derived | 9 |.
é = a . a " (9.14)
a, n = material constants
Assuming that damage (degree of progress in failure) is associated with
the local plastic deformation and that the damage rate {^) is in
propor-tion to the local plastic strain rate, the following equapropor-tion is obtained:
n
()) = ai . a"i (9.15)
a^, n^ = local material constants
Integration of equation (9.15) in the case of failure (<)> = 1) leads to:
a"i . t^ - a^-^ (9.16)
t^ = time to failure
If the relation between £^. (the minimum creep rate) and a is assumed
to be given by (9.14), substitution into equation (9.15) gives the re-lation: .
(^min)'-tf = 1 ^ (9.17)
" i
5 1
-Equation (9.17) shows that a l o g - l o g p l o t of è • versus time to f a i l u r e w i l l r e s u l t in a s t r a i g h t l i n e .
The general form i s :
log t j : = A - B log é .
" f ^ mm
(9.18)
The minimum creep rate, é . , is often approximated by ê in the secondary part of the creep curve (see Fig. 9.16) where É is assumed to be constant.
strain
time
Fig. 9.16 In the secondary part of the
creep curve é is assumed to
be constant
Expression (9.18) was used in | 9 J to describe creep results of various
metals and in I 10 I to compare c and t^ for tensile creep tests on
i_ _i f s e c f '^
lightweight and normalweight concretes.
With respect to the experimental results of this investigation as given in Chapter 8, the cyclic creep velocity at maximum stress level is con-sidered. The creep rate in the secondary part of the creep curve is related to the number of cycles to failure. A log-log plot is shown in Fig. 9.17. By means of linear regression the following requation was found:
log N = -3.25 - 0.89 ^°log t
sec
(9.19)e per sec. sec '^
The standard deviation of the random error was 0.20, and the coefficient of determination (r^) was 92%. No significant difference could be shown to exist between drying and sealed specimens, though for a given value of è a larger number of cycles to failure tends to occur in the case
sec ^ -^ of sealed specimens.
-52-As a function of time t (in sec.) formula (9.19) can be rewritten as:
lOn
log t = -4.02 - 0.89 ""log i^^^ (9.20)
For normalweight concretes subjected to long-term t e n s i l e loadings the f o l l o w i n g r e l a t i o n was found in |_ 10 J :
log t = -4.06 - 0.81 "^^log é
sec (9.21)
This relation is also shown in Fig. 9.17. There is considerable similarity between the lines for creep at constant loading and for cyclic creep.
^°Log êsec (ê per second)
- 5
ion-tension • drying • sealed -compression o drying confidence Limit^°Log N
Fig. 9.17 log t versus log N for uniaxial dynamic tension-tension tests
and tension-compression tests as measured on drying and sealed specimens. The relation found in |_ 10 | for uniaxial tensile creep is also indicated
-53-10 PRELIMINARY CONCLUSIONS
As it is the aim of this report to describe the execution of the tests and the experimental results, the theoretical approach is treated briefly. However, preliminary conclusions can be mentioned.
1. In dynamic tensile tests on drying specimens the relation between stress-strength levels and the mean number of cylces to failure can be described by:
in a a .
^^log N = 15.02 - 14.90 . ^p^M + 3.13 . JELH (9.2) ctm ctm
In alternating tension-compression (a • /f' > 0) by:
III I 1 1 w l 11
^°log N = 9.46 - 7.71 . j ^ - 3.78 . j?^ (9.4)
ctm cm
As follows from (9.2) and (9.4) an increasing stress-strength amplitude will result in a decreasing number of cycles to failure.
2. In comparison with drying specimens a statistically significant lower number of cycles to failure was found in the case of dynamic tests on sealed specimens (Fig. 9.2 and 9.4).
3. Although low minimum stress-compressive strength levels were applied in the dynamic compression-tension tests, a considerable decrease in the number of cycles to failure was found as compared with zero-tension tests. Alternation from compression to tension seems to cause addition-al damage (Fig. 9.8).
4. By means of statistical methods the scatter of the test results in the Wöhler-diagrams could be explained in part by the variability of the static strengths, causing random errors of the stress-strength levels (Fig. 9.9 to 9.15).
5. Sealed specimens exhibited more total tensile peak strains than drying specimens (Fig. 8.2 and Appendix E ) .
Bibüothaek a'd. Civieie Stevirr.veg ',
•54-6. A marked relation exists between secondary creep velocity (at tensile peak stress) and the number of cycles to failure. The confidence region proved to be relatively narrow (Fig. 9.17). The following relation was found:
^"log N = -3.25 - 0.89 ^°log é (9.19)
5 5
-REFERENCES
1. Leeuwen, J . van, Siemes , A.J.M.
Miner's rule with respect to p l a i n concrete Heron, V o l . 24, No. 1 , 1979.
2. Linger, D.A., Gillespie, H.A.
A study of the mechanism of concrete fatigue and fracture H.R.B. Research News, No. 22, February 1966, pp. 40-51.
3. Tepfers, R.
Tensile f a t i g u e strength of p l a i n concrete ACI-Journal, August 1979, pp. 919-933.
4. Raithby, K.D.
Some flexural fatigue properties of concrete - effects of age and methods of curing
First Australian Conference on Engineering Materials, University of New South Wales, 1974, pp. 211-229.
5. Frénay, J.W.I.J.
Behaviour of plain concrete subjected to uniaxial cyclic tensile loadings
Master's thesis Delft University of Technology, Department of Civil Engineering, Concrete Structures 1980 (in Dutch).
6. Siemes, A.J.M. Fatigue of concrete
CUR-VB report of the Netherlands Committee for Research, Codes and
Specifications for Concrete, Zoetermeer 1981 (to be published; in Dutch).
7. Weigler, H., Freitag, W.
Dauerschwel1- und Betriebsfestigkeit von Konstruktions-Leichtbeton Deutscher Ausschuss fur Stahlbeton, Heft 247, 1975.
8. Fedorov, V.V.
Regression problems with controllable variables subject to error Biometrika No. 61, 1974, pp. 49-56.
-56-9. Taira, S., Ohtani, R., Nakamura, S.
A damage rule of creep and low cycle fatigue at elevated temperatures Proc. Symp. Mech. Behaviour of Materials, Kyoto, Japan 1973, pp. 221 ev
10. Nishibayashi, S.
Tensile creep of concrete
Proc. Rilem Colloquium, Creep of Concrete, Leeds, England 1978.
11. Bergland, G.D.
A guided tour of the fast Fourier transform IEEE spectrum, July 1969, pp. 41-52.
-57-APPENDIX A
Test details
concrete composition: w.c.r. = 0.70; max. grain size = 16 mm
curing conditions : 28 days at 100% RH, subsequently in the
•laboratory
age at loading : on the average 40, 52, 66 and 84 days
frequency : 9Hz (sinusoidal) specimen : cylinder 0 75 x 150 (mm^) splitting test:
i
T
X
compressive test:T
static strength (age): 28 days.
b) Tegfers | 3 |
concrete composition: w.c.r. = 0.51 and 0.89; max. grain size = 16 mm
curing conditions : 20 days in fog room and then at 20 °C and 60% RH
age at loading : 63 and 132 days
frequency : lOHz (sinusoidal)
specimen : cube 150 x 150 x 150 (mm^)
-58-c) Raithby [ 4 J
concrete composition: 0.50, 0.62; max. grain size = 19 mm
curing conditions : different combinations of: in water, in the laboratory, 20 °C and 65% RH, sealed and oven dried 105 °C
age at loading : from 4 weeks to 5 years
frequency : 20Hz (sinusoidal)
specimen : beam 102 x 102 x 510 (mm^)
Ö
"Z^
"Z^
APPENDIX B
-59-batch No. mould deformation loading control during application
of loading
1-16
17-38
59-62
39-58^
63-91
plastic testing apparatus and casting method
plastic
steel
steel
steel
manually controlled application of loading in static tensile and not measured
(in static
ten-sile tests occa- dynamic tests (procedure 1 sionally measured;
XX
1
0 160 mm)
measured in static manually controlled application and dynamic tests of loading in static tensile tests;
(1 = 270 mm) in the dynamic tests the Raytheon computer controlled the loading (procedure 2^^)
measured in static machine controlled application of and dynamic tests the load in static tensile tests. (1 = 270 mm) In the dynamic tests the load was
° controlled by the microcomputer as described in Chapter 5 (proce-dure 3^^)
Note: Because of machine defects the specimens of batches No. 19, 63, 69 and 70 could not be tested.
X These experiments are described in detail in |_ 5 |.
XX 0r29Ê^y!2Ê_]:' ^'t ^'^ increasing frequency from 0.1 to 6Hz, the
dynamic loading was adjusted during the first 10 a 15 cycles.
grgcedure_2: The dynamic load was adjusted but the frequency increased during the first 15 cycles according to:
'-W^nJ^^^y-
n = l, 2
16; n > 16, f = 6Hz,grocedure_3: The dynamic loading increased until the adjusted levels were reached in about 6 cycles at a frequency of 6Hz.
APPENDIX C 6 0 -no. 33-11** 34-12 36-10 32-09 33-10 35-09 36-11 32-04 32-10 33-09 34-09 34-08 35-08 36-08 32-07 34-07 35-07 32-06 33-08 30-06 31-04 31-07 32-12 29-11 30-03 90-02d 91-08d 29-05 30-10 30-11 90-15d 91-02d 91-04d 29-09 30-08 31-05 31-11 89-12d 90-08d 91-14d 29-12 30-05 31-10 28-04 28-05 28-09 29-08 27-03 stress-strength level a . /f' min cm compression-0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.10 0.10 0.10 0.10 0.10 "max^'''ctm log cycles to failure -tension t e s t s 0.40 0.40 0.40 0.50 0.50 0.50 0.50 0.60 0.60 0.60 0.60 0.70 0.70 0.70 0.80 0.80 0.80 0.40 0.40 0.50 0.50 0.50 0.50 0.60 0.60 0.60 0.60 0.70 0.70 0.70 0.70 0.70 0.70 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.90 0.90 0.90 0.40 0.40 0.40 0.40 0.50 4.81 5.53 > 6.50 3.26 3.80 3.96 3.86 3.56 4.34 3.54 2.73 3.04 2.29 3.04 2.45 2.37 2.62 > 6.48 > 6.32 5.99 3.45 5.18 4.50 3.75 5.85 5.28 4.95 2.04 2.86 2.61 3.61 3.00 4.82 2.41 2.97 2.00 1.64 3.50 3.76 3.55 1.15 1.92 1.85 > 6.31 6.13 6.02 6.39 3.81
no. stress-strength level a . /f'
mm cm
compress ion-tens ion 27-06 28-10 30-12 88-04d 91-12d 26-02 26-05 27-05 27-12 28-11 87-Old 88-15d 25-05 25-06 25-07 25-08 25-09 87-02d 90-12d 87-04d 90-05d 24-08 24-11 24-12 87-15d 88-12d 89-05d 90-04d 91-05d 25-10 25-11 25-12 8 7 - l l d 88-Old 0 81-02ds 23-02 23-10 23-12 26-11 65-05d 6 6 - l l d 68-14d 81-Olds 8 3 - l l d s 84-02ds 20-08 20-11 20-12 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 .00 - tension 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 / f max ctm log cycles to f a i l u r e t e s t s (continuation) 0.50 0.50 0.50 0.50 0.50 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.75 0.75 0.80 0.80 0.80 0.80 0.80. 0.85 0.85 0.85 0.90 0.90 0.90 0.90 0.90 1 t e s t s 0.55 0.60 0.60 0.60 0.60 0.60 0.65 0.65 0.65 0.65 0.65 0.70 0.70 0.70 3.46 3.63 > 5.71 > 6.42 > 6.32 5.24 5.47 2.93 3.41 3.49 4.99 3.69 3.09 3.59 3.55 1.95 1.68 3.32 4.82 3.87 5.01 2.90 2.90 2.38 2.87 3.03 2.32 3.61 2.75 1.97 1.91 1.49 2.00 2.52 > 6.31 > 5.69 > 5.68 > 6.15 > 6.30 > 2.41 > 6.30 > 6.72 3.66 > 5.73 > 6.49 > 6.30 2.83 4.38
