Rapp
^ ^ p ^ ^ J I ~\ ^ ^ I T ^ Department of Civil Engineering I I I I J \^ I I L Concrete Structures
Stevin Laboratory C T Technische Hogeschool Delft
Beton
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Delft University of Technology Department of Civil Engineering
Report 5-85-1 Research No. 2.3.83.07 January 1985
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CONCRETE UNDER BIAXIAL LOADING: STATIC COMPRESSION - IMPACT TENSION
dr.ir. A.J. Zielinski
Mail ing address:
Delft University of Technology Stevin Laboratory Stevinweg 4 2628 CN Delft The Netherlands Technische Hogeschool fl^^- Bibliotheek
Afdeling: Civiele Techniek I Stevinweg 1
•S 7~Z Ó^^(9r^/ f postbus 5048
2600 GA DelftCONTENTS
Summary
1. INTRODUCTION 2. LITERATURE SURVEY
3. SCOPE OF THE EXPERIMENTAL PROGRAM 4. TESTING EQUIPMENT 4.1 Hopkinson apparatus 4.2 Measurement set-up 5. SPECIMENS 5.1 Manufacture 5.2 Properties of concrete 6. TESTING PROCEDURES
7. RESULTS OF BIAXIAL TESTS 7.1 Failure envelopes 7.2 Stress-strain curves 8. DISCUSSION 9. CONCLUSIONS NOTATION REFERENCES
SUMMARY
This investigation is focused on the behaviour of concrete under combined compression and tension.
The biaxial tests on concrete prisms (50x80x100mm) were carried out in the modified Hopkinson apparatus. Specimens were subjected to different levels of compression (a ) and thereafter subjected to either static or impact
- 1 4
tensile loading (stress rates 10 and 10 N/mm^/s respectively). Static and impact failure envelopes of concrete are determined as well as stress-strain curves.
It emerged that failure envelopes of concrete under biaxial compression-tension have similar shape under static and impact loading conditions. The static envelopes may therefore be extended to higher rates of loading with the aid of strength-loading rate relations from uniaxial tests.
The impact stress-strain curves are steeper than the static ones, and the strains corresponding to the ultimate stress % are larger under impact loading. Low compressive stresses (a < ^ f ' ) had little effect on the stress-strain curves, whereas higher stresses o decreased the ultimate stress a. and resulted in greater curvature of the stress-strain curves. Single fracture and multiple fracture modes of failure were observed in biaxial tests carried out on specimens subjected to low and to high compressive stresses respectively.
The results of this investigation are discussed in view of the scarce results obtained by other researchers.
The results obtained are explained with the aid of mechanisms of crack extension and fracture of concrete.
INTRODUCTION
The behaviour of structures under impact loading calls for special attention. Accidental impact on nuclear reactor containments, offshore structures and highway structures among others may lead to environmental disasters and lost of human 1ives.
Reliable assessment methods for the response of such structures to impact are needed in order to provide the required safety and to allow rational, economical design.
The finite element method can be applied to advanced dynamic analysis of concrete structures. It can account for the rate-sensitiveness of materials which has been investigated mainly by means of uniaxial tests.
In actual structures, however, concrete members are usually in a multiaxial state of stresses.
It is therefore necessary to know whether the results of uniaxial impact tests on concrete can be directly extended to comprise a multiaxial state of stress.
The present investigation is focused upon the behaviour of concrete in biaxial compression-tension.
Furthermore, the behaviour of low quality concretes is more affected by high rates of loading than that of high quality concretes.
The ultimate stress a can be linked with the stress rate a by the formula proposed by Mihashi and Izumi [2]:
1
"u ^ /jf_, 1+6 °U,0 ' °o
where the subscript "o" refers to the static loading conditions and B is a material parameter. The approximate value of the exponent -r—- is 0.042 for
I +p
uniaxial tension.
Results of multiaxial tests on concrete are scarce. To the author's know-ledge there is only one reported investigation [3] concerning the behaviour of concrete at the multiaxial state of stress and high rates of loading. The tests were carried out on concrete cylinders. First, confining pressure was applied; then the specimens were subjected either to axial tension or to axial compression. Three rates of straining were applied under axial loading:
- static ê = 10"^/s,(S);
- intermediate è = 10"^/s ,(III); - impact s = 1/s, (I).
Figure 2.1 shows stress-strain curves for axially loaded concrete at various levels of confinement and three loading rates. It emerges that the
com-pressive strength and the corresponding strain increase with increasing confining pressure and rate of loading. For tension the picture is less clear due to initial compressive stresses caused by confinement. The higher resistance of concrete to axial tension is manifest for higher strain rates. Takeda et al . [3] represented these results in the three dimensional stress-space - see Fig. 2.2. The effect of the rate of loading upon the failure
envelope for concrete is manifest.
Fig. 2.3 shows the relation between the octahedral shear stress and the octahedral normal stress standardized to the uniaxial compressive strength at the appropriate rate of loading, it can be concluded that the rate effect on tensile and compressive behaviour is consistent for concrete at various levels of confining pressure. This indicates that for the purpose of a general dynamic analysis of concrete structures a unique formulation of the loading rate effect can be used.
C(N/mm2) E(10--*) a. compression G(N/mm^^"93
edO-l
stress (N/mm^) b. tensionFig. 2.1 Influence of axial straining rate and magnitude of confining pressure on axial stress-strain curves of concrete [3]
OjikOalN/mm^)
Fig. 2.2 Rate effect on failure envelope of concrete in the triaxial state of stress [3].
"CocL'Cb
-Q25
Fig. 2.3 Relation between octahedral shearing stress ( T ^ ) and octahedral normal stress (^^QJ,*.) ^t various strain rates [3].
Nilsson [4] proposed a constitutive model for concrete which accounts for rate effects by means of a single rate hardening parameter. The failure conditions can be expressed as: .
% / " u , 0 ^ S ^^^2 ^" (^'') - C 3 l n ( é ^ ^ ) 2 (2.2) ef
where i is the effective strain rate and the parameters C., C„ and C, can be obtained by fitting to the experimental data.
SCOPE OF THE EXPERIMENTAL PROGRAM
The object of this investigation is to determine the behaviour of concrete under biaxial loading regime: static compression — impact tension.
Besides the impact tests the static biaxial compression-tension tests have to be performed in order to obtain reference data for defining rate effects. The behaviour under biaxial loading will be characterized by means of
failure envelopes and stress-strain curves.
Various levels of compression should be investigated in biaxial tests. The companion uniaxial tests have to be carried out.
The tests involve plain concretes of low quality of high quality, the mean cube compressive strength being ~ 20 N/mm^ and ~ 50 N/mm^ respectively. The test parameters are listed in Table 3.1.
Table 3.1 Variables of the experimental program
low (0-1/4)
intermediate (1/4-1/2) high (>1/2)
rate of tensile static (10' ) loading (N/mm^s) impact (10^)
compressive strength low ('\.20) of concrete (N/mm^) • high ('\>50)
levels of
compressive stress
(y^cyi)
Fig. 4.1 Principle of the Split Hopkinson Bar test method and the overall view of the apparatus constructed.
4. TESTING EQUIPMENT
4.1 Hopkinson apparatus
A Split-Hopkinson-Bar technique has been extensively used in the Stevin Laboratory for the testing of plain concrete and fibre-reinforced concrete under impact tensile loading [5,6].
The equipment has also been applied to testing the bond between steel and concrete at high'loading rates [7]. Here a brief description of the Hop-kinson equipment will be given.
3 The apparatus has been constructed for stress rates in the range from 10
5
to 10 N/mm^/s and maximum load duration of about 2 mill second.
It consists of this coaxial aluminium bars between which a specimen is glued (see Fig. 4.1). A tensile stress pulse can be generated by a drop
weight hitting the anvil at the bottom of the lower bar. The pulse propagates upwards at a velocity of about 5000 m/s, passes through the specimen and reaches the damper at the end of the upper bar. The 3.5m length of the lower bar ensures a uniformly distributed stress pulse. The 5m length of the upper bar prevents reflections from the fixed upper end reaching the specimen before the initial pulse has passed and fractured the specimen. Because aluminium and concrete have similar mechanical impedance, reflections at the bar-specimen-bar interfaces are minimized. The diameter of the bars is 74mm, whereas concrete prisms with a cross-section of 50x80mm have been used for biaxial tests. The reflections due to the geometrical mismatch between the bars and the specimen are virtually eliminated due to smoothing out by special adapters (074mm->- 50mmx80mm over a length of 100mm).
Almost the whole initial pulse can be transmitted through the concrete specimen into the upper bar in which the strains were measured by strain gauges. These measurements were used for calculating the loading force and the tensile stress in the specimen. The static tensile tests can also be carried out in the Hopkinson apparatus. The lower bar can be pulled by a steel cable fixed in a hydraulic jack.
The longitudinal and lateral strains were measured on concrete prisms in-strumented with 30mm long strain gauges (Tokyo Sokki Kenkyujo Co.Ltd.PL-30) on the front and back faces.
Additional prestressing equipment (see Fig. 4.2) enables biaxial tests to be performed in the Hopkinson apparatus.
Fig. 4.2 Prestressing equipment used for biaxial compression-tension tests on concrete.
This equipment consists of four steel platens (40x550x750mm) mounted on steel bars of 20mm diameter.
A flat hydraulic prestressing jack (capacity 400kN) is placed between two platens on one side of the equipment.
On the other side a force-measuring device is placed between two other platens. The compressive force is applied to the specimen through special "steel brushes" fixed to the inner pair of platens. Each of the brushes consists of 1260 single rods (4mmx4mmx100mm) spaced with 0.2mm clear dis-tance between them.
The brushes have been developed by Nelissen [8] and were found to satisfy stiffness requirements with respect to free deformations at the uniformly loaded faces of the specimen.
4.2 Measurement set-up
The measurements in biaxial tests comprise the strain in the upper bar e., , the longitudinal strain e- and lateral strain e^ on the free faces of the concrete specimen, and the prestressing force P.
The tensile stress o and the compressive stress <^p in the specimen can be determined as follows:
^AT^Al-'^Al
•"l = bT^ (4.1)
^2= A (4.2)
modulus of elasticity of aluminium barcross-section of aluminium bar
width, depth and hight of concrete prism
The measuring signals are amplified by four Tektronix TM 503 amplifiers and fed into two Nicolet Explorer II 2-channel transient recorders with a maximum measuring frequency of 2 MH and 4k core.
The results can be stored on floppy disks for further processoring by the laboratory computer HP 21 MX or for plotting by x-y recorders.
Fig. 4.3 shows a schematic view of the measuring system. where: E Al
^Al b,d,h
R | - ^ '^
F i g . 4.3 Measurement set-up
1. Tektronix TM-503 a m p l i f i e r
SPECIMENS
1 Manufacture
A high q u a l i t y and a low q u a l i t y concrete were tested in t h i s research. Table 5.1 shows two mix compositions. Portland cement type I I I (B) and r i v e r gravel were used.
Table 5.1 Mix composition
Mix A Mix B cement content 375 190 (kg/m^) water/cement ratio 0.50 0.85 (kg/kg) aggregate content 1820 1950 sand+gravel (kg/m^) aggregate grading (%) 4-8 mm 2-4 1-2 0.5-1 0.25-0.5 0.1-0.25
The specimens used in the biaxial tests were sawed from five 50mmx290mmx • 320mm concrete slabs which had been cast in a single mould. Companion specimens were cast for the standard control tests:
- 6 cubes (150mm) for compressive tests,
- 6 cubes (150mm) for tensile splitting tests, - 3 cylinders (015Ommx4OOmm) for compressive tests.,
During casting, the mix was compacted for 120 s. Demoulding took place af-ter two days, afaf-ter which the slabs, cubes and cylinders were stored in a room with 100% relative humidity. Two weeks after casting, the slabs were cut with a slow-feed diamond saw and polished. Finally, 50mmx80mmx100mm prisms were obtained. They contained two parallel 5mm deep saw-cuts situated 35mm from the bottom of specimens.
30 20 15 15 13 7 32 22 16 11 8 11
Hopkinson apparatus. About one day before testing the prisms were each in-strumented with four strain gauges.
The tests on prisms were carried out in the 5th and 6-th week after casting.
Properties of concrete
The quality of concrete was determined by means of standard laboratory tests on 150mm cubes and 01OOmmx4OOmm cylinders at the age of 28 days.
Additional tests on cubes were carried out at the age of 56 days. Three specimens were used for eacht type of tests. The rate of loading was
0.5 N/mm^/s in compression tests and 0.1 N/mm^/s in tensile splitting tests. Table 5.1 shows results of these tests.
Table 5.1 Results of static control tests.
Mean values and coefficients of variation
Cube compressive strength f' (N/mm^)
c.v. (%)
Cylinder compressive strength f' , (N/mm^)
c":V.
{%)
Cube tensile splitting strength f,nl (N/mmM c!^! (%) Mix 28d 49.5 1.6 34.3 3.0 3.05 6.1
A
56d 58.8 3.6 3.49 3.5 Mix 28d 20.7 5.8 14.0 10.7 1.57 7.0B
56d 25.8 2.6 2.03 5.2 Secant modulus of elasticityat a - 2/3 f'y^ E /N/mm2) c.v. (%) 62430 7.3 34145 5.9
TESTING PROCEDURES
The 50mmx80mmx100mm prisms for biaxial tests were first subjected to com-pression. A prism was placed between the brushes, and the pressure in the flat jack was increased to the chosen level. Next, the prism was glued between the lower and the upper bar of the Hopkinson apparatus - the applied pres-sure was about 0.1 N/mm^. The filled polyester resin F88 requires about J hour to harden. The longitudinal and lateral strains of the free faces of the specimens as well as the prestressing force were measured during
prestressing and gluing. When the vertical pressure was relieved, the specimen ceased to be loaded in this direction, since the lower bar was balanced by compensation weights.
In the static tests the axial tensile force was gradually increased up to failure of the concrete occurred. The rate of loading was approximately 0.1 N/mm^s. In the impact tests the drop-weight was used and the rate of
4
loading was about 10 N/mm/s.
The strains in the specimen, the strain in the upper aluminium bar and the prestress were measured during tensile loading and then registered on floppy disks for further analysis of the measurements.
The specimens were visually examined first after prestressing and then af-ter tensile loading in order to detect cracking outside the zone weakened by the saw-cut.
1/8
O VB
0 / f c y l
Oz/tyl
(a)
(b)F i g . 7.1 Strength of concrete under b i a x i a l compression-tension a. t e s t r e s u l t s
RESULTS OF BIAXIAL TESTS
Failure envelopes
The results obtained are summarized in Table 7.1 and 7.2 for mix A and mix B respectively.
The values of the static tensile strength f and the impact tensile strength f are given for various levels of the compressive stress a,.
The results of biaxial tests can be represented in a two-dimensional stress plane with the principal stresses ^. and o on axes. The stresses are
usually standardized with respect to compressive strength. The cylinder compressive strength f' -, will be used here as reference. This investigation
is limited to the compression-tension quadrant of the ^^''^2 stress plane. Fig. 7.1 shows results of either static compression - static tension tests or static compression - impact tension tests. The companion results of uniaxial tensile tests are also plotted.
The lines which are plotted in Fig. 7.1a correspond with the Mohr-Coulomb failure envelope:
Figure 7.1b shows this criterion with f=1/8 f' , [ 1 6 ] .
Despite considerable scatter in the results, several features can be ob-served.
The tensile strength of concrete is hardly affected by compressive stresses up to about 0.7 f'•, caused by preloading in the direction perpendicular to the direction of tensile loading.
It appears that at all levels of compressive stress tested the impact ten-sile strength of concrete is higher than the static tenten-sile strength.
There is no pronounced difference between the high-quality concrete and the low-quality concrete. Obviously the absolute strength values are higher in the case of the high-quality concrete (compare Tables 7.1 and 7.2).
-5.7 7.7 8.1 14.0 14.9 16.0 16.1 18.4 22.5 31.0 -4.0 5.0 6.7 7.5 8.5 9.8 11.0 14.8 15.6 18.9 19.7 21.9 22.0 26.7 26.8 27.2 2.80 2.07 2.13 1.97 1.58 2.52 2.81 1.65 2.61 1.77 1.49 4.02 4.98 5.03 5.82 6.02 4.30 5.39 5.13 5.04 4.67 4.01 5.26 4.12 4.23 4.82 5.12 3.81 5.32 3.32 4.39 2.40
Table 7.2 Results of tests on Mix B ( f ^ = 14.0 N/mm^) Compressive stress (N/mm^) Tensile strength (N/mm^)
static impact ^2 ^0 ^ -4.1 4.1 4.4 7.5 9.7 -4.0 4.2 5.4 6.9 7.3 9.7 9.9 0.98 1.04 1.06 0.75 0.77 0.51 0.40 0.57 1.72 2.33 2.88 1.49 1.08 2.08 1.12 1.52 1.65 1.98
Considering failure modes (see Fig. 7.2), it was observed that most of the specimens fractured at the section reduced by saw-cuts.
Some specimens subjected to high '^,. compressive stress (a \ f' -i) exhibited multiple fracture as shown in Fig. 7.3.
(a)
(b)
F i g . 7,2 F a i l u r e modes: a . simple f r a c t u r e b . m u l t i p l e f r a c t u r e
2 Stress-strain curves
The representative stress-strain curves are shown in Figs. 7.3 and 7.4 for mix A and mix B respectively. Tensile stress (o.) is plotted on the vertical axis whereas strains {t. and E O ) are plotted on the horizontal axis.
The curves refer either to impact (I) or to static (S) tensile loading tests carried out on previously compressed concrete prisms.
It should be emphasized that the strains e. and e^ are due to tensile loading only; hence for obtaining the total strain e? and eJ for biaxially loaded concrete the initial strains e? and e° due to compression have to be added. The total strains are also given.
Considering high-strength concrete (see Fig. 73) it can be observed that the tensile stress-strain curves are hardly affected by compressive pre-loading at low Op stress levels (up to about 1/3 f' , ) . They are almost linear. At higher o stress levels the stress-strain curves are less steep and bend more markedly towards the horizontal. The strain e. at failure is significantly larger than in the case of low o
From figure 7.4 emerges that the °,-e, curves for low-strength concrete are in general more strongly curved than those for high-strength concrete. The tensile strain e- at failure is smaller for biaxially loaded concrete than in the case of uniaxial loading. The latter is true for both static and impact tensile loading. The stress-strain curves obtained in this in-vestigation do not possess a descending branch, this being due to the test technique.
200
e, (10 m/m)
50,^ . -100
e^lO
W m )
Fig. 7.3 Stress-strain curves of concrete mix A
No.
Static/ Impact ap(N/mm2) e* (10"^m/m) :* (10"^m/m)1
2
3
4
5
6
7
8
9
0
s
s
s
s
0.0
8.1
14.0 22.50.0
5.0
8.5
11.0 18.9 19.746
111
201
337
130
156
182
152
351
337
19 304 637 813 28 294 321 533 1000 813o,(N/mm2)
200 ^ 150
e,(10^m/m) 50 ^ 100 £3(10 m/m)
F i g . 7.4 S t r e s s - s t r a i n curves of concrete mix B
No, 1 2 3 4 5 Static/ Impact S S I I I a2(N/mm2) 0.0 7.5 0.0 4.0 6.9 e* (10"°m/m) 62 108 94 102 91 E* (lO'^'m/m) 8 518 6 230 359
(b)
(c)
-02 1 :—I r
-O ^ • -4827p»i (-339i3kgf/cm2)
1.0 0.8 06 0.4 0 2 -0.1
Fig. 8.1 Ultimate stress envelope of concrete under combined compression tension:
a. results of Kupfer, Hilsdorf and Rusch [9] b. results of Nelissen [8]
DISCUSSION
In general, the results of static compression-tension tests are in agreement with the results of previous biaxial experiments on concrete carried out by Kupfer et al. [9], Nelissen [8] and Tusuji et al. [10] (see Fig. 8.1).
With respect to failure envelopes it can be observed that the tensile strength of concrete in biaxial compression-tension gradually decreases with the compressive stress a^ up to about 0.7 of the cylinder compressive
strength f' •,. The tensile strength can be approximated by the uniaxial tensile strength in that range of o,^. At higher stresses a^ the decrease in the tensile strength is pronounced.
The results of this investigation indicate that the strength increase due to high rates of loading is similar for concrete subjected to uniaxial ten-sion and for concrete subjected to biaxial compresten-sion-tenten-sion. It appears that the rate effects determined in uniaxial tests on concrete apply also to the strength of concrete under various multiaxial loading regimes as
suggested by Takeda et al. [3].
The rate-dependent failure envelopes for concrete in biaxial compression-tension can be constructed by combining equation 2.1 with the Mohr-Coulomb failure criterion with tension cut off. The above is illustrated in Fig. 8.2.
cctg»»
Fig. 8.2 Rate-dependent failure envelope of concrete under combined compression-tension (f/fQ=(^/°Q) ) •
whereas in this investigation the specimens were first compressed and then subjected to tensile loading.
Fig. 8.3 Radial stress along interface of a rigid either perfectly bond on or partially debonded inclusion:
a. the external compression is applied at infinity in the horizontal direction
b. the external tension is applied at infinity in the vertical direction
is found In general, little influence of compressive stresses ^p - J cvl
for the o.-E. and a.-e^ curves. The curves of low-strength concrete are little more affected by o than those of high-strength concrete. The effect of compression is pronounced at higher a„ stress levels. A signifi-cant increase of the strains E . can be observed for the static as well as impact a,-E. curves of the high-strength concrete as compared with curves at low compressive stresses. The strains E . at failure are much larger for high than for low op stress levels in the case of high-strength concrete, whereas the opposite is true for low-strength concrete.
The modes of failure observed - simple fracture at low ap and multiple fracture at high op " illustrate that combined mechanisms of compressive and tensile failure are involved in tests on concrete under biaxial loading. The following can be helpful in studying the behaviour of concrete under biaxial compression-tension. The analysis of Mushelishvili[11] and of Perlman et al. [12] can be used for determining the state of stress around a single rigid inclusion in the elastic matrix subjected to either com-pressive or tensile loading. Fig. 8.3 shows several cases of concentrations of radial stresses o /a along interfaces of inclusions. Fig. 8.4 illustrates transfer of compressive forces in a system consisting of a few rigid in-clusions embedded in the elastic matrix.
Fig. 8.4 Zones of local tension (T) and compression (C) between rigid inclusions in a matrix subjected to uniaxial compression.
sumably the mechanisms described above occur simultaneously and efficiently oppose each other, so that little effect of compressive loading on the failure envelope of concrete under biaxial compression-tension is to be observed.
The experimental investigations [13,14] on concrete in uniaxial compression have shown that bond microcracks extend along interfaces of aggregate
particles under load exceeding about 35% of the ultimate load. At about 80-90% of the ultimate load the cracks tend to propagate parallel to the direction of loading, through the mortar matrix. The above means that only high compressive stresses oo lead to the formation of partly fractured planes perpendicular to the direction of tensile loading in biaxial tests. Low Op stresses do not cause significant cracking and therefore hardly affect the ultimate tensile stress.
Considering the strains in the concrete, it emerges that the strain E ^ in the direction of compressive loading gradually increases with the compressive stress Op, whereas the strain E . in the other direction may rapidly increase due to crack formation at higher levels of Op.
The strain E . during tensile loading may be larger in combined compression tension, especially at higher levels of a„, than in uniaxial tension owing to the opening of a multitude of cracks situated in the planes perpendicular to the direction of E..
The effects of high rates of loading upon the behaviour of concrete in un-iaxial tension is extensively discussed in [15] in the light of previous investigations carried out in the Stevin Laboratory. The mechanisms of ex-tensive simoultaneous cracking and fracturing of tougher material zones have been considered as essential to the explanation of greater energy ab-sorption, higher tensile strength and larger corresponding strain of con-crete under impact loading than under static loading conditions.
Similar rate effects are observed in this investigation on concrete loaded under combined compression and tension. The present question is whether the same mechanisms of fracture at high loading rates apply to concrete sub-jected to compressive preloading as to the virgin material. In the case of uniaxial loading (see Fig. 8.5) the rapidly increasing tension can drive the cracks to rapid extension through tough aggregate particles instead of growing around them. It can also cause pronounced extension of cracks in the whole volume of highly strained material zones.
S T A T I C „„crocrocks I M P A C T
Fig. 8.5 Fundamental difference between static and impact tensile fracture.
In the case of biaxial loading of concrete under compressive stress a„ and slowly increasing tension a. further extension of bond cracks and mortar cracks will take place in the zones of tension (T). Many of these cracks can be arrested in the zones of compression (C) or by tough aggregate particles. Others can penetrate through the weakest or less precompressed parts of the matrix and form continuous fractures. Of course, more extensive cracking will occur for higher levels of compressive stress a„.
The fracture mechanisms described above are consistent with results of biaxial tests on concrete.
CONCLUSIONS
1. The increase in strength due to high rates of tensile loading was similar for concrete under uniaxial loading and for concrete subjected to biaxial compress ion-ten si on.
2. Failure envelopes of concrete under combined compression and tension have similar shape under static and impact loading conditions.
3. The strength-loading rate relationships determined in uniaxial tests can be used for extending failure envelopes of concrete under static com-pression-tension to higher rates of loading. It seems that the same
procedures can be applied to constructing rate-dependent failure envelopes of concrete under other combinations of multiaxial loading.
4. Both the static and the impact stress-strain curves of concrete in ten-sion o. were little affected by low compressive stresses (c^pS ^ f' • , ) . At higher stresses CT„ the stress-strain curves showed greater curvature. The ultimate stress o decreased and the corresponding strain e. in-creased in the case of high-strength concrete, but decreases in the case of low-strength concrete.
5. Two modes of failure could be distinguished in biaxial compression-tension tests:
- simple fracture at low compressive stresses ^ • - multiple fracture at high compressive stresses ^p •
6. The impact stress-strain curves are in general steeper than the static ones, and the strains at the ultimate stress ^. are larger under impact loading than under static loading.
7. The mechanisms of crack extension and fracture of concrete discussed are consistent with experimental evidence obtained in this investigation. 8. Further investigations on concrete under various combinations of
multi-axial loading are required for verifying the rate-sensitiveness of con-crete subjected to different loading conditions.
A « cu " static values of a, and
6
H J , O , Mo u
a . - octahedral normal stress
oct
T 4. - octahedral shear stress
oct
E - strain
E.,Ep - principal strains
E p E p - initial values of E- and £p under compressive loading
E | , £ 2 - total values of E . and Ep due to biaxial loading
£ - strain rate
ef
ê - effective strain rate
f - uniaxial tensile strength
f - static value of f
f 1 - tensile splitting strength
f' - cube compressive strength
f' 1 - cylinder compressive strength
A - cross-section
C - constant
E - modulus of elasticity
P - prestressing force
b,d,h - dimensions of test prism
c.v. - coefficient of variation
REFERENCES
1. Zielihski, A.J.,
"Concrete structures under impact loading. Rate effects" Stevin Report No. 5-84-14, Delft University, 1984. 2. Mihashi, H., Izumi, M.,
"A stochastic theory for concrete fracture"
Cement and Concrete Research, Vol. 7, 1977, pp. 411-422. 3. Takeda, I., Tachikawa, H., Fuji moto, K.
"Mechanical behaviour of concrete under higher rate of loading than in static tests"
Proc. Symposium: "Mechanical behaviour of Materials", Kyoto 1974, Vol. II, pp. 479-486.
4. Nilsson, L.
"Impact loading on concrete structures",
Department of Structural Mechanics, Chalmers University, Publication 79:1, 1979.
5. Körmeling, H.A., Zielinski, A.J., Reinhardt, H.W.
"Experiments on concrete under single and repeated impact tensile loading"
Stevin Report No. 5-80-3, Delft University, 1980. 6. Körmeling, H,A.
"Experimental results of plain and steel fibre reinforced concrete un-der uniaxial impact tensile loading"
Stevin Report No. 5-84-8, Delft University, 1984. 7. Vos, E., Reinhardt, H.W.
"Bond resistance of deformed bars, plain bars and strands under impact loading"
Stevin Report No. 5-80-6, Delft University, 1980. 8. Nelissen, L.J.M.,
"Biaxial testing of normal concrete"
Heron, Vol. 18, No. 1, Delft University, 1972. 9. Kupfer, H., Hilsdorf, H.K., Rusch, H,
"Behaviour of concrete under biaxial stresses" AC! Journal, August 1969, pp. 656-666.
Perlman, A.B., Sih, G.C,,
"Elastic problems of c u r v i - l i n e a r cracks in bonded d i s s i m i l a r materials"
Int. Journal of Engineering Sciences, Vol. 5, 1967, pp. 845-867. Hsu, T.T.C., S l a t e , F.O., Sturman, G.M., Winter, G.
"Microcracking of plain concrete and the shape of the s t r e s s - s t r a i n curve"
ACI Journal, February 1963, pp. 209-224. Shah, S.P., Slate, F.O.
"Internal microcracking, mortar-aggregate bond and the stress-strain curve of concrete"
Proc.Int.Conf.: The Structure of Concrete, Ed. Brooks & Newman, C&CA," 1968, pp. 82-92.
Zielinski, A.J.
"Model for tensile fracture of concrete at high rates of loading" Cement and Concrete Research, Vol. 14, No. 2, 1984, pp. 215-224. Walther, R.
"Uber die Berechnung der Schubtragfahigkeit von Stahl- und Spann-betonbalken - Schubbruchtheorie",
Stevin-reports published by the division of concrete structures:
SR - 1 Leeuwis, M. "Kruip en kriraponderzoek op ongewapend beton. Collec-taneum onderzoeken 1958-1970" (2 delen), out of print.
(5-71-3)
SR - 2 Froon, M. "Hoogwaardig beton" (1972). out of print. (5-72-1)
SR - 3 Walraven, J.C. "De meewerkende breedte van voorgespannen T-balken" (1973). out of print.
(5-73-1)
SR - 4 Nelissen, L.J.M. "Het gedrag van ongewapende en gewapende beton-blokken onder geconcentreerde belasting" (1973).
(5-73-7) . -. .^r SR - 5 Nelissen, L.J.M. "Stress-strain relationship of light weight
con-crete and some practical consequences" (1973). out of print. (5-73-8)
SR - 6 Bruggeling, A.S.G. "De constructieve beïnvloeding van de tljdsaf-hankelike doorbuiging van betonbalken" (1974).
(5-74-2)
SR - 7 Stroband, J., Tack, P.J. "Kolomvoetverbinding met geïnjecteerde stekeinden (1974).
(5-74-3)
SR - 8 Christiaanse, A.R., Vrande, L.W.J.W. van der, Rooden, R.J.W.M. van "Het gedrag van stalen voetplaatverbindingen" (2 delen) (1974). (5-74-4)
SR - 9 Uljl, J.A. den, BednSr, J. "Onderzoek naar het verankeringsgedrag van gebundelde staven" (1974).
(5-74-5)
SR - 10 Nelissen, L.J.M. "Twee-assig onderzoek van grindbeton" (1970). S R - 11 Meuzelaar, L.C., Smit, D.R., Brakel, J., Zwart, J.J. "Ponts S
haubans en béton précontraint" (1974). (5-74-6)
SR - 12 Bruggeling, A.S.G., Boer, L.J. den "Eigenschaften von stahlfaser-bewehrtem Kiesbeton" (1974).
(11-71-10)
SR - 13 Boer, L.J. den "Fibre reinforced concrete" (1973). out of print. Conference on properties and applications of fibre reinforced con-crete and other reinforced building materials.
SR - 14 Uljl, J.A. den "Met bamboe gewapend beton onder herhaalde belas-ting" (1976). out of print.
(5-76-1)
SR - 15 Dijk, H.A. van, Nelissen L.J.M., Stekelenburg, P.J. van "Het gedrag van kolom-balkverbindignen in gewapend beton" (1*576).
(5-76-2)
SR - 16 Brunekreef, S.H. "Gedeeltelijk voorgespannen beton; Op buiging be-last" (1977).
(5-76-8)
SR - 17 Betononderzoek 1971-1975 (met samenvatting in het Engels) (1976). SR - 18 Bruggeling, A.S.G. "Time-dependent deflection on partially
pres-tressed concrete beams" (1977). (5-77-1)
SR - 19 Veenvliet, K.Th. "Dynamisch gedrag van liggers op twee steunpunten onder invloed van een mobiele belasting" (1977).
(5-78-6)
Reinhardt, H.W. "Contribution of the fibres tó the load bearing capacity of a bar and fibre reinforceed concrete beams" (1978). (5-78-9)
Stekelenburg, P.J. van. Walraven, J.C., Mathews, M.S. "Development of a semicylindrical shaped roof in ferrocement" (1978). out of print.
(5-78-11)
Walraven, J.C. "Mechanisms of shear transfer in cracks in concrete. A survey of literature" (1978).
Reinhardt, H.W. "On the heat of hydration of cement" (1979). out of print.
(5-79-1)
Pat, M.G.M., Fontijn, H., Reinhardt, H.W. Stroeven, P. "Erosie van beton" (1979).
(5-79-30)
Walraven, J.C., Vos, E., Reinhardt, H.W. "Experiments on shear transfer in cracks In concrete. Part 1: Description of results" (1979).
(5-79-3)
Walraven, J.C. "Experiments on shear transfer in cracks in con-crete. Part 2: Analysis of results" (1979). ! ^
(5-79-10)
Gremmen, C. "Beton met grof grind als toeslagmateriaal". (5-79-5)
Körmeling, H.A., Reinhardt, H.W., Shah, S.P. "Static and dynamic testing of concrete beams reinforced with fibres and continuous bars" (1979). . • i . (5-78-10)
Huyhge, G.F., Walraven, J.C., Stroband, J. "Onderzoek naar
voorge-spannen kanaalplaten" (1980). • (5-80-2) • • • - :
Körmeling, H.A., Zielinski, A.J., Reinhardt, H.W. "Experiments on concrete under single and repeated uniaxial impact tensile loading" (1980).
(5-80-3)
Vos, E., Reinhardt, H.W. "Bond resistance cf deformed bars, olaln bars and strands under impact loading" (1980).
(5806) . -Betononderzoek 1976-1980 (1980).
Reinhardt, H.W. "Schaalwetten bij proeven met betonconstructies" (1980).
(5-80-9)
Walraven, J.C. "Aggregate Interlock: a theoretical and experimental analysis" (dissertatie) (1980).
Walraven, J.C. "The Influence of depth on the shear strength of lightweight structural members without shear reinforcement" (1980). (5-78-4)
Bruggeling, A.S.G., Oostlander, L.J. "Concentrated load on a thick-walled cylinder" (1980).
(5-81-1)
Pat, M.G.M. "Kruipspreiding. Deel 1: Proefresultaten" (1980). (5-80-1)
Pat, M.G.M., Reinhardt, H.W. "Variability of creep of concrete -analysis of the results" (1980).
(5-80-8) cv:- -•' . :: '.:-. ••
Z i e l i n s k i , A . J . " E x p e r i m e n t s o n m o r t a r u n d e r s i n g l e sftid r e p e a t e d uniaxial impact tensile loading" (1981).
(5-81-3) ... , .-. II -
V-Cornellssen, H.A.W., Timmers, G. "Fatigue of plain concrete in uni-axial tension and in alternating tension-compression - experiment and results" (1981).
(5-81-7) • • Stroband, J., Kolpa, J.J. "The behaviour of reinforced concrete column-to-beam joints. Part 2: Corner joints subjected to positive moments,
(5815) " " •
-Stroband, - J. , Kolpa, J.J. "The behaviour of reinforced concrete ••: column-to-beam joints. Part 1: Corner joints subjected to a negative moment".
(5-83-9) ; , • -• Betononderzoek 1980-1982.
Zielinski, A.J. Behaviour of concrete at high rates of tensile loading, A theoretical and experimental approach.
(5-83-5)
Uijl, J.A. den "Tensile stresses in the transmission zones of hol-low-core slabs prestressed with pretensioned strands" (1983).
(5-83-10)
Cornelissen, H.A.W. "Constant-amplitude tests on plain concrete In uniaxial tension and tension-compression" (1984)
(5-84-1)
Zorn, N.F. "Stress wave propagation on reinforced concrete piles during driving" (1983).
(5-83-21)
Zorn, N.F. "Cracking and induced steel stresses of reinforced and prestressed piles during driving" (1984)
(5-84-6)
Körmeling, H.A. "Experimental results of plain and steel fibre reinforced concrete under uniaxial impact tensile loading" (1984). (5-84-8)
Zielinski, A.J. "Fracture of concrete and mortar under uniaxial Impact tensile loading" (dissertatie) (1982).
Vos, E. "Influence of loading rate and radial pressure on bond in reinforced concrete" (dissertatie) (1983).
Körmeling, H.A. "impact tensile behaviour of steel fibre concrete at very low temperatures" (1984).
(5-84-13)
Zielinski, A.J. "Concrete structures under impact loading. Rate effects" (1984).
