4.1. Distribution of Aggregates across the Beam Cross-Section
Test beam specimens of test series A and C were cut into four parts, each 250 mm long, after completion of the test and the distribution of aggregate particles were examined. The cutting plane was divided into four equal sub-areas measuring 37.5 mm in height and 95 mm in width, and the aggregate concentration was measured on each sub-area.
Figure 12a,b show the average aggregate concentrations for the four sub-areas of the cutting planes and the locations of the centroids of aggregates for test series A and C, respectively. The calculated aggregate concentrations are listed in
Table 5 where
represents the location of the centroid of the aggregate areas in the cutting plane and A-1, A-2, A-3, and C-1, C-2, C-3 represent the cutting planes for test series A and C, respectively. The location of centroid
was calculated by taking the first moment of each aggregate area on each cutting plane. It is observed from
Figure 12 and
Table 5 that the aggregate concentration is the lowest at the top sub-area and increases toward the bottom, and the centroid of the aggregate areas on the cutting plane is below the centroid of the test beam, at 46.1% of the beam depth from the bottom surface. This implies that the variation in aggregate concentration within the beam depth came about during placement of the fresh concrete and caused the variation of shrinkage strain within the beam depth.
Figure 12.
Average percentage of aggregate areas in each sub-area: (a) Test series A; (b) Test series C.
Figure 12.
Average percentage of aggregate areas in each sub-area: (a) Test series A; (b) Test series C.
Table 5.
Aggregate concentration for each sub-area.
Table 5.
Aggregate concentration for each sub-area.
Sub-Area Location | Percentage Ratio (%) |
---|
Test Specimen in Test Series A | Test Specimen in Test Series C |
---|
A-1 | A-2 | A-3 | Average | C-1 | C-2 | C-3 | Average |
---|
Top | 21.5 | 21.3 | 18.9 | 20.8 | 21.8 | 19.8 | 17.4 | 19.6 |
Top-mid | 26.1 | 23.2 | 24.0 | 23.6 | 21.1 | 27 | 30.3 | 26.1 |
Bot-mid | 34.6 | 26.8 | 24.4 | 27.3 | 26.6 | 21.8 | 27.8 | 25.4 |
Bottom | 33.1 | 28.6 | 32.8 | 28.2 | 30.5 | 31.4 | 27.8 | 29.9 |
| 47.8 | 46.8 | 44.8 | 46.4 | 46 | 46.3 | 45.1 | 45.8 |
4.2. Evaluation of Shrinkage Variation by Hobbs’s Equation
Shrinkage strains within the beam depth of the test specimens were predicted by Hobbs’s equation [
10], which takes into account the effects of aggregate volume concentration on shrinkage strain. The uniform parts of the shrinkage strains shown in
Figure 11a were used as the reference shrinkage strain for test series A and C. For test series B and D, two sub-areas, above and below the centroid of the test beam were considered, and 5% of the aggregate concentration was subtracted and added, respectively, to the design aggregate concentration of the mix because aggregate concentrations on the cutting plane were not measured for these series.
Hobbs’s equation is expressed as:
where
,
, and
are shrinkage strain, shrinkage of the cement paste fraction, and aggregate volume concentration. The shrinkage of the cement paste fraction
in Equation (2) was calculated by substituting the values of
and
into Equation (2), where the uniform part shrinkage strain was used for
and the aggregate volume fraction of the mix design of
Table 1 was used for
. Thus, this calculation approximates the shrinkage character of the test beam using the uniform part of the shrinkage strain. Aggregate volume concentrations
for the four sub-areas of test series A, B, C, and D were calculated by computing the aggregate volumes corresponding to the aggregate concentration in
Table 5 from the total aggregate volumes determined in the mix designs. The values for the aggregate volumes and aggregate volume concentrations for each sub-area are listed in
Table 6 where
is the aggregate volume in a sub-area. For test series B and D, the aggregate volumes
were not available (see
Table 6) because the aggregate concentration was not measured. In these cases, a ±5% fluctuation was imposed on the aggregate volume concentration
.
Table 6.
Aggregate volume concentration for four test series A, B, C, and D.
Table 6.
Aggregate volume concentration for four test series A, B, C, and D.
Location | Test Series A | Test Series B | Test Series C | Test Series D |
---|
| | | | | | | |
---|
Top | 6340 | 0.69 | – | 0.70 | 5950 | 0.68 | – | 0.69 |
Top-mid | 7190 | 0.73 | – | 7620 | 0.75 | – |
Bot-mid | 8320 | 0.78 | – | 0.80 | 7710 | 0.76 | – | 0.79 |
Bottom | 8590 | 0.79 | – | 9070 | 0.81 | – |
Average | – | 0.75 | – | 0.75 | – | 0.75 | – | 0.74 |
Figure 13 compares the measured shrinkage strains at the three strain gage locations, the predicted shrinkage strains based on Hobbs’s equation for the sub-areas within the beam depth, and the uniform shrinkage strains for the four test series. For series A, B, and D, the predicted shrinkage strains at the top and bottom sub-areas are slightly smaller and larger than the measured shrinkage strains at the top and bottom surfaces of test specimens, respectively. In contrast, for series C the predicted shrinkage strains at the top and bottom sub-areas are larger and smaller than the measured shrinkage strains at the top and bottom surfaces of test beam specimens, respectively.
Figure 13.
Comparison of measured and predicted shrinkage strains within beam depth for test series A, B, C, and D (test-based): (a) Test series A; (b) test series B; (c) test series C; (d) test series D.
Figure 13.
Comparison of measured and predicted shrinkage strains within beam depth for test series A, B, C, and D (test-based): (a) Test series A; (b) test series B; (c) test series C; (d) test series D.
The shrinkage strains were calculated by the ACI 209 model with consideration of the mix designs and the curing conditions of the test series, to generalize the predicting process for shrinkage strains within the beam depth. Two sub-areas, above and below the centroid of the test beam, were considered, and 5% of aggregate volume concentration was subtracted and added, respectively, to the design aggregate volume concentration of the mix. The shrinkage strains accounting for the change of aggregate concentration within the beam depth were calculated by Hobbs’s equation.
Figure 14a–d compares the measured shrinkage strains at the top and bottom surfaces of test specimens with the predicted shrinkage strains at the two sub-areas for the four test series of A, B, C, and D, respectively. The mismatch between the measured and predicted shrinkage strains is primarily due to the difference between the measured uniform shrinkage strains and the shrinkage strains by the ACI model.
Figure 13 and
Figure 14 demonstrate that the variation of shrinkage strain within the beam depth is caused by the variation of the aggregate volume concentration within the beam depth, and the variation of shrinkage strain within the beam depth can be estimated by accounting for the change of aggregate volume concentration in Hobbs’s equation.
Figure 14.
Comparison of measured and predicted shrinkage strains within beam depth for test series A, B, C, and D (ACI-based): (a) Test series A; (b) test series B; (c) test series C; (d) test series D.
Figure 14.
Comparison of measured and predicted shrinkage strains within beam depth for test series A, B, C, and D (ACI-based): (a) Test series A; (b) test series B; (c) test series C; (d) test series D.