ABSTRACT
Reinforced soil structures find use in many locations now as the inclusion of reinforcement in soil improves the load carrying capacity of soil, thereby controlling the settlements within permissible limits. The reinforced soil foundations may be frequently subjected to cyclic and vibratory loads and as such the improvements in the soil properties under such loading conditions need be studied. With this objective, a large number of cyclic plate tests were conducted on the soil deposited in the test tank under controlled density and geogrid reinforcements included at the most appropriate locations and spacing. Empirical equations have been established on the basis of the test data obtained, which can be suitably utilized to design the reinforced soil structures (foundations). The study shows that geogrid reinforced soil possesses increased bearing capacity and reduced settlements. The damping capacity of the soil is improved and the soil can sustain larger number of loading cycles before bearing capacity failure. This study establishes that use of geogrid reinforcements is very useful in improving the overall behaviour of soil under cyclic loads particularly in ground improvement for roads, airfields, etc.

* Civil Engg. Department, NIT Hamirpur (Himachal Pradesh), India.

1. INTRODUCTION
Reinforcement of soil using geogrid has become an established technique to improve its properties under static loads. Since the reinforced soil is finding frequent use in embankments and foundations that are often subjected to cyclic loads, it becomes imperative to study the soil properties under cyclic loading conditions. The present study is conducted with a view to establish theproperties of geogridreinforced soil under cyclic loads.
The static strength characteristics of reinforced soil have been studied by Binquet and Lee (1975 a & b). Basset and Last (1978), Akinmusuru and Akinbolade (1981), Fragaszy and Lawton (1984), Guido et al. (1986), Khing et al. (1994), Sitharam et al. (1995), Adams and Collin (1997), Consoli et al. (2003). Kazuya Yashuhara et al. (1988) and Puri et al. (1991) conducted cyclic plate load tests on small sized reinforced soil samples and reported improvement in soil behaviour. Shin et al. (2002) conducted laboratory model tests to determine permanent settlement due to cyclic load of railroad bed and exploring possibility of using geogrid layers as reinforcement to reduce settlement of subbase layer. They concluded that practically all permanent settlement due to cyclic load is completed after application of 105 cycles of load. However, data based upon fullscale laboratory model tests on reinforced soil is not available and as such actual behaviour of reinforced soil structures in field cannot be established. The properties and behaviour of geogridreinforced soil deposited under controlled density in large test tanks and subjected to cyclic loads have been studied.
2. EXPERIMENTAL PROGRAMME
The experimental programme includes conducting cyclic plate load tests on rigid model footing of size 150 mm × 150 mm placed on unreinforced and geogridreinforced soil deposited in steel tank of size 900 mm × 900 mm × 1000 mm height, at a relative density of 70%. The reinforced soil consisted of 2, 3, 4, 6 and 8 geogrid reinforcement layers of size varying from one to five times footing width. The vertical spacing between the reinforcement layers and the depth of the first reinforcement layer below the footing base was kept 0.25 B each decided on the basis of the result data available in literature (Binquet and Lee, 1975 a & b). The experimental data of the cyclic plate load tests have also been analyzed to incorporate the effect of mean effective confining pressure and the area of footing.
(i) Soil used: Soil used in experimental investigations was dry poorly graded sand. The particle size distribution curve is shown in Fig. 1. According to Indian Standard on Soil Classification for General Engineering Purpose (IS:14981970), the test soil is classified as poorly graded sand (SP). The maximum and minimum void ratios were determined in accordance with procedure laid down in Indian Standard IS:2720 (Part XIV1983). Fig. 2 shows the direct shear tests conducted on sand at 70 per cent relative density. The relevant physical and mechanical properties of soil are listed in Table 1.
Table 1. Physical and Mechanical properties of Soil 
S.No. 
Property 
Value 
1 
Soil type 
SP (as per IS:14981970) 
2 
Effective size (D10) 
0.188 mm 
3 
Uniformity coefficient (Uc) 
1.51 
4 
Coefficient of curvature (Cc) 
1.10 
5 
Mean specific gravity 
2.65 
6 
Void ratio 
emin = 0.533, emax = 0.928 
7 
Angle of internal friction (f) 
40°, at 70% relative density 
(ii) Geogrid reinforcements: The reinforcing material use in model tests was geogrid Nelton CE121. The physical, chemical and mechanical properties of the reinforcement are listed in Table 2 :
Table 2. *Properties of Reinforced Material, geogrid Nelton CE121 
S.No. 
Property 
Value 
1 
Physical 
Colour – Black
Roll width – 2 m
Roll length – 15 m
Mesh aperture – 8 mm × 6 mm
Mesh thickness – 3.3 mm
Structural weight – 720 g/m2 
2 
Chemical 
100 per cent Highdensity polyethylene
Excellent resistance to chemical
and microbiological agents.
Poor resistance to oxidising agents
and ultraviolet light. 
3 
Mechanical 
Maximum load – 7.68 kN/m 
*Properties given by manufacturer. 
The stress strain characteristics of geogrid Nelton CE 121 have been studied by conducting tensile strength test in the universal testing machine. Fig. 3. shows the stressstrain curve of reinforcing material. Geogrid specimen of size 50 mm × 200 mm has been tested in universal testing machine at a strain rate of 1.45 mm/minute.
3. INTERPRETATION OF TEST RESULTS
(i) Bearing Capacity Ratio: The bearing capacity ratio (BCR) for a given sand bed is ratio of the ultimate bearing pressures of footing on reinforced (quR) to unreinforced (quU) sand bed. Fig. 4 shows a plot of bearing capacity ratio with respect to number of reinforcement layers of size varying from B to 5B for 150 mm square footing over reinforced sand bed at 70 per cent relative density. This plot shows that bearing capacity increases with increase in number of geogrid layers for all reinforcement sizes. For example, BCR increases from 1.25 to 2.06, for 150 mm square reinforcement sizes, when the number of reinforcement layers increases from 2 to 8. Similar trend is observed for other reinforcement sizes at 70 per cent relative density, at 50 per cent relative density and for the footing of size 300 mm square (not shown here). However, increase in BCR is not significant when the number of reinforcement layers increases beyond 6 as observed from Fig. 4. Thus, for any reinforcement size, six geogrid layers gives significant increase in bearing capacity, beyond which the increase in BCR is not appreciable. Fig. 4 can be converted into mathematical equation of the form:
y = (l + ax)/(b + cx) ... Equn. 1
where, y = bearing capacity ratio; x = ratio of size of reinforcement to size of footing (varying from 1 to 5); a, b and c = constants given in Table 3.

Table 3. Conversion of Fig. 4 into Mathematical Equations 
For number of reinforcement layers, n = 8 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
1.5310 
0.7583 
49.53 
0.993,925,6 
b 
1.0030 
0.2281 
22.73 
0.916,851,1 
c 
0.2881 
0.1802 
62.54 
0.990,782,4 
For number of reinforcement layers, n = 6 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
1.3470 
0.5897 
43.77 
0.993,703,9 
b 
0.9839 
0.1778 
18.08 
0.900,830,3 
c 
0.2810 
0.1568 
55.81 
0.990,908,3 
For number of reinforcement layers, n = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
1.2718 
0.4697 
36.95 
0.993,728,7 
b 
0.9904 
0.1400 
14.14 
0.886,870,6 
c 
0.2918 
0.1374 
47.08 
0.991,293,6 
For number of reinforcement layers, n = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.8645 
0.1789 
20.69 
0.993,256,2 
b 
1.0020 
0.0644 
6.42 
0.877,014,2 
c 
0.2221 
0.0622 
28.00 
0.990,759,4 
For number of reinforcement layers, n = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.6704 
0.3684 
54.96 
0.994,404,7 
b 
1.0280 
0.1214 
11.81 
0.823,070,8 
c 
0.2393 
0.1738 
72.63 
0.993,131,7 

The bearing capacity ratio increases with increase in size of reinforcement as shown in Fig. 5. For example, for 150 mm square footing placed over sand bed (70 per cent relative density) reinforced with 4 reinforcement layers of size varying from B to 5B, the BCR values vary from 1.88 to 3.16. The increase in BCR is not significant when the reinforcement size increases beyond 4B. Thus, it can be concluded that increase in BCRis not significant when the number of reinforcement layers increases beyond six and the size of reinforcement increases beyond 4B. Binquet and Lee (1975, a & b), Temel Yetimoglu et al. (1994) have reported a similar trend of increase in BCR with respect to the number of reinforcement layers for static plate load tests on reinforced sand beds. Fig. 5 can be converted into mathematical equation of the form :
y = y0 + a ln(x – x0) ... Equn. 2
where, y = bearing capacity ratio; x = number of reinforcement layers (2, 3, 4, 6 and 8); x0, y0 and a = constants given in Table 4.
Table 4. Conversion of Fig. 5 into Mathematical Equations 
For reinforcement size/footing size = 5 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
1.3010 
0.4513 
34.69 
0.975,548,3 
x0 
0.8353 
0.6871 
82.26 
0.961,900,5 
y0 
1.2260 
0.8961 
73.11 
0.988,260,8 
For reinforcement size/footing size = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
1.1190 
0.3507 
31.35 
0.974,008,3 
x0 
0.8010 
0.6012 
75.06 
0.960,228,2 
y0 
1.252 
0.6896 
55.09 
0.987,385,2 
For reinforcement size/footing size = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.9878 
0.2685 
27.19 
0.972,833,8 
x0 
0.7733 
0.5083 
65.73 
0.958,979,4 
y0 
1.2540 
0.5242 
41.80 
0.986,703,6 
For reinforcement size/footing size = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.9370 
0.1693 
18.07 
0.987,065,1 
x0 
1.2410 
0.4799 
38.68 
0.975,954,7 
y0 
0.7916 
0.3684 
46.54 
0.994,225,6 
For reinforcement size/footing size = 1 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.6083 
0.3049 
50.12 
0.987,439,3 
x0 
1.2720 
1.3530 
100.63 
0.976,469,8 
y0 
0.8324 
0.6658 
79.98 
0.994,403,9 
(ii) Settlement Ratio: The settlement ratio (SR) for a reinforced sand bed is the ratio of settlements of footing on reinforced and unreinforced sand beds; observed at a pressure equal to ultimate bearing capacity of the latter. For 150 mm square footing placed over reinforced sand bed, and on the sand bed reinforced with two geogrid layers of size 150 mm × 10 mm (Fig. 6)
SR = Sru/Sou = 0.736

Fig. 6 shows a plot of settlement ratio with respect to number of reinforcement layers of size varying from B to 5B for 150 mm square footing over reinforced sand beds at 70 per cent relative density. These curves show that settlement ratio decreases with an increase in number of reinforcement layers for all sizes of reinforcement, which seems to be due to greater mobilization of interface friction between the soil and reinforcements. Thus, SR decreases from 0.74 to 0.64 for 150 mm square reinforcement size, when the number of reinforcement layers increases from 2 to 8. Similar trend is observed for other reinforcement sizes at 70 per cent relative density. But the rate of decrease in SR is not significant when the number of reinforcement layers increases beyond 6. Thus, for any reinforcement size, six reinforcement layers provide optimum decrease in settlement after which the rate of decrease in SR is not significant. Fig. 6 can be converted into mathematical equation of the form:
y = aeb/(x+c) ... Equn. 3
where, y = settlement ratio; x = ratio of size of reinforcement to size of footing (varying from 1 to 5); a, b and c = constants given in Table 5.
Table 5. Conversion of Fig. 6 Into Mathematical Equations 
For number of reinforcement layers, n = 8 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.2821 
0.0080 
2.854 
0.976,124,1 
b 
1.6030 
0.1393 
8.693 
0.997,254,3 
c 
1.2670 
0.0840 
62.546.631 
0.994,714,0 
For number of reinforcement layers, n = 6 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.3088 
0.0141 
4.549 
0.973,998,7 
b 
1.4600 
0.2207 
15.12 
0.996,965,7 
c 
1.2420 
0.1442 
11.61 
0.994,182,3 
For number of reinforcement layers, n = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.3501 
0.0161 
4.597 
0.972,166,5 
b 
1.3150 
0.2264 
17.22 
0.996,575,4 
c 
1.2540 
0.1663 
13.26 
0.993,368,2 
For number of reinforcement layers, n = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.4164 
0.0263 
6.317 
0.968,731,2 
b 
1.0810 
0.3123 
21.90 
0.995,917,3 
c 
1.2350 
0.2775 
22.46 
0.992,050,4 
For number of reinforcement layers, n = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.5979 
0.0264 
4.420 
0.940,414,8 
b 
0.4217 
0.1775 
42.08 
0.993,105,1 
c 
0.8214 
0.2857 
34.78 
0.988,226,6 
The settlement ratio decreases with increase in size of reinforcement as shown in Fig. 7. For 150 mm square footing placed over sand be (70 per cent relative density) reinforced with 6 reinforcement layers of size B to 5B, SR values decrease from 0.64 to 0.40. The decrease in SR is not significant when reinforcement size increases beyond 4B. Therefore, it can be concluded that rate of decrease in SR is not significant when number of reinforcement layers increases beyond six and reinforcement size increases beyond 4B. Similar settlement behaviour of geogridreinforced sand beds has been reported by Yousef (1995) under static loads. Fig. 7 can be converted into mathematical equation of the form :
y = aeb/(x + c) ... Equn. 4
where, y = settlement ratio; x = number of reinforcement layers (2, 3, 4, 6 and 8); a, b and c = constants given in Table 6.

Table 6. Conversion of Fig. 7 into Mathematical Equations 
For reinforcement size/footing size = 5 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.1964 
0.0538 
27.38 
0.994,042,1 
b 
6.6500 
3.3490 
50.36 
0.998,975,6 
c 
4.0790 
1.4000 
34.31 
0.997,375,8 
For reinforcement size/footing size = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.1974 
0.0651 
32.97 
0.995,375,9 
b 
7.7870 
4.5530 
58.47 
0.999,142,1 
c 
4.7890 
1.8710 
39.08 
0.997,659,8 
For reinforcement size/footing size = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.2443 
0.0566 
23.18 
0.994,223,8 
b 
6.3350 
3.0730 
48.50 
0.998,919,1 
c 
4.4860 
1.4750 
32.88 
0.997,082,4 
For reinforcement size/footing size = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.2676 
0.0547 
20.44 
0.994,893,6 
b 
6.5160 
2.9230 
44.86 
0.998,993,2 
c 
4.9310 
1.4870 
30.17 
0.997,164,9 
For reinforcement size/footing size = 1 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
0.4580 
0.0287 
6.263 
0.983,646,4 
b 
2.2190 
0.6170 
27.80 
0.997,088,6 
c 
2.8370 
0.5801 
20.44 
0.993,107,5 
iii) Damping Capacity: The damping capacity of sand bed is ratio of area of first ten hysteresis loops (up to the pressure at failure of unreinforced sand bed) to the total area under the pressure settlement curves upto the same pressure. The areas, in this study, were measured by an electronic digital planimeter. The damping capacity ratio is the ratio of the damping capacities of the reinforced and unreinforced sand beds. For example, the damping capacity ratio of sand bed reinforced with two layers of size 150 mm square at 70 per cent relative density is 1.17. Fig. 8 shows a plot of damping capacity ratio with respect to number of reinforcement layers of size varying from B to 5B for 150 mm square footing over reinforced sand bed at 70 per cent relative density. It is observed from these curves that damping capacity ratio increases with an increase in number of reinforcement layers for all sizes of reinforcement. For example, damping capacity ratio increases from 1.17 to 1.28 for 150 square reinforcement, when the number of reinforcement layers increases from 2 to 8. Similar behavior is observed for other reinforcement sizes at 70 per cent relative density. However, the rate of increase in damping capacity ratio is not significant when the number of reinforcement layers increases beyond 6. Fig. 8 can be converted into mathematical equation of the form:
y = (Bmax1 x/(Kd1 + x)) + (Bmax2x/(Kd2 + x)) ... Equn. 5
where, y = damping capacity ratio; x = ratio of size of reinforcement to size of footing (varying from 1 to 5); Bmax1, Kd1, Bmax2 and Kd2 = constants given in Table 7.

Table 7. Conversion of Fig. 8 into Mathematical Equations 
For number of reinforcement layers, n = 8 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.2650 
3.365 × 10^{7} 
2.661 × 10^{9} 
1.000,000,0 
K_{d1} 
0.9918 
6.977 × 10^{4} 
7.035 × 10^{6} 
1.000,000,0 
B_{max2} 
1.3620 
3.365 × 10^{7} 
2.472 × 10^{9} 
1.000,000,0 
K_{d2} 
0.9919 
6.472 × 10^{4} 
6.524 × 10^{6} 
1.000,000,0 
For number of reinforcement layers, n = 6 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
2.1100 
2.2917 × 10^{1} 
1.382 × 10^{3} 
0.999,808,4 
K_{d1} 
0.6868 
1.3660 × 10^{1} 
1.989 × 10^{3} 
0.998,032,1 
B_{max2} 
1.715 × 10^{5} 
5.3740 × 10^{12} 
3.133 × 10^{9} 
1.000,000,0 
K_{d2} 
6.3550 × 10^{6} 
1.9920× 10^{14} 
3.134 × 10^{9} 
1.000,000,0 
For number of reinforcement layers, n = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.6850 
2.0220 × 10^{1} 
1.200 × 10^{3} 
0.999,666,4 
K_{d1} 
0.4010 
1.0270 × 10^{1} 
1.562 × 10^{3} 
0.996,785,4 
B_{max2} 
1.8750 × 10^{2} 
3.0640 × 10^{6} 
1.634 × 10^{6} 
1.000,000,0 
K_{d2} 
3.9160 × 10^{2} 
6.4140 × 10^{7} 
1.638 × 10^{6} 
1.000,000,0 
For number of reinforcement layers, n = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.3650 
1.6340 × 10^{1} 
1.197 × 10^{3} 
0.999,558,3 
K_{d1} 
0.1983 
8.8900 × 10^{0} 
4.482 × 10^{3} 
0.995,900,4 
B_{max2} 
7.5850 
4.9890 × 10^{3} 
6.577 × 10^{4} 
0.999,997,0 
K_{d2} 
93.270 
6.7500× 10^{4} 
7.236 × 10^{4} 
0.999,997,3 
For number of reinforcement layers, n = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
0.9769 
2.7850 × 10^{1} 
2.851 × 10^{3} 
0.999,871,1 
K_{d1} 
5.7780 × 10^{ 10} 
1.5010 × 10^{1} 
2.597 × 10^{12} 
0.998,410,2 
B_{max2} 
1.1890 
2.4680 × 10^{1} 
2.075 × 10^{3} 
0.998,435,3 
K_{d2} 
6.3290 
7.4320 × 10^{2} 
1.174 × 10^{4} 
0.999,879,0 
With increase in size of reinforcement, damping capacity ratio generally increases as shown in Fig. 9. For 150 mm square footing placed over sand bed at 70 per cent relative density reinforced with 6 reinforcement layers of sizes varying from B to 5B, the damping capacity ratio varies from 1.27 to 2.06. The increase in damping capacity ratio is not appreciable when size of reinforcement increases beyond 4B. It can be concluded that the rate of improvement in damping capacity is not significant when the number of reinforcement layers increases beyond six and size of reinforcement increases beyond 4B.

Patel and Paldas (1983) have reported improvement in damping capacity of sand bed reinforced with fibreglass woven roving under cyclic loading conditions. Fig. 9 can be converted into mathematical equation of the form :
y = (Bmax1 x/(Kd1 + x)) + (Bmax2 x/(Kd2 + x)) ... Equn. 6
where, y = damping capacity ratio; x = number of reinforcement layers (2, 3,4, 6 and 8); Bmax1, Kd1, Bmax2 and Kd2 = constants given in Table 8.
Table 8. Conversion of Fig. 9 into Mathematical Equations 
For reinforcement size/footing size = 5 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.8290 
2.597 × 10^{1} 
1.420 × 10^{3} 
0.999,798,7 
K_{d1} 
0.6317 
2.141 × 10^{1} 
3.389 × 10^{3} 
0.998,368,4 
B_{max2} 
6.0230 × 10^{4} 
5.723 × 10^{10} 
9.502 × 10^{7} 
1.000,000,0 
K_{d2} 
1.0410 × 10^{6} 
9.891 × 10^{11} 
9.503 × 10^{7} 
1.000,000,0 
For reinforcement size/footing size = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.5530 
2.1670 × 10^{1} 
1.395 × 10^{3} 
0.999,738,8 
K_{d1} 
0.4041 
1.9600 × 10^{1} 
4.849 × 10^{3} 
0.997,951,0 
B_{max2} 
5.999 × 10^{5} 
2.9940 × 10^{12} 
4.991 × 108 
1.000,000,0 
K_{d2} 
7.821 × 10^{6} 
3.9040 × 10^{13} 
4.992 × 10^{8} 
1.000,000,0 
For reinforcement size/footing size = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.4130 
4.3830 × 10^{0} 
3.102 × 10^{2} 
0.993,813,7 
K_{d1} 
0.3342 
6.3970 × 10^{0} 
1.914 × 10^{3} 
0.978,758,5 
B_{max2} 
4.4700 × 10^{6} 
2.0030 × 10^{8} 
4.482 × 10^{3} 
0.999,950,0 
K_{d2} 
5.6830 × 10^{7} 
2.5480 × 10^{9} 
4.483 × 10^{3} 
0.999,950,0 
For reinforcement size/footing size = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
1.4160 
2.2530 × 10^{1} 
1.591 × 10^{3} 
0.999,751,2 
K_{d1} 
0.4682 
2.2830 × 10^{1} 
4.876 × 10^{3} 
0.998,031,8 
B_{max2} 
6.0240 × 10^{4} 
5.4640 × 10^{10} 
9.070 × 10^{3} 
1.000,000,0 
K_{d2} 
1.0510 × 10^{6} 
9.5310 × 10^{11} 
9.071 × 10^{3} 
1.000,000,0 
For reinforcement size/footing size = 1 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
B_{max1} 
0.6930 
4.8720 × 10^{1} 
7.030 × 10^{3} 
0.999,950,0 
K_{d1} 
0.3329 
2.6890 × 10^{2} 
8.079 × 10^{4} 
0.999,950,0 
B_{max2} 
0.6499 
4.8720 × 10^{1} 
7.497 × 10^{3} 
0.999,950,0 
K_{d2} 
0.3348 
2.8590 × 10^{2} 
8.542 × 10^{4} 
0.999,949,9 
(iv) Loading Cycle Ratio: The number of loading cycles at ultimate bearing capacity of unreinforced and reinforced sand beds has been plotted for various reinforcement layers of different sizes as shown in Fig. 10. It is observed that the number of loading cycles required to cause the bearing capacity failure of the sand bed increases with the increase in size of geogrid reinforcement for different reinforcement layers from 2 to 8. Fig. 10 can be converted into mathematical equation of the form:
y = y0 + a ln (x – x0) ... Equn. 7

where, y = loading cycle ratio; x = ratio of size of reinforcement to size of footing (varying from 1 to 5); x0, y0 and a = constants given in Table 9.
Table 9. Conversion of Fig. 10 into Mathematical Equations 
For number of reinforcement layers, n = 8 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
13.1900 
1.1090 
8.404 
0.970,112,2 
x^{0} 
0.6442 
0.1288 
20.00 
0.965,244,8 
y^{0} 
15.880 
01.9350 
12.19 
0.986,241,1 
For number of reinforcement layers, n = 6 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
9.8280 
1.0170 
10.34 
0.954,078,5 
x^{0} 
0.4561 
0.1228 
26.93 
0.952,260,2 
y^{0} 
17.770 
1.6360 
9.205 
0.975,725,3 
For number of reinforcement layers, n = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
8.2690 
0.3178 
3.843 
0.952,342,8 
x^{0} 
0.4400 
0.0444 
10.09 
0.950,941,4 
y^{0} 
16.810 
0.5074 
3.019 
0.974,384,0 
For number of reinforcement layers, n = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
8.3780 
0.8515 
10.16 
0.973,087,7 
x^{0} 
0.6915 
0.1641 
23.73 
0.967,848,9 
y^{0} 
13.160 
0.1512 
11.49 
0.987,891,1 
For number of reinforcement layers, n = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
8.2393 
1.3520 
16.40 
0.996,391,2 
x^{0} 
1.7880 
0.5277 
29.51 
0.992,906,1 
y^{0} 
5.2740 
3.0680 
58.17 
0.998,559,1 
Similarly Fig. 11 shows that the number of loading cycles required to cause the bearing capacity failure of the sand bed increases with the number of layers for different reinforcement sizes. For example, the number of loading cycles required to cause the bearing capacity failure of the sand bed increases from 12 to 21 for 150 mm square reinforcement, when the number of reinforcement layers increases from 2 to 8. However, Similarly Fig. 11 shows that the number of loading cycles required to cause the bearing capacity failure of the sand bed increases with the number of layers for different reinforcement sizes. For example, the number of loading cycles required to cause the bearing capacity failure of the sand bed increases from 12 to 21 for 150 mm square reinforcement, when the number of reinforcement layers increases from 2 to 8. However, the rate of increase in loading cycle ratio is not significant when the number of reinforcement layers increases beyond 6. For 150 mm square footing placed over sand bed at 70 per cent relative density reinforced with 6 reinforcement layers of sizes varying from B to 5B, the number of loading cycles required to cause bearing capacity failure of the sand bed increases from 20 to 35. Fig. 11 can be converted into mathematical equation of the form:
y = y0 + a ln(x – x0) ... Equn. 8
where, y = loading cycle ratio; x = number of reinforcement layers (2, 3, 4, 6 and 8); x0, y0 and a = constants given in Table 10.

Table 10. Conversion of Fig. 11 into Mathematical Equations 
For reinforcement size/footing size = 5 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
17.770 
4.1510 
23.37 
0.9939483 
x^{0} 
1.7750 
0.8701 
46.42 
0.9831131 
y^{0} 
1.3610^{} 
10.240 
752.3 
0.9971210 
For reinforcement size/footing size = 4 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
15.530 
3.5990 
23.18 
0.9934656 
x^{0} 
1.8040 
0.8390 
46.51 
0.9820837 
y^{0} 
0.6796 
8.8010 
1295.0 
0.9968835 
For reinforcement size/footing size = 3 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
15.418 
4.8640 
31.56 
0.9956095 
x^{0} 
2.1830 
1.3150 
60.25 
0.9868509 
y^{0} 
2.280 
12.420 
544.7 
0.9979260 
For reinforcement size/footing size = 2 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
12.150 
4.9140 
40.44 
0.9954394 
x^{0} 
2.1430 
1.6630 
77.60 
0.9864525 
y^{0} 
0.4806 
12.500 
2600 
0.9978444 
For reinforcement size/footing size = 1 
Parameter 
Value 
Standard Error 
CV(%) 
Dependencies 
a 
9.5710 
4.5420 
47.45 
0.9965981 
x^{0} 
2.4430 
2.1490 
87.983 
0.9892505 
y^{0} 
1.2100 
11.900 
983.50 
0.9983968 
Therefore, it can be concluded that geogrid reinforcements create significant improvement in the capacity of the sand bed to sustain larger loads (till bearing capacity failure) with decreased settlements, increased damping capacity and larger number of loading cycles.
4. DISCUSSION SUMMARY
The result of the cyclic plate load tests indicate that by reinforcing the sand bed with geogrid layers, there is significant improvement in the ultimate bearing capacity and the total settlements are significantly reduced. With the increase in size and number of reinforcement layers, larger increase in ultimate bearing capacity and more reduction in total settlement is achieved. The damping capacity of sand bed is improved upon reinforcing it with geogrid layers, the improvement being more with larger number and size of reinforcement layers. Reinforced sand can sustain larger number of loading cycles before bearing capacity failure which shows its improved cyclic loading characteristics. These results indicate that the geogrid reinforcements can be effectively used in ground improvement for roads, airfields and other ground structures subjected to cyclic and vibratory loads.
5. CONCLUSIONS
1. The ultimate bearing capacity of sand bed is improved upon reinforcing with geogrid layers. The improvement in ultimate bearing capacity is up to about four times depending upon the size and the number of reinforcement layers. There is greater improvement in the ultimate bearing capacity with increase in the number and size of reinforcement layers. For size of reinforcements greater than four times the footing width and the number of reinforcement layers more than six, the rate of improvement in ultimate bearing capacity is not significant i.e., the optimum benefit is obtained by providing this particular size and number of reinforcement layers.
2. The total settlements are reduced up to less than half upon providing geogrid reinforcements in sand bed. The reduction in the total settlement is more with increase in size and numbers of layers. The optimum benefit is achieved by providing six geogrid layers of size four times the width of footing.
3.The damping capacity of sand bed is improved upon reinforcing with geogrid layers. The improvement is more with increase in number and size of reinforcement layers. The optimum benefit is obtained for six geogrid layers of size four times the width of footing.
4. The number of loading cycles required to cause the bearing capacity failure of sand bed increases to 3.5 times those for the unreinforced sand bed. The increase in number of loading cycles is more for larger number and larger sizes of reinforcement layers. Normally, the optimum number of loading cycles is achieved for six geogrid, layers with size four times the width of footing.
5. The geogrid reinforcements can be effectively used in ground improvement for roads, airfields and other ground structures subjected to cyclic and vibratory loads.
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