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Mechanical behaviour of fresh concrete
CEMENT and CONCRETERESEARCH. Vol. I I , pp. 323-339, 1981. Printed in the USA. 0008/8846/81/030323-17502.00/0 Copyright (c) 1981 Pergamon Press, Ltd.
MECHANICAL BEHAVIOUR OF FRESH CONCRETE
A. Alexandridis Engineer, Stone and Webster Toronto, Canada N. J. Gardner Professor, Department of Civil Engineering University of Ottawa Ottawa, Canada KIN 9B4
(Communicated by R.E. Philleo) (Received Dec. 19, 1980) ABSTRACT The shear strength characteristics of fresh concrete were studied through the use of a triaxial compression apparatus on concrete cylinders i00 mm in diameter and 200 mm high at 21°C and 4°C for set times ranging from 40 minutes to 160 minutes. The test results were analyzed by the shear strength theories of Mohr-Coulomb and Rowe for each tlme-temperature combination. The angle of internal friction was found to be a constant property of the mix in the range of 37 @ to 41 @ for a 10% failure strain when analyzed by Mohr-Coulomb. Rowe's analysis gave friction angles of 18 @ to 21 @ for a 10% failure strain. Both theories indicate that the cohesion of fresh concrete is initially zero and increases with set time.
La r~sistance au cisaillement du b~ton frais a ~t~ ~tudi~e avec l'aide d'un appareil triaxlal sur des ~chantillons de i00 mm de diam~tre par 200 mm en grandeur ~ 21°C et 4°C pour une dur~e de 40 minutes ~ 160 minutes. Par la suite les r~sultats ont ~t~ analys~s employant les theories de Mohr-Coulomb et Rowe et ont ~t~ exprlm~s comme les deux param~tres de coh~sloD et de frottement pour chaque combinaison de temperature et de dur~e. On a d~duit que l'angle de frottement ~tait une propri~t~ constante du m~lange et r~sidait entre 37 ° e t 41 ° pour une d~formatlon finale de i0% utilisant la th~orie de Mohr-Coulomb. L'analyse de Rowe a donn~ des angles de frottement de 18 ° ~ 21 ° pour une d~formation finale de 10%. Les deux theories ont indiqu~ que la cohesion ~tait initialement z~ro et qu'elle s'~tait agrandit avec le temps.
323
324
Vol. I I , A. A l e x a n d r i d i s ,
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N.J. Gardner
Introduction When fresh concrete is first mixed, it possesses in addition to viscosity significant shear strength so that when it is poured from its mixing container into a form it does not fully take the shape of the form unless it is tamped or vibrated. However, when removed from its container, fresh concrete is unable to maintain the mould shape and will slump. The properties of fresh concrete therefore, lie between those of a liquid and those of a solid substance. Fresh concrete may be visualized as particles of inert aggregate which are held or suspended in a deformable matrix of cement paste and air bubbles. The paste matrix itself is composed of water and grains of cement. Given time and the proper environmental conditions, the cement paste matrix is converted by means of a physicochemical process between the cement grains and the water into a rocklike mass of particles. The hardened particulate material which replaces the original cement grains is called cement gel and it occupies the space in the mixture originally occupied by the cement grains and the water. To date, a significant amount of research has been aimed at understanding the mechanical properties of concrete in the hardened state. Problems pertaining to the pumpability, workability, placing of fresh concrete and the lateral pressures developed in formwork are examples in which little is understood of how fresh concrete behaves. Engineers who need to undertake calculations for such problems must resort to empirical formulae, charts and tables. Such a procedure however, may lead to gross inaccuracies when applied to concretes whose mix proportions and conditions at mixing differ significantly from those corresponding to the concrete for which the procedure was derived. Consequently, a more fundamental understanding of the mechanical behaviour of fresh concrete is necessary. Fresh concrete like soil is a particulate system composed of particles weakly bonded together and submerged in a liquid medium. It possesses shear strength as a result of (I) the frictional resistance and interlocking between the aggregate and the cement particles and (2) the bonding together of the aggregate particles by the cement during hydration. The former component of shear strength is termed internal friction and requires strain to be mobilized. The latter component is termed cohesion. True cohesion results from cement hydration and therefore depends on the time since addition of water to the mix and the temperature at which the mix is allowed to hydrate. The mechanical behaviour of fresh concrete is very similar to that of a cohesive soil in that it possesses cohesion, internal friction and pore water pressure. Hence it is advantageous to review the basic theories which explain the behaviour of particulate systems. Theoretical BackBround Stress in a Particulate System Fresh concrete may be thought of as a particulate system; a system composed of particles which are not strongly bonded together like the grains of a solid yet neither free to move like the elements of a fluid, which are submerged in a liquid medium. A system such as this may transmit forces through (i) the voids (pore pressure), (2) particle contact points (effective stresses) and (3) the cross-sections of the particles. It has been shown by Terzaghi that for a particulate system the total stress on any plane not cutting across any particles is given with sufficient accuracy by the following equation
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325 SHEAR STRENGTH, TRIAXlAL COMPRESSION, FRESH CONCRETE
where
o = total stress; o' = effective stress and U = pore pressure.
The above equation is based on the assumption that the ratio (Ac/A) 2 is much smaller than unity where A c denotes the area of contact points on the plane and A denotes the total area of the plane. Shear Strength Theories Mohr-Coulomb Theory The conventional expression used in soll mechanics to describe the shear strength of a soil, at failure, is the Coulomb equation T = C + a n tan~ where
• C on tan ~
= = = =
[I]
shear stress cohesion normal stress coefficient of friction.
Soll mechanics practice is to represent the stress conditions at failure as a series of Mohr's Circles on a single set of axes; the envelope of such stress circles is known as the Mohr-Coulomb Failure Envelope. It is important to note that for saturated samples the shear strength depends on effective stress and not total stress and Coulomb's equation must be written in terms of effective stress parameters to become 3' = C' + Sn' tan~'
[2]
Primes indicate effective stress parameters. The Mohr-Coulomb theory has a fundamental meaning provided the assumption is made that no volume occurs. With volume change included in the C' - ~' parameters, the theory is essentially a useful engineering tool, but nothing more. Since the volume changes depend on the principle stress system imposed which differs between types of tests, universal C'-~' values cannot be found for a material independent of the test method. Rowe's Theory Consider the force P required to cause the body in Figure la to move along a plane inclined at an angle B to the direction of the force P (5). Assuming that the material exhibits both cohesive and frictional components of shearing resistance then, at the instant of slip, the forces can be resolved to give P - C
x +
Q + C x tan8 = tan (~u where
8)
[3]
C~ = a cohesive stress quantityJprojected length in P direction #u = the angle of friction of the material Q = force on body normal to P.
Using this equation, the equations of state for a simple geometrical packing of spheres may be derived. Two such packings of spheres are the body centered cubic and the rhombic which correspond to the loosest and the most dense states respectively. Consider a system of spheres in a body centered cubic packing as shown in Figure lb. Movement of one sphere relative to another during the application of a deviator stress can only take place by sliding. Applying Equation [3] at a point of contact of the spheres, the following condition of sliding is arrived at
326
Vol. A. A l e x a n d r i d i s ,
N.J.
V~L,
P
,f2 L~- ° /
L_~3
Body Centered Cubic Packing of Spheres b) plan
forces on on inclined plane
L~
2
~-~
3
> >
L ~"
L,
rio.
~L, X
,/2 L3
o)
II,
Gardner
'
L
2
t
L~ ÷ 0"~'
~i.. L3
),-~,--,g LI
2
_,
c) section X - X
d) failure mechanism
Derivation
FIG. 1 of Rowe's Equation
LI/4 - C ~ I ~ 3 L3/2 + C~l~3tan(B)
= tan(~
+ B)
[4]
where C~ is a cohesion parameter per unit area in direction ~i; L1 and L 3 are the loads per contact in the respective directions. Since L 1 = oie3z 3 and L 3 = o ~ i ~ 3 , Equation [4] may be rewritten in terms of effective stresses. 0{~ 3 - 4C'~ 1 2o'~ 1 + 4 C ~ I tan(B) = t a n ( ~
+ B)
[5]
noting that 2~i/~ 3 = tan(s) an angle related to the packing of the spheres and substituting into-Equation [5] gives Ùi = o ~ t a n ( ~ ) t a n ( ~
+ B) + 2C'tan(~)W
[i + tan(B)tan(~'W + B)]
[6]
Figure id shows the mechanism where by failure takes place. Note that the spheres move apart in the horizontal direction and move into the resultant space in the vertical direction. Considering 61 and 6 3 to be the deflections resulting from a change in load, B the instantaneous and B o the original angles of the slip planes, respectively 61 = 2d(sin(~o) 6 3 = 2d(cos(B) but
~61 = 2dcos(~) 61' = -aload
aB x -a l o- a d
- sin (B)) - cos (Bo)) a63
and
6 ~ = aload
2dsin(B)
a~ x -a l -o a d
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327
SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE
6; therefore but
m
= tan (8)
$~ - 6~/£ 3
and
e i = 61/£ 1
Hence in terms of strain
%
1
_--r = ~
tan(a)tan(B)
= v'
[7]
e1
where v' is the instantaneous Poisson's ratio. The r a t i o o f t h e work done p e r u n i t v o l u m e (RWD) on t h e a s s e m b l y o f s p h e r e s b y t h e m a j o r p r i n c i p l e s t r e s s t o t h e work done on t h e m i n o r p r i n c i p l e s t r e s s b y t h e a s s e m b l y d u r i n g an i n c r e ment o f e x p a n s i o n i s g i v e n by
°iel
KWD " ~
°i
1
= O~ -2V --r =
tan(¢' + ~) 2C'
tan(8)
O~p ~tanC8) + t a n ( , ; +
8)
[8]
Working through the same analysis for a rhombic packing of spheres will also give Equation 8. Rowe demonstrated the validity of the theory by performing triaxial tests on cubic and rhombic packings of spheres made of steel. Rowe hypothesized that the form of Equation 8 derived for regular packings of uniform spheres should also apply to some extent for a mixture of packings of various size particles. If such a particulate system is loaded, then the particles would slide past each other with the individual values of 8 being such as to minimize the rate of internal work done. This condition may be a (RWD) I , expressed8 into [8] as we haa~e = 0 for which 8 = (45 - ~ ). Substituting this value of
°i' i ¢' o~ 2-~-T = tan2(45 + # )
4C' o' + o'~ tan(45 + # )
[9]
All of the variables in Equation 9 may be measured independently and the behaviour of the system predicted. In order to verify the applicability of [9], Rowe performed a large number of triaxial compression tests on cohesionless particles of loose and dense sands which showed good agreement between the experimental results and Equation 9. By rearranging, Equation 9 may be put in a more useful form for the purpose of analysis
°i = KlO 3, + K 2 ~-r where KI ~ 2 tan 2 (45 + 2~! K2
8C' tan (45 + -~) ~2
It can be noted that Equation 9 relates the stresses imposed upon the system to the deformation of the system and hence is analogous to Hookes Law for an elastic material. Consequently if C~' and @U' are known then for any given value of v' the stress a I' corresponding to a lateral stress 03' is known implicitly.
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A. A l e x a n d r i d i s , N.J. Gardner Literature Review In his research on the rheology of fresh concrete, L'Hermite (2), used the triaxial cylinder torsion test to determine points on the Coulomb shear strength envelope. For a concrete mix with a continuously graded aggregate, 300 kg of cement per cubic metre and a water/cement ratio of 0.65. L'Hermite obtained the following values of the coefficient of friction and the cohesion: = 35 ° ~r = 24° (coefficient of friction with respect to the post peak strength) Cohesion = 5.02 kPa The cohesion was measured separately by means of a tensile test between two boxes. A more complete description of the test procedure however, was not included in the paper. L'Hermite concluded that: The rheological properties of fresh concrete, the results which can be expressed in mechanical units, can be measured by simple experiments; The coefficient of friction tan~, decreases as the quantity of mixing water increases; and, - The deformation at failure appears to depend on the grading of the aggregate.
-
-
In 1962, Ritchie (4), published a paper on the triaxial testing of fresh concrete in which it was stated that the workability of a concrete mix is a result of the fundamental rheological properties of the fresh mix. Ritchie measured directly the angle of internal friction of fresh concrete using the three dimensional stress system of the triaxial test. A variety of concrete mixes ranging from high to low workability were tested. Undrained triaxial tests were performed on i00 mm dia. x 200 mm cylindrical specimens at a strain rate of 2.1% per minute until they were seen to fail or the strain exceeded 20%. Cell pressures ranging from 35 to 420 kPa were used. The corresponding Mohr circles were drawn along with the best line giving the angle of internal friction. Table i summarizes Ritchie's results.
and
Ritchie concluded that: (i) For mixes having the same compacting factor, the angle of internal friction increases as the aggregate/cement ratio increases; (2) As the water/cement ratio increases, the lubricating effect of the paste layer between the aggregate increases resulting in decreased values of internal friction.
In 1968, Olsen (3) completed a study to relate the lateral pressures of concrete on formwork to the undrained shear strength of the concrete. Olsen measured and recorded the variations in the undrained shear strength of fresh concrete with time using triaxial tests. He then proceeded to relate the stress-strain relationships of the concrete specimens from the triaxial tests to the strains and corresponding pressures imposed on the formwork. To evaluate the shear strength, Olsen performed undrained triaxial tests for set times (time from addition of water to the mix) ranging from 20 min to 180 min and confining pressures ranging from 140 kPa to 560 kPa. A standard mix having a ratio of cement to sand to coarse aggregate of 1.0:1.5:1.5 was used with a water cement ratio of 0.4. The results of the triaxial tests are s-mmarized in Table 2. From Olsen's results, the following generalized conclusions can be made: (i) For low set times, the shear strength of fresh concrete consists mainly of cohesive bond in the cement paste; (2) As the setting continues, the cement paste becomes less plastic and
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329 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE
TABLE 1 Ritchie's Test Results
[Ref. 4]
Apparent cohesion (kPa)
Workability
Compacting factor
1:3
low medium high
0.85 0.92 0.95
0.452 0,477 0.485
85 125 125
3.5 2.0 1.5
12 ii 8
14 35 28
1:4-1/2
low medium high
0.85 0.92 0.95
0.512 0.549 0.561
30 50 70
7.5 6.5 4.0
28 28 25
21 28 49
1:6
low medium high
0.85 0.92 0.95
0,557 0.665 0.690
zero 60 60
9.0 4.5 2.5
32 30 *
56 56
1:7-1/2
low medium hiBh
0.85 0.92 0.95
0.676 0.775 0.805
zero 20 40
i0.0 5.0 4.5
34 34 *
70 49
Mix
Slump (mm)
Vebe time (s)
Angle of internal friction (degrees)
Water/ cement ratio
TABLE 2 Olsen's Test Results Set Time minutes
Cohesion kPa
20 30 45 60 75 90 120 180
20 27 27 34 35 36 46 64
[Ref. 3]
Internal Friction Angle degress - minutes 1° 1° i° 1@ 3= 3° 3° 5@
34' 22' 47' 47' 02' 26' 18' 31'
the mobility of the aggregate particles decreases resul=ing in an increasing angle of internal friction; and (3) Cohesion steadily increases with set time as cement combines with water to form the binding medium. It should be emphasized that all three investigators performed undrained tests but neglected however to take pore pressure measurements. Consequently all of the test results are in terms of total stresses and an effective stress analysis cannot be carried out. Since the undrained values C and ¢ are pore pressure dependent, and the pore pressures developed under field conditions may differ significantly from those in the laboratory, it is difficult to appreciate the applicability of their test results. It is also evident from the literature of the development of pressures in formwork that the stiffening of concrete is largely dependent on the temperature of the concrete. The effect of temperature on the shear strength of fresh concrete was not mentioned or investigated by any of the three in-
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A. Alexandridis, N.J. Gardner Experimental Investigation To determine the mechanical properties of fresh concrete three types of standard test were carried out at various times after addition of water to the concrete and at two different temperatures. The types of tests used were triaxial compression tests at various cell pressures, unconfined compression tests and K o tests. A more extensive description of the experimental procedure and the results is given by Alexandridis (l). The schedule of the experimental program is given in Table 3. TABLE 3 Schedule of Test Program Test Conditions Test Set Time
Temperature
Confining Pressure
Triaxial Compression
40, 80 min 120, 160
21°C, 4°C
0, 35, 70, 105, 140 kPa
Unconfined Compression
40, 80 min 120, 160
21°C, 4°C
0
K°
40 or 20 min 80, 120, 160
21°C, 4°C
Not applicable
Concrete Mix Characteristics Two concrete mix proportions were used in this investigation corresponding to the following two groups of tests: Triaxial Tests and Unconfined Compression Test. A single concrete mix was used having ratios of cement to sand to coarse aggregate of 1.0:2.7:2.1 by weight and a water/cement ratio of 0.57. This resulted in a concrete with a 50 mm slump and an average 28-day compressive strength of 25.9 MPa. ~ Tests. For these tests, a mix having ratios of cement to sand to coarse aggregate of 1.0:1.5:3.0 by weight was used along with a water/cement ratio of 0.55. The resulting concrete had a 63 mm slump and a 28-day average compressive strength of 34.7 MPa. In all tests, Type i Normal Portland Cement was used and crushed air dried limestone gravel and quartz sand. The aggregate may be classified as angular in appearance. In an attempt to maintain temperature and humidity uniformity throughout the testing period, all materials were stored in containers prior to mixing and kept in the room where the testing was to be conducted. Triaxial Compression Test In a triaxial test a cylindrical specimen is initially subjected to an equal all around pressure and then failed through application of an axial load. The specimen is enclosed by a rubber membrane which is sealed at the bottom to a pedestal and at the top to a metal cap. The entire assemblage is contained in a chamber into which air or water is allowed to enter at the desired pressure. This pressure acts on the sides and top of the specimen through the rubber membrane and cap. Additional axial load may be applied to the specimen by means of a piston passing through the top of the chamber and resting on the metal cap. Porous stone is placed beneath the sample and is connected to the outside of the chamber by tubing. This allows the pressure of the water in the sample to be measured if drainage is not allowed. Alternatively, if drainage is allowed, the quantity of water passing into or out of the sample can be monitored.
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331 SHEAR STRENGTH, TRIAXlAL COMPRESSION, FRESHCONCRETE
I
I
I - Cono'ele Somple 2 - Trlol~ol Cell 3 - Trioxiol Piston Rod 4 - Proving Ring ond Diol Gouge 5 - ~ f l l c l i o n Gouge 6 - To~ Cap 7 - Rebb~r Mmnlwoee
;7
Air ~pply
8 - V.tiol~le 50~ed F.teclric Motor oj. Pressure Regu~atin~l Vah,e
--
I 0 - Pore Prestvfe Cell
I I - Tronsducer 12- u-Tube Press.re Meosurm~ Device
II
FIG. 2 Triaxial Compression Set-up Figure 2 shows a sketch of the triaxial compression set up used during the investigation. Testing was done on i00 mm x 200.-, cylindrical concrete specimens. In all, forty undrained triaxial tests were successfully performed for sct times, defined as time after addition of water to mix, of 40, 80, 120 and 160 minutes, cell pressures of O, 30, 70, 105 and 140 kPa and temperatures of 21°C and 4°C. The results taken for every combination of set time and temperature, with cell pressure maintained constant, were temperature, deviator stress, axial strain, lateral strain and pore water pressure. A typical set of results is given in Figures 3A and 3B. Unconfined Compression Tests During the triaxial testing it was determined that the fresh concrete dilated during the application of the deviator load and that negative pore water pressures developed in the samples even though the cell pressure was set equal to zero. Consequently, it became impossible to perform an unconfined compression test for the purpose of getting a better estimation for the cohesion of fresh concrete as defined by the Mohr-Coulomb failure theory. To overcome this, a series of drained unconfined compression tests were performed on unsupported cylindrical samples of fresh concrete° The samples were formed by placing concrete in a plexiglass cylinder 82 m~ in diameter and 178 m high° The concrete was placed in four layers and each layer received fifty tamps as was done for the triaxial tests. The concrete was then allowed to set for the appropriate time period after which the cylinder was carefully slipped off and the sample compressed to failure at a strain rate of 0.14 --- per minute. To measure the lateral strain, a strip of wet paper marked off I n . - . w a s placed around the midsection of the sampleo For the unconfined compression tests the measurements stress, axial strain and lateral strain.
taken were axial
332
Vol. 11, No. 3 A. A l e x a n d r i d i s , N.H. Gardner
FIG. 3A.
Stress difference versus axial strain for 21°C and 160 mln. SET TIME.
FIG. 3B.
Pore pressure versus axial strain for 21°C and 160 min. SET TIME.
*o
6o
to
~o
o
AX',AJ. ST~AL~ FLG
3A
STRF.55 01FFER[NCF. V[RSU$ AXIAL. STRALN FOR ZI"C AND 1 6 0 ~ n , ~ T TIMF.
8O
•
QO
40
lo
-ZO
~
O; • 0
kPo
-40
FIG. 3 Typical Results of Triaxial Tests for Concrete at 21°C and 160 min. Set Time. K o Tests. Values of K o for fresh concrete were obtained by applying vertical stresses through a piston in series with a loading ring onto fresh concrete samples 81.8 mm in diameter and 177.8 mm high. These samples were restrained from moving laterally by means of a plexiglass cylinder casing whose diameter was 1.5 mm larger than that of the piston° Water was allowed to drain out of the concrete during the application of vertical stresses° A circular sponge was placed between the piston and the concrete to prevent the loss of the finer sand and cement particles. The lateral strains and the corresponding stresses as a result of vertical loading were measured directly by means of two strain gauges on either side of the cylinder attached at its midheight. The vertical load was applied in successive increments of 55°6 N and the corresponding strain gauge readings were taken only after the excess pore pressures had dissipated and the load dial gauge had stabilizedo In Figure 4, K o has been plotted as a function of the effective vertical stress and for set times ranging from 20 minutes to 160 minutes, The plots have been done for temperatures of 4°C and 21°Co
Vol. I I , No. 3
333
SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE 240 ~0~ 21"("
M,~
/'
#
zl°c 4OM~ 21~ I
I
200
/\ °
FIG. 4 K o versus Effective Vertical Stress.
120
~
so
I
I
I'\
iS0
~
"
,,'
/#
It;
,
t\
,
i
",,i
\t,'k
"X,l',,t,
40 4"C
,s 4~
0
I Oi
oil
k----
120klm
I 03
40~n 4"C
I 04
I 05
0i6
i , O#
K.
Analysis of Experimental Data In general in particle mechanics there does not exist an equivalent of Hookes law in which stress is related to deformation° Soil mechanics practice has concerned itself wlth the failure properties of the system regardless of deformation° However, Rowe's recent theory does allow the effective stresses on the sample to be related to the deformation of the sample. Consequently in this section the failure behaviour of the fresh concrete will be compared to both the Mohr-Coulomb and Rowes theories and in addition the Rowe theory will be compared to the pre-failure behaviour. Failure Behavlour Choice of Failure Criteria In the triaxial testing of soils it is generally accepted that a test sample has failed when the maximum stress difference a I - o~ has been reached. For highly plastic materials, this condition of failure is sometimes not achieved until the test sample has undergone large axial strain. For this reason, failure has also come to be arbitrarily defined as the instant a test sample achieves 20% axial strain. For fresh concrete in formwork, it may be inappropriate to use 20% axial strain as a failure criterion because such large strains should never be encountered in real practice. It was therefore decided to adopt two failure criteria for analyzing the test results: (i) (2)
.The maximum stress difference or 20% axial strain° The maximum stress difference or i0% axial strain.
Mohr-Coulomb Analysis Calculation of the Mohr-Coulomb parameters C' - #' was performed by plotting the t w o failure criteria for each temperature on a p'-q' diagram oi - o t oi - o (p , = --~----, q , . --~----'). Figure 5 shows such a plot for 21 o Co The four lines corresponding to ~et times of 40, 80, 120 and 160 minutes, represents a linear regression of the failure stresses for cell pressures of O, 35, 70, i05, 140 kPao For such a plot the slope a and the intercept A' on the q' axis can be related to the Mohr-Coulomb parameters
334
Vol. I I , No, 3
A. A]exandridis, N.J. Gardner I
250~ •
200 •
160 M,n 4 0 M,n
~,
q20M,7
\
, FIG. 5 p'-q' Plot for 21°C and 10% Failure Strain.
"61 "o I00
50
0 bt"
I
t
I
I
50
~O
1SO
200
p' • Cr'~'O'i 2
•
1
1
250
300
WPo
sln¢' = tan~ = q'/p' and A' C !
The lines were regressed the friction angles.
=
C0S¢ v
through the origin to get an initial estimate of
Using these initial estimates of the friction angles and the results of the unconfined compression tests the effective cohesions were calculated C' = ~qu tan (45 - ~) Hence values were calculated for the A' intercepts on the p'-q' plots. The p'-q' lines were then linearly regressed through the appropriate q' intercepts to give a refined value for the friction angles. The results of the Mohr-Coulomb analysis have been s u ~ r l z e d in Table 4. Rowe's Analysis Figure 6 represents a plot t ! failure criterion on ol/v - o 3! figure for set times or 40, 80. linearly regressed through five
of the trlaxlal test results at 21°C for the 10% planes. Four lines have been drawn in each 120 and 160 minutes. Each llne has been points corresponding to 0, 35, 30, 105 and
~u
140 kPa cell pressures. The slopes represent K I - 2tan2(45 +~2--) in Rowe's equation corresponding to the set times for which the lines have been drawn. The o~/v, intercepts represent K 2 = 8C' tan(45 +-2~). Knowing the values of K I an~ K 2, the friction angle ¢~ and the cohesion C~ corresponding to Rowe's t~eory can be evaluated° Initlally it w a s a s s u m e d that the cohesion was negligibly small and the lines were made to pass through the origin of the coordinate system. From the slopes~ initial est~-mtes of ¢~ were made from which the cohesion C~ and subsequently K 2 were estimated For 21°C. The lines were then linearly regressed a second time through their appropriate ol/v' intercepts to obtain more accurate values of ¢~ at 21°Co At 4eCp accurate values of the instantaneous Polsson's ratio uv could not be obtained because
Vol. I I , No. 3
335 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE TABLE 4 Results of Mohr-Coulomb Analysis
Temperature 21°C
40C
S~ (a)
Set Time
Effective Friction Angle¢'
S;
40 mln
41°C
0.4 °
1.90 kPa
80
39 °
0.6 °
4.69
120 160
39 ° 41 °
1.2 ° 1.6 °
5.74 6.30
40
40 °
0.4 °
1.47
80
39 °
006 °
2.87
120 160
38 °
107 ° 0.9 °
1.68 2.38
37 °
*
Cohesion C'
standard deviation Failure Criterion; Maximum Stress Difference or 10% Axial Strain
Temperature
21°C
4°
Set Time
Effective Friction Angle¢'
* S;
Cohesion C'
40 min 80 120 160
38 ° 36 ° 36 ° 39 °
0.9 ° 1.7 ° io4 ° 2o7 °
lo61 kPa 4006 4097 5o67
40 80 120 160
38 ° 36 ° 35 ° 34 °
i.i ° 1o5 ° i.i 0.5 °
1.96 2.59 1.54 2.24
S; - standard deviation (b)
Failure Criterion; Maximum Stress or 20% Axial Strain
500
f
4O0
-5i~,300
~ ' 2 o . .
O' 1
__--r V.S.
at 21°C.
FIG. 6 03
F
200
-\~
o- eo.,.
A- 40Mm
for 10% Failure Strain I00
0
60
0
wOO
120
0"~" , iPo FIG. 6
U;O" VS
o','
FOR I 0 % FAILURE: STRAIN AT 21"C
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Vol. I I , No. 3 A. Alexandridis, N.J. Gardner
FIG. 7 250
!
Variation of °--~l V.S.~, Concrete.
o~ for
o~ = O, Set Time
40 mln. at 21°C. 200
150 i 30
I 4o (:T]'
,
J 5o
kPo
of the highly plastic nature of the concrete specimens. Consequently, K 2 and correspondingly C~ could not be estimated. Therefore, a second linear regression was not possible at 4@C. However, because the difference in ~ at 21°C between a linear regression through the origin and a linear regression through K 2 is in the order of l ° to 2 °, it was felt that the first estimates of ~ at 4°C adequately represented the true angles of internal friction° The results of Rowe's analysis have been summarized in Table 5. Pre-failure Behaviour Using Rowe's theory it should be possible to plot all the results of tests at one set time/temperature combination as a single straight llne. The problem experimentally is in obtaining values of the instantaneous Poisson's Ratio to any reasonable degree of accuracy at low strains. Consequently this part of the investigation was not wholly successful° Figure 7 shows the results for the specimens tested at a set time of 40 mins. at a temperature of 21°Co It can be seen that the experimental results do approximate to a straight line but it would be unrealistic to make extensive conclusions due to the lack of confidence in the basic data concerning lateral strain° Discussion Figures 5, 6 and 7 indicate that fresh concrete exhibits shear strength characteristics in accordance with both Coulomb's equation ~' = C' + o~ tan~' oI and Rowe's equatlon ~-r = KlO ~ + K 2 where K I and K 2 are functions of the effective friction angle and the effective cohesion° However, caution must be exercised when one interprets this statement with respect to Coulomb's theory. The decrease in pore pressure with axial strain as shown in Figure 3B resulting from an applied deviator stress, implies that the concrete mix used in this investigation behaved in a dilatant fashion when being sheared~ The test samples therefore, could not be distorted without an increase in the distance between the particles unless the individual particles themselves were to break. Because of the dilatancy, only a portion of the work done by the applied stresses went into shearing the test samples. The remainder of the work went into rearranging the particles during dilatancy. Coulomb's strength parameters C' and ~' therefore, do not strictly represent the cohesion and the
Vol. I I , No. 3
337 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE TABLE 5 Results of Rowe's Analysis
Temperature
21°C
4°C
S~ a)
Cohesion
18 ° 18 °
2o0 °
20 °
2.3 °
19 °
2.8 °
0.15 kPa 0.53 0.40 1.57
3.2 ° 3.0 ° 6o6 ° 2.8 °
-
Effective Friction Angle @~v
40 min 80 120 160 40 80 120 160
22 ° 22 ° 23 ° 20 °
1.8 °
Ct
- standard deviation of ¢~ Failure Criterion: Maximum Stress Difference or i0% Axial Strain
Temperature
21°C
4°
S'
S'* @u
Set Time
Effective Friction Angle ~ ,
S¢~
C'
19 ° 21 ° 18 ° 21 °
2.4 ° 7.5 ° 4.3 ° 5.7 °
0.15 kPa 0.53 1.40 1.57
40
20 °
200 °
80
20 °
307 °
120 160
21 ° 19 °
1.7 °
-
Set Time
40 min 80 120 160
,*
2.5 °
Cohesion
!
- standard deviation of @u
(b~ ~ Failure Criterion: Maximum Stress Difference or 20% Axial Strain angle of internal friction respectively in the full sense of their definitions. Instead, the values assigned to C' and 4' are such as to make Coulomb's straight llne fit the best results° Dilatancy however, poses no problem with respect to Rowe's theory where Rowe's shear strength equation was derived for a particulate system as opposed to a continuum whose volume remains constant. The strength parameters C~ and ~ correspond directly to the shearing resistance offered by cohesion and internal friction respectively° From Table 4 Coulomb's friction angle for the concrete mix lles in the range of 37 ° to 41 ° for a 10% failure strain and 34 ° to 39 ° for a 20% failure strain° These values correspond to a dense arrangement of particles and are supported by the fact that concrete aggregates tend to be closely packed and angular, achieving a high degree of particle interlock. In Table 5, Rowe's friction angle is seen to lie in the range of 18 ° to 21 ° for a 10% failure strain and 19 ° to 23 ° for a 20% failure strain° In the verification of his theory, Rowe evaluated the friction angles for clean sands and silts and found them to lle between 23 ° and 30°; the lower friction angles corresponding to the coarser material° Since concrete is composed of aggregates in the coarse sand and gravel sizes, the friction angles obtained in
338
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N.J. Gardner
this investigation are consistant with those in Rowe's investigationo It may be seen that Rowe's friction angles are approximately half the value of Coulomb's friction angles. This supports the idea that Coulomb's friction angles not only include the effects of frictional resistance against shearing but also to resistance in expanding against the principle stresses acting on the test sample~ No general trend is evident for the observed changes in the friction angles for either an increasing set time or for temperature change. The reason is that the component of shear resistance resulting from the translation and frictional resistance between aggregate and cement particles which is inincluded in the angle of internal friction, depends on surface roughness and the load per particle° Since these are independent of temperature change and set time, so should be the angle of internal friction. Since the angle of internal friction is an inherent property of a concrete mix and remains essentially constant with time and temperature, it is the increase in cohesion as a result of the hydration process which allows fresh concrete to develop significant shear strength with time and solidify. This increase in cohesion with time is clearly depicted in Tables 4 and 5 for both Coulomb's theory and Rowe's theory at 4°C and 21°C0 Since hydration is a chemical reaction, the rate of increase in cohesion with time is seen to be significantly lower at 4°C as compared to 21°Co The lateral pressure corresponding to the "at rest" state may be evaluated when the coefficient K o for fresh concrete is known= The variation of K o for the given mix with set time, effective vertical stress and temperature can be seen in Figure 4° It may be seen that K o is variable and highly dependent on set time at 21°C and to a lesser degree at 4°C. Initially, fresh concrete being in a highly plastic state behaves much like a liquid. Any vertical stresses are correspondingly transformed into lateral stresses° Consequently, K o approaches unity° As the fresh concrete hydrates, lateral strains as a result of vertical stress become restricted and approach those of Poisson's ratio for cured concrete. The value of K o therefore decreases signficantly with set time. This decrease is of a smaller order at the colder temperature because of the hydration process being retarded. Conclusions (I) The shear strength properties of fresh concrete may be expressed by either the Mohr-Coulomb or Rowe's theory of shear strength. A/though Coulomb's theory is fundamentally incorrect, it is able to adequately express the results of shear tests performed on fresh concrete. (2) The shear strength of fresh concrete is both time and temperature dependent. (3) Immediately following mixing, the shear strength of fresh concrete is due mainly to the internal friction resulting from particle interaction which remains constant with set time and temperature changeo (4) The hardening of concrete with set time is due mainly to the development of cohesion as a result of the hydration process° At lower temperatures, hydration is retarded and cohesion develops at a slower rate causing the fresh concrete to behave fluid like for a longer period of time° (5) The coefficient K o for fresh concrete is highly variable and depends on set time, temperature and to a lesser degree on the applied vertical stress. Its value is lower at warm temperatures and decreases with set time~ Acknowledgements The authors wish to acknowledge the financial support of the National Sciences and Engineering Research Council of Canada under Grant A5646.
VO]. I I , No. 3
339 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE References
io
Ao Alexandridis, "Mechanical Properties of Fresh Concrete", MoA°Sco Thesis, University of Ottawa, 1979, ppo158o
2°
R. L'Hermlte, "Vibration and the Rheology of Freshly Mixed Concrete", Revue des Materiaux de Construction, No° 405, 1949, pp.179-87o
3o
R.Ho Olsen, "Lateral Pressure of Concrete on Formwork", PhoD. Thesis, Oklahoma State University, 1968, pp.122°
4.
AoG.B° Ritchie, "The Trlaxial Testing of Fresh Concrete", Magazine of Concrete Research, Vol. 14, No° 40, 1962o
5.
P.W. Rowe, "The Stress-Dilatancy Relation for Static Equilibrium of an Assembly of Particles in Contact", Proceedings of the Royal Society Series A, Volo 269, 1962, ppo500-527°
MECHANICAL BEHAVIOUR OF FRESH CONCRETE
A. Alexandridis Engineer, Stone and Webster Toronto, Canada N. J. Gardner Professor, Department of Civil Engineering University of Ottawa Ottawa, Canada KIN 9B4
(Communicated by R.E. Philleo) (Received Dec. 19, 1980) ABSTRACT The shear strength characteristics of fresh concrete were studied through the use of a triaxial compression apparatus on concrete cylinders i00 mm in diameter and 200 mm high at 21°C and 4°C for set times ranging from 40 minutes to 160 minutes. The test results were analyzed by the shear strength theories of Mohr-Coulomb and Rowe for each tlme-temperature combination. The angle of internal friction was found to be a constant property of the mix in the range of 37 @ to 41 @ for a 10% failure strain when analyzed by Mohr-Coulomb. Rowe's analysis gave friction angles of 18 @ to 21 @ for a 10% failure strain. Both theories indicate that the cohesion of fresh concrete is initially zero and increases with set time.
La r~sistance au cisaillement du b~ton frais a ~t~ ~tudi~e avec l'aide d'un appareil triaxlal sur des ~chantillons de i00 mm de diam~tre par 200 mm en grandeur ~ 21°C et 4°C pour une dur~e de 40 minutes ~ 160 minutes. Par la suite les r~sultats ont ~t~ analys~s employant les theories de Mohr-Coulomb et Rowe et ont ~t~ exprlm~s comme les deux param~tres de coh~sloD et de frottement pour chaque combinaison de temperature et de dur~e. On a d~duit que l'angle de frottement ~tait une propri~t~ constante du m~lange et r~sidait entre 37 ° e t 41 ° pour une d~formatlon finale de i0% utilisant la th~orie de Mohr-Coulomb. L'analyse de Rowe a donn~ des angles de frottement de 18 ° ~ 21 ° pour une d~formation finale de 10%. Les deux theories ont indiqu~ que la cohesion ~tait initialement z~ro et qu'elle s'~tait agrandit avec le temps.
323
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N.J. Gardner
Introduction When fresh concrete is first mixed, it possesses in addition to viscosity significant shear strength so that when it is poured from its mixing container into a form it does not fully take the shape of the form unless it is tamped or vibrated. However, when removed from its container, fresh concrete is unable to maintain the mould shape and will slump. The properties of fresh concrete therefore, lie between those of a liquid and those of a solid substance. Fresh concrete may be visualized as particles of inert aggregate which are held or suspended in a deformable matrix of cement paste and air bubbles. The paste matrix itself is composed of water and grains of cement. Given time and the proper environmental conditions, the cement paste matrix is converted by means of a physicochemical process between the cement grains and the water into a rocklike mass of particles. The hardened particulate material which replaces the original cement grains is called cement gel and it occupies the space in the mixture originally occupied by the cement grains and the water. To date, a significant amount of research has been aimed at understanding the mechanical properties of concrete in the hardened state. Problems pertaining to the pumpability, workability, placing of fresh concrete and the lateral pressures developed in formwork are examples in which little is understood of how fresh concrete behaves. Engineers who need to undertake calculations for such problems must resort to empirical formulae, charts and tables. Such a procedure however, may lead to gross inaccuracies when applied to concretes whose mix proportions and conditions at mixing differ significantly from those corresponding to the concrete for which the procedure was derived. Consequently, a more fundamental understanding of the mechanical behaviour of fresh concrete is necessary. Fresh concrete like soil is a particulate system composed of particles weakly bonded together and submerged in a liquid medium. It possesses shear strength as a result of (I) the frictional resistance and interlocking between the aggregate and the cement particles and (2) the bonding together of the aggregate particles by the cement during hydration. The former component of shear strength is termed internal friction and requires strain to be mobilized. The latter component is termed cohesion. True cohesion results from cement hydration and therefore depends on the time since addition of water to the mix and the temperature at which the mix is allowed to hydrate. The mechanical behaviour of fresh concrete is very similar to that of a cohesive soil in that it possesses cohesion, internal friction and pore water pressure. Hence it is advantageous to review the basic theories which explain the behaviour of particulate systems. Theoretical BackBround Stress in a Particulate System Fresh concrete may be thought of as a particulate system; a system composed of particles which are not strongly bonded together like the grains of a solid yet neither free to move like the elements of a fluid, which are submerged in a liquid medium. A system such as this may transmit forces through (i) the voids (pore pressure), (2) particle contact points (effective stresses) and (3) the cross-sections of the particles. It has been shown by Terzaghi that for a particulate system the total stress on any plane not cutting across any particles is given with sufficient accuracy by the following equation
Vol. I I , No. 3
325 SHEAR STRENGTH, TRIAXlAL COMPRESSION, FRESH CONCRETE
where
o = total stress; o' = effective stress and U = pore pressure.
The above equation is based on the assumption that the ratio (Ac/A) 2 is much smaller than unity where A c denotes the area of contact points on the plane and A denotes the total area of the plane. Shear Strength Theories Mohr-Coulomb Theory The conventional expression used in soll mechanics to describe the shear strength of a soil, at failure, is the Coulomb equation T = C + a n tan~ where
• C on tan ~
= = = =
[I]
shear stress cohesion normal stress coefficient of friction.
Soll mechanics practice is to represent the stress conditions at failure as a series of Mohr's Circles on a single set of axes; the envelope of such stress circles is known as the Mohr-Coulomb Failure Envelope. It is important to note that for saturated samples the shear strength depends on effective stress and not total stress and Coulomb's equation must be written in terms of effective stress parameters to become 3' = C' + Sn' tan~'
[2]
Primes indicate effective stress parameters. The Mohr-Coulomb theory has a fundamental meaning provided the assumption is made that no volume occurs. With volume change included in the C' - ~' parameters, the theory is essentially a useful engineering tool, but nothing more. Since the volume changes depend on the principle stress system imposed which differs between types of tests, universal C'-~' values cannot be found for a material independent of the test method. Rowe's Theory Consider the force P required to cause the body in Figure la to move along a plane inclined at an angle B to the direction of the force P (5). Assuming that the material exhibits both cohesive and frictional components of shearing resistance then, at the instant of slip, the forces can be resolved to give P - C
x +
Q + C x tan8 = tan (~u where
8)
[3]
C~ = a cohesive stress quantityJprojected length in P direction #u = the angle of friction of the material Q = force on body normal to P.
Using this equation, the equations of state for a simple geometrical packing of spheres may be derived. Two such packings of spheres are the body centered cubic and the rhombic which correspond to the loosest and the most dense states respectively. Consider a system of spheres in a body centered cubic packing as shown in Figure lb. Movement of one sphere relative to another during the application of a deviator stress can only take place by sliding. Applying Equation [3] at a point of contact of the spheres, the following condition of sliding is arrived at
326
Vol. A. A l e x a n d r i d i s ,
N.J.
V~L,
P
,f2 L~- ° /
L_~3
Body Centered Cubic Packing of Spheres b) plan
forces on on inclined plane
L~
2
~-~
3
> >
L ~"
L,
rio.
~L, X
,/2 L3
o)
II,
Gardner
'
L
2
t
L~ ÷ 0"~'
~i.. L3
),-~,--,g LI
2
_,
c) section X - X
d) failure mechanism
Derivation
FIG. 1 of Rowe's Equation
LI/4 - C ~ I ~ 3 L3/2 + C~l~3tan(B)
= tan(~
+ B)
[4]
where C~ is a cohesion parameter per unit area in direction ~i; L1 and L 3 are the loads per contact in the respective directions. Since L 1 = oie3z 3 and L 3 = o ~ i ~ 3 , Equation [4] may be rewritten in terms of effective stresses. 0{~ 3 - 4C'~ 1 2o'~ 1 + 4 C ~ I tan(B) = t a n ( ~
+ B)
[5]
noting that 2~i/~ 3 = tan(s) an angle related to the packing of the spheres and substituting into-Equation [5] gives Ùi = o ~ t a n ( ~ ) t a n ( ~
+ B) + 2C'tan(~)W
[i + tan(B)tan(~'W + B)]
[6]
Figure id shows the mechanism where by failure takes place. Note that the spheres move apart in the horizontal direction and move into the resultant space in the vertical direction. Considering 61 and 6 3 to be the deflections resulting from a change in load, B the instantaneous and B o the original angles of the slip planes, respectively 61 = 2d(sin(~o) 6 3 = 2d(cos(B) but
~61 = 2dcos(~) 61' = -aload
aB x -a l o- a d
- sin (B)) - cos (Bo)) a63
and
6 ~ = aload
2dsin(B)
a~ x -a l -o a d
Vol. I I , No. 3
327
SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE
6; therefore but
m
= tan (8)
$~ - 6~/£ 3
and
e i = 61/£ 1
Hence in terms of strain
%
1
_--r = ~
tan(a)tan(B)
= v'
[7]
e1
where v' is the instantaneous Poisson's ratio. The r a t i o o f t h e work done p e r u n i t v o l u m e (RWD) on t h e a s s e m b l y o f s p h e r e s b y t h e m a j o r p r i n c i p l e s t r e s s t o t h e work done on t h e m i n o r p r i n c i p l e s t r e s s b y t h e a s s e m b l y d u r i n g an i n c r e ment o f e x p a n s i o n i s g i v e n by
°iel
KWD " ~
°i
1
= O~ -2V --r =
tan(¢' + ~) 2C'
tan(8)
O~p ~tanC8) + t a n ( , ; +
8)
[8]
Working through the same analysis for a rhombic packing of spheres will also give Equation 8. Rowe demonstrated the validity of the theory by performing triaxial tests on cubic and rhombic packings of spheres made of steel. Rowe hypothesized that the form of Equation 8 derived for regular packings of uniform spheres should also apply to some extent for a mixture of packings of various size particles. If such a particulate system is loaded, then the particles would slide past each other with the individual values of 8 being such as to minimize the rate of internal work done. This condition may be a (RWD) I , expressed8 into [8] as we haa~e = 0 for which 8 = (45 - ~ ). Substituting this value of
°i' i ¢' o~ 2-~-T = tan2(45 + # )
4C' o' + o'~ tan(45 + # )
[9]
All of the variables in Equation 9 may be measured independently and the behaviour of the system predicted. In order to verify the applicability of [9], Rowe performed a large number of triaxial compression tests on cohesionless particles of loose and dense sands which showed good agreement between the experimental results and Equation 9. By rearranging, Equation 9 may be put in a more useful form for the purpose of analysis
°i = KlO 3, + K 2 ~-r where KI ~ 2 tan 2 (45 + 2~! K2
8C' tan (45 + -~) ~2
It can be noted that Equation 9 relates the stresses imposed upon the system to the deformation of the system and hence is analogous to Hookes Law for an elastic material. Consequently if C~' and @U' are known then for any given value of v' the stress a I' corresponding to a lateral stress 03' is known implicitly.
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A. A l e x a n d r i d i s , N.J. Gardner Literature Review In his research on the rheology of fresh concrete, L'Hermite (2), used the triaxial cylinder torsion test to determine points on the Coulomb shear strength envelope. For a concrete mix with a continuously graded aggregate, 300 kg of cement per cubic metre and a water/cement ratio of 0.65. L'Hermite obtained the following values of the coefficient of friction and the cohesion: = 35 ° ~r = 24° (coefficient of friction with respect to the post peak strength) Cohesion = 5.02 kPa The cohesion was measured separately by means of a tensile test between two boxes. A more complete description of the test procedure however, was not included in the paper. L'Hermite concluded that: The rheological properties of fresh concrete, the results which can be expressed in mechanical units, can be measured by simple experiments; The coefficient of friction tan~, decreases as the quantity of mixing water increases; and, - The deformation at failure appears to depend on the grading of the aggregate.
-
-
In 1962, Ritchie (4), published a paper on the triaxial testing of fresh concrete in which it was stated that the workability of a concrete mix is a result of the fundamental rheological properties of the fresh mix. Ritchie measured directly the angle of internal friction of fresh concrete using the three dimensional stress system of the triaxial test. A variety of concrete mixes ranging from high to low workability were tested. Undrained triaxial tests were performed on i00 mm dia. x 200 mm cylindrical specimens at a strain rate of 2.1% per minute until they were seen to fail or the strain exceeded 20%. Cell pressures ranging from 35 to 420 kPa were used. The corresponding Mohr circles were drawn along with the best line giving the angle of internal friction. Table i summarizes Ritchie's results.
and
Ritchie concluded that: (i) For mixes having the same compacting factor, the angle of internal friction increases as the aggregate/cement ratio increases; (2) As the water/cement ratio increases, the lubricating effect of the paste layer between the aggregate increases resulting in decreased values of internal friction.
In 1968, Olsen (3) completed a study to relate the lateral pressures of concrete on formwork to the undrained shear strength of the concrete. Olsen measured and recorded the variations in the undrained shear strength of fresh concrete with time using triaxial tests. He then proceeded to relate the stress-strain relationships of the concrete specimens from the triaxial tests to the strains and corresponding pressures imposed on the formwork. To evaluate the shear strength, Olsen performed undrained triaxial tests for set times (time from addition of water to the mix) ranging from 20 min to 180 min and confining pressures ranging from 140 kPa to 560 kPa. A standard mix having a ratio of cement to sand to coarse aggregate of 1.0:1.5:1.5 was used with a water cement ratio of 0.4. The results of the triaxial tests are s-mmarized in Table 2. From Olsen's results, the following generalized conclusions can be made: (i) For low set times, the shear strength of fresh concrete consists mainly of cohesive bond in the cement paste; (2) As the setting continues, the cement paste becomes less plastic and
Vol. I I , No. 3
329 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE
TABLE 1 Ritchie's Test Results
[Ref. 4]
Apparent cohesion (kPa)
Workability
Compacting factor
1:3
low medium high
0.85 0.92 0.95
0.452 0,477 0.485
85 125 125
3.5 2.0 1.5
12 ii 8
14 35 28
1:4-1/2
low medium high
0.85 0.92 0.95
0.512 0.549 0.561
30 50 70
7.5 6.5 4.0
28 28 25
21 28 49
1:6
low medium high
0.85 0.92 0.95
0,557 0.665 0.690
zero 60 60
9.0 4.5 2.5
32 30 *
56 56
1:7-1/2
low medium hiBh
0.85 0.92 0.95
0.676 0.775 0.805
zero 20 40
i0.0 5.0 4.5
34 34 *
70 49
Mix
Slump (mm)
Vebe time (s)
Angle of internal friction (degrees)
Water/ cement ratio
TABLE 2 Olsen's Test Results Set Time minutes
Cohesion kPa
20 30 45 60 75 90 120 180
20 27 27 34 35 36 46 64
[Ref. 3]
Internal Friction Angle degress - minutes 1° 1° i° 1@ 3= 3° 3° 5@
34' 22' 47' 47' 02' 26' 18' 31'
the mobility of the aggregate particles decreases resul=ing in an increasing angle of internal friction; and (3) Cohesion steadily increases with set time as cement combines with water to form the binding medium. It should be emphasized that all three investigators performed undrained tests but neglected however to take pore pressure measurements. Consequently all of the test results are in terms of total stresses and an effective stress analysis cannot be carried out. Since the undrained values C and ¢ are pore pressure dependent, and the pore pressures developed under field conditions may differ significantly from those in the laboratory, it is difficult to appreciate the applicability of their test results. It is also evident from the literature of the development of pressures in formwork that the stiffening of concrete is largely dependent on the temperature of the concrete. The effect of temperature on the shear strength of fresh concrete was not mentioned or investigated by any of the three in-
330
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A. Alexandridis, N.J. Gardner Experimental Investigation To determine the mechanical properties of fresh concrete three types of standard test were carried out at various times after addition of water to the concrete and at two different temperatures. The types of tests used were triaxial compression tests at various cell pressures, unconfined compression tests and K o tests. A more extensive description of the experimental procedure and the results is given by Alexandridis (l). The schedule of the experimental program is given in Table 3. TABLE 3 Schedule of Test Program Test Conditions Test Set Time
Temperature
Confining Pressure
Triaxial Compression
40, 80 min 120, 160
21°C, 4°C
0, 35, 70, 105, 140 kPa
Unconfined Compression
40, 80 min 120, 160
21°C, 4°C
0
K°
40 or 20 min 80, 120, 160
21°C, 4°C
Not applicable
Concrete Mix Characteristics Two concrete mix proportions were used in this investigation corresponding to the following two groups of tests: Triaxial Tests and Unconfined Compression Test. A single concrete mix was used having ratios of cement to sand to coarse aggregate of 1.0:2.7:2.1 by weight and a water/cement ratio of 0.57. This resulted in a concrete with a 50 mm slump and an average 28-day compressive strength of 25.9 MPa. ~ Tests. For these tests, a mix having ratios of cement to sand to coarse aggregate of 1.0:1.5:3.0 by weight was used along with a water/cement ratio of 0.55. The resulting concrete had a 63 mm slump and a 28-day average compressive strength of 34.7 MPa. In all tests, Type i Normal Portland Cement was used and crushed air dried limestone gravel and quartz sand. The aggregate may be classified as angular in appearance. In an attempt to maintain temperature and humidity uniformity throughout the testing period, all materials were stored in containers prior to mixing and kept in the room where the testing was to be conducted. Triaxial Compression Test In a triaxial test a cylindrical specimen is initially subjected to an equal all around pressure and then failed through application of an axial load. The specimen is enclosed by a rubber membrane which is sealed at the bottom to a pedestal and at the top to a metal cap. The entire assemblage is contained in a chamber into which air or water is allowed to enter at the desired pressure. This pressure acts on the sides and top of the specimen through the rubber membrane and cap. Additional axial load may be applied to the specimen by means of a piston passing through the top of the chamber and resting on the metal cap. Porous stone is placed beneath the sample and is connected to the outside of the chamber by tubing. This allows the pressure of the water in the sample to be measured if drainage is not allowed. Alternatively, if drainage is allowed, the quantity of water passing into or out of the sample can be monitored.
Vol. I I , No. 3
331 SHEAR STRENGTH, TRIAXlAL COMPRESSION, FRESHCONCRETE
I
I
I - Cono'ele Somple 2 - Trlol~ol Cell 3 - Trioxiol Piston Rod 4 - Proving Ring ond Diol Gouge 5 - ~ f l l c l i o n Gouge 6 - To~ Cap 7 - Rebb~r Mmnlwoee
;7
Air ~pply
8 - V.tiol~le 50~ed F.teclric Motor oj. Pressure Regu~atin~l Vah,e
--
I 0 - Pore Prestvfe Cell
I I - Tronsducer 12- u-Tube Press.re Meosurm~ Device
II
FIG. 2 Triaxial Compression Set-up Figure 2 shows a sketch of the triaxial compression set up used during the investigation. Testing was done on i00 mm x 200.-, cylindrical concrete specimens. In all, forty undrained triaxial tests were successfully performed for sct times, defined as time after addition of water to mix, of 40, 80, 120 and 160 minutes, cell pressures of O, 30, 70, 105 and 140 kPa and temperatures of 21°C and 4°C. The results taken for every combination of set time and temperature, with cell pressure maintained constant, were temperature, deviator stress, axial strain, lateral strain and pore water pressure. A typical set of results is given in Figures 3A and 3B. Unconfined Compression Tests During the triaxial testing it was determined that the fresh concrete dilated during the application of the deviator load and that negative pore water pressures developed in the samples even though the cell pressure was set equal to zero. Consequently, it became impossible to perform an unconfined compression test for the purpose of getting a better estimation for the cohesion of fresh concrete as defined by the Mohr-Coulomb failure theory. To overcome this, a series of drained unconfined compression tests were performed on unsupported cylindrical samples of fresh concrete° The samples were formed by placing concrete in a plexiglass cylinder 82 m~ in diameter and 178 m high° The concrete was placed in four layers and each layer received fifty tamps as was done for the triaxial tests. The concrete was then allowed to set for the appropriate time period after which the cylinder was carefully slipped off and the sample compressed to failure at a strain rate of 0.14 --- per minute. To measure the lateral strain, a strip of wet paper marked off I n . - . w a s placed around the midsection of the sampleo For the unconfined compression tests the measurements stress, axial strain and lateral strain.
taken were axial
332
Vol. 11, No. 3 A. A l e x a n d r i d i s , N.H. Gardner
FIG. 3A.
Stress difference versus axial strain for 21°C and 160 mln. SET TIME.
FIG. 3B.
Pore pressure versus axial strain for 21°C and 160 min. SET TIME.
*o
6o
to
~o
o
AX',AJ. ST~AL~ FLG
3A
STRF.55 01FFER[NCF. V[RSU$ AXIAL. STRALN FOR ZI"C AND 1 6 0 ~ n , ~ T TIMF.
8O
•
QO
40
lo
-ZO
~
O; • 0
kPo
-40
FIG. 3 Typical Results of Triaxial Tests for Concrete at 21°C and 160 min. Set Time. K o Tests. Values of K o for fresh concrete were obtained by applying vertical stresses through a piston in series with a loading ring onto fresh concrete samples 81.8 mm in diameter and 177.8 mm high. These samples were restrained from moving laterally by means of a plexiglass cylinder casing whose diameter was 1.5 mm larger than that of the piston° Water was allowed to drain out of the concrete during the application of vertical stresses° A circular sponge was placed between the piston and the concrete to prevent the loss of the finer sand and cement particles. The lateral strains and the corresponding stresses as a result of vertical loading were measured directly by means of two strain gauges on either side of the cylinder attached at its midheight. The vertical load was applied in successive increments of 55°6 N and the corresponding strain gauge readings were taken only after the excess pore pressures had dissipated and the load dial gauge had stabilizedo In Figure 4, K o has been plotted as a function of the effective vertical stress and for set times ranging from 20 minutes to 160 minutes, The plots have been done for temperatures of 4°C and 21°Co
Vol. I I , No. 3
333
SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE 240 ~0~ 21"("
M,~
/'
#
zl°c 4OM~ 21~ I
I
200
/\ °
FIG. 4 K o versus Effective Vertical Stress.
120
~
so
I
I
I'\
iS0
~
"
,,'
/#
It;
,
t\
,
i
",,i
\t,'k
"X,l',,t,
40 4"C
,s 4~
0
I Oi
oil
k----
120klm
I 03
40~n 4"C
I 04
I 05
0i6
i , O#
K.
Analysis of Experimental Data In general in particle mechanics there does not exist an equivalent of Hookes law in which stress is related to deformation° Soil mechanics practice has concerned itself wlth the failure properties of the system regardless of deformation° However, Rowe's recent theory does allow the effective stresses on the sample to be related to the deformation of the sample. Consequently in this section the failure behaviour of the fresh concrete will be compared to both the Mohr-Coulomb and Rowes theories and in addition the Rowe theory will be compared to the pre-failure behaviour. Failure Behavlour Choice of Failure Criteria In the triaxial testing of soils it is generally accepted that a test sample has failed when the maximum stress difference a I - o~ has been reached. For highly plastic materials, this condition of failure is sometimes not achieved until the test sample has undergone large axial strain. For this reason, failure has also come to be arbitrarily defined as the instant a test sample achieves 20% axial strain. For fresh concrete in formwork, it may be inappropriate to use 20% axial strain as a failure criterion because such large strains should never be encountered in real practice. It was therefore decided to adopt two failure criteria for analyzing the test results: (i) (2)
.The maximum stress difference or 20% axial strain° The maximum stress difference or i0% axial strain.
Mohr-Coulomb Analysis Calculation of the Mohr-Coulomb parameters C' - #' was performed by plotting the t w o failure criteria for each temperature on a p'-q' diagram oi - o t oi - o (p , = --~----, q , . --~----'). Figure 5 shows such a plot for 21 o Co The four lines corresponding to ~et times of 40, 80, 120 and 160 minutes, represents a linear regression of the failure stresses for cell pressures of O, 35, 70, i05, 140 kPao For such a plot the slope a and the intercept A' on the q' axis can be related to the Mohr-Coulomb parameters
334
Vol. I I , No, 3
A. A]exandridis, N.J. Gardner I
250~ •
200 •
160 M,n 4 0 M,n
~,
q20M,7
\
, FIG. 5 p'-q' Plot for 21°C and 10% Failure Strain.
"61 "o I00
50
0 bt"
I
t
I
I
50
~O
1SO
200
p' • Cr'~'O'i 2
•
1
1
250
300
WPo
sln¢' = tan~ = q'/p' and A' C !
The lines were regressed the friction angles.
=
C0S¢ v
through the origin to get an initial estimate of
Using these initial estimates of the friction angles and the results of the unconfined compression tests the effective cohesions were calculated C' = ~qu tan (45 - ~) Hence values were calculated for the A' intercepts on the p'-q' plots. The p'-q' lines were then linearly regressed through the appropriate q' intercepts to give a refined value for the friction angles. The results of the Mohr-Coulomb analysis have been s u ~ r l z e d in Table 4. Rowe's Analysis Figure 6 represents a plot t ! failure criterion on ol/v - o 3! figure for set times or 40, 80. linearly regressed through five
of the trlaxlal test results at 21°C for the 10% planes. Four lines have been drawn in each 120 and 160 minutes. Each llne has been points corresponding to 0, 35, 30, 105 and
~u
140 kPa cell pressures. The slopes represent K I - 2tan2(45 +~2--) in Rowe's equation corresponding to the set times for which the lines have been drawn. The o~/v, intercepts represent K 2 = 8C' tan(45 +-2~). Knowing the values of K I an~ K 2, the friction angle ¢~ and the cohesion C~ corresponding to Rowe's t~eory can be evaluated° Initlally it w a s a s s u m e d that the cohesion was negligibly small and the lines were made to pass through the origin of the coordinate system. From the slopes~ initial est~-mtes of ¢~ were made from which the cohesion C~ and subsequently K 2 were estimated For 21°C. The lines were then linearly regressed a second time through their appropriate ol/v' intercepts to obtain more accurate values of ¢~ at 21°Co At 4eCp accurate values of the instantaneous Polsson's ratio uv could not be obtained because
Vol. I I , No. 3
335 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESHCONCRETE TABLE 4 Results of Mohr-Coulomb Analysis
Temperature 21°C
40C
S~ (a)
Set Time
Effective Friction Angle¢'
S;
40 mln
41°C
0.4 °
1.90 kPa
80
39 °
0.6 °
4.69
120 160
39 ° 41 °
1.2 ° 1.6 °
5.74 6.30
40
40 °
0.4 °
1.47
80
39 °
006 °
2.87
120 160
38 °
107 ° 0.9 °
1.68 2.38
37 °
*
Cohesion C'
standard deviation Failure Criterion; Maximum Stress Difference or 10% Axial Strain
Temperature
21°C
4°
Set Time
Effective Friction Angle¢'
* S;
Cohesion C'
40 min 80 120 160
38 ° 36 ° 36 ° 39 °
0.9 ° 1.7 ° io4 ° 2o7 °
lo61 kPa 4006 4097 5o67
40 80 120 160
38 ° 36 ° 35 ° 34 °
i.i ° 1o5 ° i.i 0.5 °
1.96 2.59 1.54 2.24
S; - standard deviation (b)
Failure Criterion; Maximum Stress or 20% Axial Strain
500
f
4O0
-5i~,300
~ ' 2 o . .
O' 1
__--r V.S.
at 21°C.
FIG. 6 03
F
200
-\~
o- eo.,.
A- 40Mm
for 10% Failure Strain I00
0
60
0
wOO
120
0"~" , iPo FIG. 6
U;O" VS
o','
FOR I 0 % FAILURE: STRAIN AT 21"C
336
Vol. I I , No. 3 A. Alexandridis, N.J. Gardner
FIG. 7 250
!
Variation of °--~l V.S.~, Concrete.
o~ for
o~ = O, Set Time
40 mln. at 21°C. 200
150 i 30
I 4o (:T]'
,
J 5o
kPo
of the highly plastic nature of the concrete specimens. Consequently, K 2 and correspondingly C~ could not be estimated. Therefore, a second linear regression was not possible at 4@C. However, because the difference in ~ at 21°C between a linear regression through the origin and a linear regression through K 2 is in the order of l ° to 2 °, it was felt that the first estimates of ~ at 4°C adequately represented the true angles of internal friction° The results of Rowe's analysis have been summarized in Table 5. Pre-failure Behaviour Using Rowe's theory it should be possible to plot all the results of tests at one set time/temperature combination as a single straight llne. The problem experimentally is in obtaining values of the instantaneous Poisson's Ratio to any reasonable degree of accuracy at low strains. Consequently this part of the investigation was not wholly successful° Figure 7 shows the results for the specimens tested at a set time of 40 mins. at a temperature of 21°Co It can be seen that the experimental results do approximate to a straight line but it would be unrealistic to make extensive conclusions due to the lack of confidence in the basic data concerning lateral strain° Discussion Figures 5, 6 and 7 indicate that fresh concrete exhibits shear strength characteristics in accordance with both Coulomb's equation ~' = C' + o~ tan~' oI and Rowe's equatlon ~-r = KlO ~ + K 2 where K I and K 2 are functions of the effective friction angle and the effective cohesion° However, caution must be exercised when one interprets this statement with respect to Coulomb's theory. The decrease in pore pressure with axial strain as shown in Figure 3B resulting from an applied deviator stress, implies that the concrete mix used in this investigation behaved in a dilatant fashion when being sheared~ The test samples therefore, could not be distorted without an increase in the distance between the particles unless the individual particles themselves were to break. Because of the dilatancy, only a portion of the work done by the applied stresses went into shearing the test samples. The remainder of the work went into rearranging the particles during dilatancy. Coulomb's strength parameters C' and ~' therefore, do not strictly represent the cohesion and the
Vol. I I , No. 3
337 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE TABLE 5 Results of Rowe's Analysis
Temperature
21°C
4°C
S~ a)
Cohesion
18 ° 18 °
2o0 °
20 °
2.3 °
19 °
2.8 °
0.15 kPa 0.53 0.40 1.57
3.2 ° 3.0 ° 6o6 ° 2.8 °
-
Effective Friction Angle @~v
40 min 80 120 160 40 80 120 160
22 ° 22 ° 23 ° 20 °
1.8 °
Ct
- standard deviation of ¢~ Failure Criterion: Maximum Stress Difference or i0% Axial Strain
Temperature
21°C
4°
S'
S'* @u
Set Time
Effective Friction Angle ~ ,
S¢~
C'
19 ° 21 ° 18 ° 21 °
2.4 ° 7.5 ° 4.3 ° 5.7 °
0.15 kPa 0.53 1.40 1.57
40
20 °
200 °
80
20 °
307 °
120 160
21 ° 19 °
1.7 °
-
Set Time
40 min 80 120 160
,*
2.5 °
Cohesion
!
- standard deviation of @u
(b~ ~ Failure Criterion: Maximum Stress Difference or 20% Axial Strain angle of internal friction respectively in the full sense of their definitions. Instead, the values assigned to C' and 4' are such as to make Coulomb's straight llne fit the best results° Dilatancy however, poses no problem with respect to Rowe's theory where Rowe's shear strength equation was derived for a particulate system as opposed to a continuum whose volume remains constant. The strength parameters C~ and ~ correspond directly to the shearing resistance offered by cohesion and internal friction respectively° From Table 4 Coulomb's friction angle for the concrete mix lles in the range of 37 ° to 41 ° for a 10% failure strain and 34 ° to 39 ° for a 20% failure strain° These values correspond to a dense arrangement of particles and are supported by the fact that concrete aggregates tend to be closely packed and angular, achieving a high degree of particle interlock. In Table 5, Rowe's friction angle is seen to lie in the range of 18 ° to 21 ° for a 10% failure strain and 19 ° to 23 ° for a 20% failure strain° In the verification of his theory, Rowe evaluated the friction angles for clean sands and silts and found them to lle between 23 ° and 30°; the lower friction angles corresponding to the coarser material° Since concrete is composed of aggregates in the coarse sand and gravel sizes, the friction angles obtained in
338
Vol. I I , A. A l e x a n d r i d i s ,
No. 3
N.J. Gardner
this investigation are consistant with those in Rowe's investigationo It may be seen that Rowe's friction angles are approximately half the value of Coulomb's friction angles. This supports the idea that Coulomb's friction angles not only include the effects of frictional resistance against shearing but also to resistance in expanding against the principle stresses acting on the test sample~ No general trend is evident for the observed changes in the friction angles for either an increasing set time or for temperature change. The reason is that the component of shear resistance resulting from the translation and frictional resistance between aggregate and cement particles which is inincluded in the angle of internal friction, depends on surface roughness and the load per particle° Since these are independent of temperature change and set time, so should be the angle of internal friction. Since the angle of internal friction is an inherent property of a concrete mix and remains essentially constant with time and temperature, it is the increase in cohesion as a result of the hydration process which allows fresh concrete to develop significant shear strength with time and solidify. This increase in cohesion with time is clearly depicted in Tables 4 and 5 for both Coulomb's theory and Rowe's theory at 4°C and 21°C0 Since hydration is a chemical reaction, the rate of increase in cohesion with time is seen to be significantly lower at 4°C as compared to 21°Co The lateral pressure corresponding to the "at rest" state may be evaluated when the coefficient K o for fresh concrete is known= The variation of K o for the given mix with set time, effective vertical stress and temperature can be seen in Figure 4° It may be seen that K o is variable and highly dependent on set time at 21°C and to a lesser degree at 4°C. Initially, fresh concrete being in a highly plastic state behaves much like a liquid. Any vertical stresses are correspondingly transformed into lateral stresses° Consequently, K o approaches unity° As the fresh concrete hydrates, lateral strains as a result of vertical stress become restricted and approach those of Poisson's ratio for cured concrete. The value of K o therefore decreases signficantly with set time. This decrease is of a smaller order at the colder temperature because of the hydration process being retarded. Conclusions (I) The shear strength properties of fresh concrete may be expressed by either the Mohr-Coulomb or Rowe's theory of shear strength. A/though Coulomb's theory is fundamentally incorrect, it is able to adequately express the results of shear tests performed on fresh concrete. (2) The shear strength of fresh concrete is both time and temperature dependent. (3) Immediately following mixing, the shear strength of fresh concrete is due mainly to the internal friction resulting from particle interaction which remains constant with set time and temperature changeo (4) The hardening of concrete with set time is due mainly to the development of cohesion as a result of the hydration process° At lower temperatures, hydration is retarded and cohesion develops at a slower rate causing the fresh concrete to behave fluid like for a longer period of time° (5) The coefficient K o for fresh concrete is highly variable and depends on set time, temperature and to a lesser degree on the applied vertical stress. Its value is lower at warm temperatures and decreases with set time~ Acknowledgements The authors wish to acknowledge the financial support of the National Sciences and Engineering Research Council of Canada under Grant A5646.
VO]. I I , No. 3
339 SHEAR STRENGTH, TRIAXIAL COMPRESSION, FRESH CONCRETE References
io
Ao Alexandridis, "Mechanical Properties of Fresh Concrete", MoA°Sco Thesis, University of Ottawa, 1979, ppo158o
2°
R. L'Hermlte, "Vibration and the Rheology of Freshly Mixed Concrete", Revue des Materiaux de Construction, No° 405, 1949, pp.179-87o
3o
R.Ho Olsen, "Lateral Pressure of Concrete on Formwork", PhoD. Thesis, Oklahoma State University, 1968, pp.122°
4.
AoG.B° Ritchie, "The Trlaxial Testing of Fresh Concrete", Magazine of Concrete Research, Vol. 14, No° 40, 1962o
5.
P.W. Rowe, "The Stress-Dilatancy Relation for Static Equilibrium of an Assembly of Particles in Contact", Proceedings of the Royal Society Series A, Volo 269, 1962, ppo500-527°