Soils and Foundations 2013;53(2):349–356 The Japanese Geotechnical Society
Soils and Foundations www.sciencedirect.com journal homepage: www.elsevier.com/locate/sandf
Influence of model footing diameter and embedded depth on particle size effect in centrifugal bearing capacity tests Y. Toyosawaa, K. Itoha, N. Kikkawaa, J.-J. Yangb,n, F. Liuc a Construction Safety Research Group, National Institute of Industrial Safety and Health, 1-4-6, Kiyose, Tokyo 204-0024, Japan Key Laboratory of Marine Environment and Ecology, Ocean University of China, Ministry of Education, Qingdao 266100, PR China c Beijing Municipal Engineering Research Institute, Beijing 100037, PR China
b
Received 5 July 2010; received in revised form 12 October 2012; accepted 8 November 2012 Available online 11 March 2013
Abstract The influence of the model footing diameter and embedded depth on the bearing capacity of circular shallow footings was studied by centrifugal model testing in order to determine a model footing size and embedded depth against particle size in a model ground. In the series of 37 tests, the ground was made by river sand whose particle size was adjusted by sieving to a mean particle size of 0.6 mm. The diameter of the model footing and the embedded depth were considered as influential parameters in this study. The diameter of the model footings varied from 5 to 40 mm and the ratio of the footing diameter to the mean particle size was calculated as 8.3–66.7. The ratio of the embedded depth to the footing diameter was 0, 0.5 and 1.0. As a result, the bearing capacity in the same equivalent diameter of footing was not dependent on the diameter of model footing when the ratio of footing diameter to particle size is more than 50 with any ratio of embedded depth to footing diameter. Our results that the proposed relationship between the ratio of footing diameter to the particle size and the ratio of the embedded depth to the footing diameter can be used to choose a reasonable model footing diameter, embedded depth and the particle size of ground material for centrifugal model tests. & 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Centrifugal model test; Particle size effect; Bearing capacity; Footing diameter; Embedded depth; IGC: E-3/E-14
1. Introduction In centrifugal model tests, the footing and soil particles should be modeled in size by a factor of N under N–g centrifugal acceleration. However, if the size of soil particles are reduced by a factor of N, the model soil will have very different stress–strain characteristics compared with the n
Corresponding author. Tel./fax: þ86 0532 667 81773. E-mail addresses:
[email protected] (Y. Toyosawa),
[email protected] (K. Itoh),
[email protected] (N. Kikkawa),
[email protected] (J.-J. Yang),
[email protected] (F. Liu). Peer review under responsibility of The Japanese Geotechnical Society.
prototype soil. Therefore, often the same soil as in prototype is used, and only the model footings are made smaller by a factor of N (Kusakabe, 1993; Okamura et al., 2004). In such cases, the particle size of soil used will be significant compared with the dimensions of the model footing. That is, the effect of particle size in centrifugal bearing capacity tests raises doubts as to the reliability of such tests. Therefore, it is necessary to investigate how particle size affects the bearing capacity of the centrifugal model tests in terms of various model footing diameter, Dm, and different embedded depth of footing, dm. The effect of particle size on bearing capacity is due to the shear band thickness (Okamura et al., 2004; Tatsuoka et al., 1992, 1991, 1997; Siddiquee et al., 1992), since the shear band thickness is proportional to the particle size, D50. When the particle size is sufficiently small compared with the model
0038-0806 & 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sandf.2012.11.027
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footing size, the bearing capacity in a centrifugal model ground should be the same as that in the prototype ground. However, when the particle size is not so small in comparison with the model footing size, the bearing capacity in a centrifugal model ground can be affected by the shear band formation. This is called the ‘‘particle size effect’’. In addition, if the model ground is loose, the shear band cannot be observed clearly. The footing shape, i.e. whether it is rectangular or circular, also affects the bearing capacity since rectangular footings are usually used in plane-strain problems, with the effect that the movement of soil particles is constrained in 2 dimensional directions and the shear band formation is more affected than when circular footings are used. The embedded depth is also an important factor for the bearing capacity of shallow footing, since the shear bands do not generate large confining pressure. Thus, the particle size effect is less obvious with increasing embedded depth. Generally, the particle size effects are obvious under conditions in which a shear band can easily be generated in the ground (Tatsuoka et al., 1997) and the ratio of the footing size to particle size is small (Okamura et al., 2004). Therefore, the ratio of the footing size to particle size, the embedded depth of footing, the density of sandy model ground and the footing shape are considered to be factors that influence the bearing capacity of the model ground in the centrifugal model tests. Gemperline (1988) and Okamura et al. (2004) reported that the particle size effects became less obvious as the density of sand decreased. Okamura et al. (1993) showed that particle size has a smaller effect on circular footings than on strip footings. However, previous research only qualitatively described this phenomenon and did not
quantitatively study the relationship between the influenceable factors and the particle size effects. In the research reported in the literature on circular footings, by Ovesen (1975, 1979) and Xu and Zhang (1996), the particle size effect was evaluated by merely comparing the loadsettlement curves from a few experimental cases under conditions of no embedment. In this research, the particle size effect has been investigated using quantitative indicators for 37 cases of various diameters and depth of embedment. Gemperline and Hon (1988), Kimura et al. (1985), Pu and Hao (1988), Pu and Ko (1988), and Yamaguchi et al. (1976) researched the influence of embedment on bearing capacity by centrifugal model tests. Liu et al. (2007) carried out bearing capacity tests using a centrifuge for several densities of sand, footing shapes and embedment while the particle size effects on bearing capacity were studied. However, few studies have been done on the influence of embedment on the bearing capacity with the particle size effect. Yang et al. (2007) studied the particle size effects in the case of no embedment and introduced the index which was used to quantify the particle size effects for rectangular footing. In this paper, a series of 37 bearing capacity tests with circular footing was conducted on dense sand by centrifuge and the effects of model footing diameter and embedded depth of footing on the bearing capacity are discussed in detail using quantitative indexes (such as Dqu, Dm/D50, dm/Dm, etc.). The relationship of both ratios well expressed the extent and degree in the bearing capacity.
100
Percent finer by weight (%)
90 80 70 60 50 40 30 20 10 0 0.01
0.1
1
10
Grain (mm)
Fig. 1. Test container (diameter: 500 mm and depth: 300 mm).
Fig. 2. Curve of grain size distribution of river sand after sieving. Table 1 Properties of the river sand after sieving. Soil particle Maximum dry density, rdmax density, rs (g/cm3) (g/cm3)
Minimum dry Water Mean particle density, rdmin content, w size, D50 (g/cm3) (%) (mm)
2.692
1.296
1.585
0.13
0.6
Table 2 Properties of sandy ground in the gravity field. Relative density, Dr (%)
Dry density, rd (g/cm3)
Void ratio, e
98–99
1.578–1.582
0.702–0.706
Y. Toyosawa et al. / Soils and Foundations 53 (2013) 349–356
2. Design of tests The model footing tests were conducted in the JNIOSH NIIS Mark-II Centrifuge at the National Institute of Occupational Safety and Health, Japan, which was described in detail by Yang et al. (2007). The rigid cylinder soil container had an inner diameter of 500 mm and a depth of 300 mm as shown in Fig. 1. River sand from Japan was used. On delivery, the material was washed, dried, and made free of silt, clay and organic matter. The grading chosen contained only those particles that passed through a 2.0-mm opening sieve and were retained on a 0.25-mm opening sieve. The sand was spread on the clean floor and dried naturally in the room for several days. The water content of the sand was confirmed constant throughout the series of the tests. The physical properties and grain size distribution were as shown in Table 1 and Fig. 2. Strength parameters of this sand, under almost same condition of the centrifuge tests, obtained from direct shear tests are f0 ¼ 371 and c0 ¼ 2 kPa. The depth of the model ground was 240 mm and was divided into 12 sublayers, each of which weighed 6.1 kg. Hand tamping was used to densify the sand. For each sublayer, predetermined quantities of sand were spread, trimmed and tamped by hand to produce a final sublayer thickness of about 20 mm. The operation was repeated and the ground was made. The properties of the sandy ground in the gravity field are given in Table 2. Under centrifugal acceleration, the sandy ground was densified, but previous researchers showed that
351
the increase of the density was insignificant, so its effect on the test results was neglected if the relative density was high and close to the maximum density (Ueno, 2001; Yang and Toyosawa, 2003; Toyosawa et al., 2004). To understand the
Fig. 4. Model circular footings.
Table 3 Test programs. Model footing diameter, Dm (mm)
Embedded depth, dm/Dm
Centrifugal acceleration, N (g)
Equivalent footing diameter, NDm (mm)
5 10 20 30 40
0/0.5/1.0 0/0.5/1.0 0/0.5/1.0 0/0.5/1.0 0
40/60 20/50 10/25/50 6.7/16.7/33.3/50 5/12.5/25/37.5
200/300 200/500 200/500/1000 200/500/1000/1500 200/500/1000/1500
footing diameter Dm footing
embedded depth
dm model ground centrifugal acceleration
Ng
Fig. 3. Model sandy ground and the loading system.
Fig. 5. Symbols of the footing diameter, embedded depth and centrifugal acceleration.
Y. Toyosawa et al. / Soils and Foundations 53 (2013) 349–356
the loading jack through a load cell as shown in Fig. 3. The self-weight of the model footing and the loading jack was taken into account when processing the test data. As shown in Fig. 4, aluminous (duralumin) circular footings were used in the tests, and were of diameter Dm ¼ 5 mm, 10 mm, 20 mm, 30 mm and 40 mm. In the model, because the diameter of the container was 12.5 times or more that of the footing and the depth of the ground was 6 times or more
900 800 D =40mm m Dm=10mm 700 600 Dm=20mm 500 400 300 Dm=30mm 200 100 0 0 0.1 0.2 0.3 0.4
3000
1600
Dm=20mm
Dm=10mm 1200
q (kPa)
Dm=20mm Dm=30mm
800
Dm=40mm
0
0.2
0.6
Dm=10mm Dm=20mm
0.4
Dm=10mm
2100
5000
Dm=30mm
1400
0
0.6
0
0.2
0.4
500 0
0
0.2
0.4
Dm=20mm
2000
0.2
Sm/Dm
7000
0.6
Dm=20mm
Dm=30mm
3000
Dm=20mm
q (kPa)
5000
4000
4000 3000
2000
2000
1000
1000
0
0.8
6000
Dm=10mm
5000
q (kPa)
q (kPa)
1000
0.6
3000
0
6000
Dm=30mm
0.4
Dm=30mm
0
0.6
2500 Dm=10mm Dm=20mm
0.6
4000
Sm/Dm
Dm=5mm
0.4
1000
Sm/Dm
1500
0.2
Sm/Dm
700
Dm=30mm
2000
0
6000
3500
q (kPa)
q (kPa)
0.4
0
2800 Dm=20mm
0.2
1000
Sm/Dm
Dm=5mm
0
Dm=40mm
Dm=30mm 0
Sm/Dm 2000 1800 1600 1400 1200 1000 800 600 400 200 0
2000
400
q (kPa)
q (kPa)
particle size effect, a high relative density condition, which is strongly dilative, was adopted for these experiments. In cases where the embedded depth was not zero, after the ground was completed, the loading system was mounted onto the test container and the footing was adjusted to just touch the ground. Then, predetermined quantities of sand were weighed, spread on the ground and tamped by the same method described above. The footing was rigidly connected to
q (kPa)
352
0
0.2
0.4
0.6
0.8
Sm/Dm
Sm/Dm
0
Dm=30mm
0
0.2
0.4
0.6
Sm/Dm
4000 Dm=40mm
q (kPa)
3000 Dm=30mm
2000 1000 0
0
0.1
0.2
0.3
0.4
Sm/Dm Fig. 6. Load–settlement curves for NDm ¼ 200 mm, 500 mm, 1000 mm and 1500 mm. (a) NDm ¼200 mm, dm/Dm ¼0 (b) NDm ¼ 200 mm, dm/Dm ¼ 0.5 (c) NDm ¼ 200 mm, dm/Dm ¼1.0 (d) NDm ¼ 500 mm, dm/Dm ¼0 (e) NDm ¼ 500 mm, dm/Dm ¼ 0.5 (f) NDm ¼500 mm, dm/Dm ¼1.0 (g) NDm ¼ 1000 mm, dm/Dm ¼ 0 (h) NDm ¼ 1000 mm, dm/Dm ¼ 0.5 (i) NDm ¼ 1000 mm, dm/Dm ¼ 1.0 and (j) NDm ¼ 1500 mm, dm/Dm ¼ 0.
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Fig. 6 shows the relationship between the bearing pressure and relative displacement (footing settlement Sm/footing width Dm) for NDm ¼ 200 mm, 500 mm, 1000 mm and 1500 mm. The ratio of the embedded depth to the footing width dm/Dm was 0, 0.5 and 1.0. Although the load–settlement curves were expected to be constant independent of the variation of NDm, the curves varied for different Dm. The difference between the load– settlement curves decreased with the increase of footing diameter, Dm. As shown in Fig. 6(a), (d), (g) and (j), the curves of Dm ¼ 30 mm and 40 mm were almost the same and were greatly different from the curves of Dm ¼ 10 mm and 20 mm for dm/Dm ¼ 0. However, for dm/Dm ¼ 0.5 and 1.0, the curves of Dm ¼ 20 mm and 30 mm were almost identical and differed significantly from the curves of Dm ¼ 5 mm and 10 mm, as shown in Fig. 6(b), (c), (e), (f), (h) and (i). On the other hand, the peak of the load–settlement curve was observed in the case of the small embedment (dm/Dm ¼ 0 and 0.5) for the footings with the diameter of 200 mm and 500 mm in prototype. The clear heaving around the circular footing observed after the tests, indicated that general shear failure occurred. On the other hand, when the footing diameter was increased to 1000 mm and 1500 mm in prototype, the peak was not clearly recognized; the mode of failure changed from general shear failure to local shear failure. In the case of dm/Dm ¼ 1.0, however, no peak in the load– settlement curve was recognizable for any footing diameter except for the 200 mm in prototype; the mode of failure was local shear failure or punching shear failure. As mentioned above, for the same ratio of footing diameter to particle size, the particle size effects were more marked when the ground was able to easily generate a shear band. It is considered that as the embedment, dm, or equivalent footing diameter, NDm, increased, local shear failure (progressive failure) became more obvious and the general shear failure was hard to form, and the particle size effects became less obvious as a result.
3000 Dm =30mm Dm =20mm
2500 qu (kPa)
3.1. Influence of particle size effects on load-settlement curves
When the peak of the load–settlement curve could be observed, the peak was defined as the ultimate bearing capacity, qu. In other cases, a hyperbolic function was used to fit the load–settlement curve, and the intersection point of the initial tangent and the asymptote was defined as qu (Yang, 1994). Fig. 7 presents the relationship between the ultimate bearing capacity, qu, and the equivalent footing diameter, NDm. The ultimate bearing capacity, qu, increased with increasing in equivalent footing diameter, NDm, and decreased with increasing model footing diameter, Dm. In the case of dm/Dm ¼ 0 (Fig. 6(a)), the ultimate bearing capacity of Dm ¼ 30 mm and Dm ¼ 40 mm was almost the same, namely, when the footing diameter, Dm, was larger than 30 mm, Dm/D50 was more than 50, the effects of Dm/D50 on the centrifuge test results could be neglected. In the cases of dm/Dm ¼ 0.5 and 1.0, when the footing
2000 Dm =10mm
1500
Dm =40mm
1000 500 0
0
500
1000 1500 NDm (mm)
2000
5000
qu (kPa)
3. Discussion on test results
3.2. Influence of particle size effects on ultimate bearing capacity
4000
Dm =20mm
3000
Dm =10mm Dm =30mm Dm =5mm
2000 1000 0
0
500 1000 NDm (mm)
1500
6000 Dm =20mm
5000 Dm =10mm qu (kPa)
the maximum footing diameter, it was considered that there was no boundary effect (Yang and Toyosawa, 2003; Meyerhof, 1948). In order to increase the roughness, sandpaper was glued to the footing base and the side surface was made harsh. The test programs are shown in Table 3. In this study, the model footing diameter, Dm, the embedded depth of footing, dm, and the centrifugal acceleration, N, were varied in order to investigate the particle size effects. These symbols are summarized in Fig. 5. The model footing was placed at the center of the model ground surface and was loaded vertically at a constant settlement rate of 1% footing-diameter/min after the acceleration reached the target.
353
4000 3000
Dm =5mm
Dm =30mm
2000 1000 0
0
500 1000 NDm (mm)
1500
Fig. 7. Relationship between qu and NDm. (a) dm/Dm ¼ 0, (b) dm/Dm ¼ 0.5 and (c) dm/Dm ¼ 1.0
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diameter Dm increased to 20 mm, Dm/D50 increased to 33.3, and the particle size effects could be eliminated. 3.3. Influence of particle size effects on settlement The relationship between footing settlement ratio, (Sm/Dm)u, at the ultimate bearing capacity (same value of qu which is illustrated in Section 3.2 was used) and NDm are summarized in Fig. 8. (Sm/Dm)u decreased as the footing diameter Dm increased and the differences of (Sm/Dm)u in the same Dm also decreased with an increase in the footing diameter. The settlement presented the same trend as the ultimate bearing capacity. As shown in Fig. 8, in the case of dm/Dm ¼ 0, when Dm/D50 was more than 50, the particle size effects could be ignored and in the cases of dm/Dm ¼ 0.5 and 1.0, Dm/D50 should exceed 33.3. 3.4. Influence of embedment on particle size effects Fig. 9 shows the relationship between the ultimate bearing capacity, qu, and the acceleration, N (g). For the
same prototype, ultimate bearing capacity, qu, decreased as the acceleration decreased. When the footing diameter increased to a certain value, the ultimate bearing capacity became a constant and the particle size effects could be neglected. Yang et al. (2007) introduced the index Dqu to quantitatively measure the extent of the influence of particle size effects on the test results. The index Dqu is expressed as the Eq. (1). Dqu ¼
9qu qu0 9 qu0
ð1Þ
where, Dqu: index to express the particle size effect, qu: ultimate bearing capacity, qu0: constant ultimate bearing capacity with no particle size effect. The value of qu, was used as the value of qu0 when qu converged to an approximately constant value and was the minimum value. Therefore, when dm/Dm ¼ 0, result value of qu at the Dm ¼ 40 test, was used as qu0. When dm/Dm ¼ 0.5 3500
0.4
3000
qu (kPa)
(Sm/Dm)u
Dm =20mm 0.2 Dm =30mm 0.1 0
2000 1500
500
1000
NDm =500mm
1500
NDm =200mm
500 0
0
NDm =1000mm
1000
Dm =40mm 2000
0
20
40
60
N(g)
NDm (mm) 0.4
6000
Dm =5mm 0.3
5000
Dm =10mm
qu (kPa)
(Sm/Dm)u
small
NDm =1500mm
2500
Dm =10mm
0.3
Dm
big
Dm =30mm 0.2 Dm =20mm
big
4000
NDm =1000mm
3000
NDm =500mm
Dm
small
2000
0.1 1000
0
0
500 1000 NDm (mm)
0
1500
10
20
30
40
50
60
7000
Dm =5mm
0.5
6000
0.4
big
5000
Dm =10mm
0.3
qu (kPa)
(Sm/Dm)u
0
N(g)
0.6
Dm =30mm
0.2 Dm =20mm
500 1000 NDm (mm)
small
NDm =1000mm
4000 NDm =500mm
3000
NDm =200mm
1000
0
Dm
2000
0.1 0
NDm =200m
1500
Fig. 8. Relationship between (Sm/Dm)u and NDm. (a) dm/Dm ¼0, (b) dm/Dm ¼ 0.5 and (c) dm/Dm ¼ 1.0.
0
0
10
20
30
40
50
60
N(g) Fig. 9. Relationship between qu and N. (a) dm/Dm ¼ 0, (b) dm/Dm ¼ 0.5 and (c) dm/Dm ¼1.0.
Y. Toyosawa et al. / Soils and Foundations 53 (2013) 349–356
or 1.0, result value of qu at the Dm ¼ 30 test, was used as qu0. On the basis of the results in Fig. 9, the relationship between Dqu and Dm/D50 was obtained as in Fig. 10. It was shown that Dqu decreased as Dm/D50 increased for all embedment values. When the ratio of footing diameter to mean particle size, Dm/D50, increased to 50, the corresponding value of Dqu was less than 2% in the case of dm/Dm ¼ 0, and in the cases of dm/Dm ¼ 0.5 and 1.0, when Dm/D50 reached 33.3, the corresponding value of Dqu was less than 2%. It could be concluded that the particle size effects were less obvious as embedment increased. Fig. 11 presents the relationship between the value of Dqu and the embedded depth ratio dm/Dm, further illustrating the influence of embedment on the particle size effects. For the same Dm/D50, the value of Dqu decreased as embedment
355
increased and the influence of the particle size effects on the test results became less obvious. It is reasonable to conclude that the particle size effects can be ignored when the ratio of footing diameter to the mean particle size, Dm/D50, increased to 33.3 in both cases of dm/Dm ¼ 0.5 and 1.0. As dm/Dm increased from 0.5 to 1.0, the value of Dqu decreased from 2% to 1%. In other words, the particle size effects are less significant in the case of dm/Dm ¼ 1.0 than dm/Dm ¼ 0.5. Based on the data in Figs. 10 and 11, the relationship between the Dm/D50 and the dm/Dm was obtained for the value of Dqu equal to 0% and 2% as shown in Fig. 12. The figure indicated the extent and degree of bearing capacity with the diameter of model footing. According to the chosen ratio of footing diameter to mean particle size, Dm/D50, and the ratio of the embedded depth to the footing width, dm/Dm, in the figure, it was possible to distinguish whether any particle size effects occurred in the tests, and to assess the
70 70
60
NDm=500mm
60
40 30 NDm=200mm 20
40 Dm /D50 =16.7
30 20
10 0
Dm /D50 =8.3
50
NDm=1000mm
Δqu (%)
Δqu (%)
50
NDm=1500m 0
20
40 Dm /D50
60
10
80
0
Dm /D50 =50
Dm /D50 =33.3 0
0.2
0.4
0.6 0.8 dm /Dm
1
1.2
70 70
60
60 50
NDm=200mm
40
Δqu (%)
Δqu (%)
50
30 20
0
10
20
30 20
10 NDm=500mm 0
Dm /D50 =16.7
40
NDm=1000mm 30 40 Dm /D50
50
Dm /D50 =33.3
10 0
60
90
Dm /D50 =50 0
0.2
0.4
0.6 0.8 dm /Dm
1
1.2
50
80
40
70
Dm /D50 =33.3
Δqu (%)
Δqu (%)
60 50 NDm=500mm
40
30 20
30 20 NDm=200mm
10 0
0
10
20
10 Dm /D50 =50
NDm=1000mm
30 40 Dm /D50
50
60
Fig. 10. Relationship between Dqu and Dm/D50. (a) dm/Dm ¼0, (b) dm/Dm ¼ 0.5 and (c) dm/Dm ¼ 1.0.
0
0
0.2
0.4
0.6 0.8 dm /Dm
1
1.2
Fig. 11. Relationship between Dqu and dm/Dm. (a) NDm ¼ 200 mm, (b) NDm ¼ 500 mm and (c) NDm ¼ 1000 mm.
Y. Toyosawa et al. / Soils and Foundations 53 (2013) 349–356
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100
Dm /D50
80
Δqu=0%
60 Δqu=2%
40
Particle size effects were obvious
20 0
Particle size effects can be neglected
0
0.2
0.4 0.6 dm /Dm
0.8
1
Fig. 12. Particle size effects.
extent of their influence. The figure could also be used to choose a reasonable model footing size or particle size with an embedded depth of footing. 4. Conclusions A series of bearing capacity tests for circular footings in the centrifuge was conducted to study the influence of the model footing diameter and the embedded depth on the bearing capacity, with the following results. (1) Against a given value of prototype diameter, the ultimate bearing capacity decreased with as the diameter of the circular model footing (Dm) increased, and also with increases in the embedded depth (dm). However, when the ratio of footing diameter to particle size (Dm/D50) was more than 50, the ultimate bearing capacity in the same equivalent diameter of footing was not dependent on the diameter of model footing with the embedded depth of 0 m. In addition, when this ratio was more than 33, the ultimate bearing capacity was not affected by the diameter of model footing, with the ratio of the embedded depth to the footing diameter as 0.5 and 1.0. (2) The relationship between the ratio of footing diameter to particle size and the ratio of embedded depth to footing diameter provided a useful index for determining the model footing diameter, embedded depth and particle size of model ground in the centrifugal circular bearing capacity tests.
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