Behaviour of eccentrically inclined loaded rectangular foundation on reinforced sand

The concept of ultimate bearing capacity (UBC) of shallow foundations under varied loads is important for foundation engineers. In the design of foundations, the soil classification, behaviour and UBC are the most essential factors. There are two criteria to select the UBC for shallow foundations, namely the stability against shear failure and the settlement within the safe limit.

A foundation under load will undergo settlement due to the horizontal and vertical movement of soil particles below the foundation. In the case of centric vertical load on the footing, stress distribution will be uniform below the footing and the footing will undergo equal settlement at both edges. On the other hand, if the load is eccentric, the stress distribution below the footing will be non-uniform causing unequal settlement at two edges which will result in the tilt of footing. The tilt will increase with the increasing eccentricity to width ratio (e/B). When the eccentricity to width ratio (e/B) is greater than 1/6, the edge of the footing away from the load will lose it in contact with the soil, which will result in the reduction of effective width of footing and hence reduction of UBC of the foundation. Nowadays the use of geogrid has increased due to its high tensile strength at low strain, open grid structure that causes bonding between geogrid and foundation soil, long service life, lightweight. High modulus polymer materials such as polypropylene and polyethylene are used to manufacture the geogrid. Geogrid may be of two types, i.e. biaxial and uniaxial geogrid, depending on the nature of manufacturing.

In soil engineering practice, several techniques for soil improvement have been used to modify the soil behaviour and reduce the foundation settlement. In the past few years, it was found that the value of UBC increases with the use of reinforcing materials such as geotextile, geosynthetics and geogrid. Many types of research have been done on the behaviour of UBC of shallow foundation supported by unreinforced and reinforced sand using a laboratory model (Meyerhof [15], Das et al. [5], Yoo [31], Shin et al. [27, 28], Michalowski [16], Patra et al. [21, 22], El Sawwaf [25], Sitharam and Sireesh [29], Ornek [19]). From these studies, it can be concluded that the reinforced soil improves the UBC and reduces the foundation settlement. Recently, the effects of load eccentricity, inclination and combined effects of both eccentric and inclined loads on the UBC of shallow foundation located over reinforced and unreinforced sand bed have been studied experimentally, analytically and numerically by different investigators (Hjiaj [9], Kumar et al. [11,12], Foye et al. [6], Padmini et al. [20], Loukidis et al. [14], Latha and Somwanshi [13], Nazir and Azzam [17], Sawwaf and Nazir [26], Patra et al. [23], Rabi Narayan Behera [3], Krabbenhoft, et al. [10], Ornek [19], Badakhshan and Noorzad [1, 2], Ganesh et al. [7]). All researchers concluded that the value of UBC rises with increasing the number of geogrid reinforcing layers because the soil-reinforcement contains a series of interlocking cells that can keep the soil within its pockets and making an interlocking action between soil and geogrid. The interlocking action permits the geogrid to bear the horizontal stresses beneath the foundation and transfer them to the adjacent stable layer of the soil, which improves the vertical behaviour of the model foundation.

Several researchers studied the effect on strip footing subjected to centrically inclined loads. Little work is reported in the literature on the problem of strip footings subjected to eccentric-inclined loads. It is seen that to date no study is performed to address the effects of eccentrically inclined loads in both dimensions, i.e. lateral and longitudinal. The purpose of the study is to investigate the behaviour of model rectangular footing placed over multiple layers of the geogrid reinforced sand subjected to eccentric-inclined loads in both dimensions. The non-dimensional parameters called bearing capacity ratio (BCR) and settlement ratio (SR) are also introduced in this study. The test results demonstrate the potential benefit of using soil reinforcement with embedment depth under eccentric and inclined loads. The study concluded that the UBC increases with the increasing number of reinforcing layers or increasing in the area of footing or both. Furthermore, the optimum layer of reinforcement was found that N = 3 because the UBC increases at the third layer of reinforcement and after that, the rate of load increment becomes much less.

Laboratory model test

2.1

Model box

Fig. 1 shows the schematic view of the experimental model apparatus used in this study. The test tank is rectangular, having dimensions of 1.24 m × 0.90 m × 0.93 m (L ′ B ′ H). All sides of the test tank were braced using horizontal steel beams. The horizontal steel beams were fitted at mid-depth of the test tank to avoid bulging during the model plate load test.

2.2

Model footing

Rectangular shape model footing was made from a mild steel plate, having dimensions of 0.24 m × 0.18 m × 0.015 m (L ′ B ′ T). Holes were made at the desired centric and eccentric points to accommodate the ball-bearing as shown in Fig. 1(b). The holes represented the eccentric points e_L and e_B in both dimensions, i.e. lateral and longitudinal. Fig. 1(c) shows the ultimate load-bearing capacity of model footing subjected to axial centric and eccentric-inclined load.

Ultimate load per unit length q_uq_u on a model footing: (a) axial centric load; (b) eccentric-inclined load.

2.3

Loading arrangements

In this study, manually operated screw jack loading was used. The ultimate load was calculated with the help of a pre-calibrated proving ring. The displacement of footing was noted with the help of two dial gauges having the least count 0.01 mm arranged on the footing diagonally. The load was transferred to the model footing with the help of ball-bearing. Fig. 2(a) shows the test arrangements of embedded model footing. During testing, three fixed frames were used for axial vertical and inclined loading (α = 0°, 7° and 14°) as shown in Fig. 2(b).

Test arrangements of embedded model footing.

Experimental test setup: (a) horizontal frame; (b) 7° inclined frame; (c) 14° inclined frame.

2.4

Test material

2.4.1

Sand

The sand was collected locally from the riverbed of Yamuna River, Yamuna Nagar (India). It was free from roots and organic matters, etc. Physical properties and gradation curves of subjected soil have been tabulated and shown in Table 1 and Fig. 3, respectively.

Table 1

Properties of Yamuna sand.

Sand Properties	Value
Soil type	SP
Specific gravity, G	2.65
Effective particle size, D₁₀ (mm)	0.17
Uniformity coefficient, C_u	3.53
Coefficient of curvature, C_c	0.80
Working dry density, γ_d (kN/m³)	15.71
Maximum dry unit weight, γ_d(max) (kN/m³)	17.5
Minimum dry unit weight, γ_d(min) (kN/m³)	12.9
Angle of internal friction, (ϕ°)	36
Maximum void ratio, e_max	0.77
Minimum void ratio, e_min	0.50

2.4.2

Geogrid

In this study, polypropylene (PP) bi-axial geogrid (Bx20/20) having an ultimate tensile strength of about 7 kN/m² was used for reinforcing the sand. The geogrid was provided by Maruti Rubplast Private Limited. The geogrid had a square-shaped aperture (0.039 m ′ 0.039 m) and provides an equal ultimate tensile strength in all directions.

Testing program and methodology

3.1

Placement of sand

The sand was placed in the test tank in a layer of 0.02 m from a fixed height of 0.20 m by rainfall method. To achieve the desired relative density, it was determined a priori by performing a series of trials with different heights of fall. In each trial, the densities were monitored by collecting samples in a small container of known volume placed at a different location in the test tank. In all model tests, the average unit weight and relative density of the sand were kept as 15.71 kN/m³ and 68%, respectively. While doing the experiment the tank was fully emptied before filling sand for the next experiment. So, the case of the densification of sand at the bottom layers can be discarded.

3.2

Placement of geogrid

The factors that affect the behaviour of shallow foundation resting on reinforced sand are u_r/B, h_r/B and b_r/B. After the brief study, it has been established that the values of (u_r/B) and (h_r/B) range from 0.25 to 0.5 (Guido et al. [8]) and 0.25 to 0.4 (Shin and Das [27]), respectively. Omar et al. [18] reported that (b_r/B) should be kept as 8 for strip footing and 4.5 for square footing. During testing width of geogrid was kept as 0.80 m. The first layer of geogrid was placed at 0.08 m beneath the model foundation and afterward, other layers of geogrid were placed at 0.06 m distance apart from each other vertically as shown in Fig. 4(a). The reinforced depth beneath the footing can be expressed in terms of Eq. (1). (1) $d = u_{r} + (N - 1) h_{r}$ d = {u_r} + (N - 1){h_r} where u_r is the depth of first reinforced layer beneath the model foundation, h_r is the distance between consecutive layers of geogrid and N is the number of geogrid reinforced layers. The placement of geogrid is shown in Fig. 4(b).

3.3

Testing parameters

Three-hundred and fifteen numbers of experiments were conducted in total to examine the effects of model footing with varied parameters including the number of the geogrid reinforced layers (N), embedment ratio (D_f /B) and eccentricity ratio (e/B, e/L) with a load of inclination (a).

Result and discussion

A series of model tests were conducted using rectangular footing with embedment ratio (D_f /B = 0, 0.5, 1.0) on unreinforced and reinforced sand bed subjected to eccentric and inclined loads in both dimensions. Settlement corresponding to each load increment was noted with the help of two sensitive dial gauges having least count of 0.01 mm arranged on the footing diagonally. The tests were continued until the applied load was reduced. At the end of each test, the sand was excavated to maintain the relative density for the next experiment. The load–settlement curves have been plotted at the end of the experimental analysis process. UBC for each test was determined from the load–settlement curve by using the double-tangent method [30]. Furthermore, BCR and SR are used to express and compare the test data from different loading conditions.

4.1

Model test results

Various tests were performed to examine the behaviour of model footing with variable parameters such as depth of embedment (D_f /B), eccentricity (e/L, e/B) and a load of inclination (α). The effects of eccentric and inclined load on model footing were studied through the model plate load tests. Some tests were repeated twice to check the obtained test data. Settlement corresponding to each load increment has been noted and the load–settlement curve has been plotted. The UBC has been found from the load–settlement curve by using the double-tangent method and shown in (D_f /B = 1) in Fig. 5. In this method, two tangent lines have been plotted, one at the start of the load–settlement curve and the second at the end of the load–settlement curve, then the intersection point of two tangent lines considered as UBC [4]. Fig. 5 shows the variation of the load–settlement curve with embedded depth (D_f /B, 0, 0.5, 1.0) of model footing at axial centric load. From the graph, it can be seen that the UBC increases with the depth of embedment and the settlement is also increased with the same.

Variation of L–S curve for D_f/B at axial centric loading (e = 0, a = 0°).

4.2

Analysis of the test results

In the present study, a non-dimensional parameter called bearing capacity ratio (BCR) is used to calculate the effects of improvement utilising reinforcement layers on increasing the UBC. It is defined as the ratio of UBC of reinforced soil to the UBC of unreinforced soil (Eq. (2)). To analyse the footing settlement, SR is proposed and defined as the ratio of footing settlement in reinforced soil to the unreinforced soil condition (Eq. (3)). (2) $BCR = \frac{q_{u (reinforced)}}{q_{u (unreinforced)}}$ BCR = {{{q_{u(reinforced)}}} \over {{q_{u(unreinforced)}}}} (3) $SR = \frac{S_{u (reinforced)}}{S_{u (unreinforced)}}$ SR = {{{S_{u(reinforced)}}} \over {{S_{u(unreinforced)}}}} where q_u is the ultimate bearing capacity and S_u is the footing settlement at the ultimate bearing capacity. The factor q_u(e-i)/q_{u (e-i=0)} is also calculated for both unreinforced sand reinforced sand conditions, which are presented in Tables 2 to 5, where q_u(e-i) is the ultimate bearing capacity of eccentrically inclined load and q_u(e-i=0) is the ultimate bearing capacity of centrically applied loads.

Table 2

Results of eccentrically loaded (e/L) rectangular footing in unreinforced and reinforced sand conditions.

D_f/B	α°	N	q_u	S_u	BCR	SR	$\frac{q_{u (e - i)}}{q_{u (e - i = 0)}}$ {{{q_{u(e - i)}}} \over {{q_{u(e - i = 0)}}}}
0	0	0	212	2.2	1.00	1.00	1.00
0	1	0	240	3.8	1.13	1.73	1.00
0	2	0	282	4.1	1.33	1.86	1.00
0	3	0	348	4.7	1.64	2.14	1.00
0	4	0	368	6	1.74	2.73	1.00
0	0	0.05	190	2.3	1.00	1.00	0.90
0	1	0.05	210	3.3	1.11	1.43	0.88
0	2	0.05	278	3.5	1.46	1.52	0.99
0	3	0.05	320	4.2	1.68	1.83	0.92
0	4	0.05	330	5	1.74	2.17	0.90
0	0	0.1	152	2.5	1.00	1.00	0.72
0	1	0.1	190	2.6	1.25	1.04	0.79
0	2	0.1	252	2.8	1.66	1.12	0.89
0	3	0.1	298	4	1.96	1.60	0.86
0	4	0.1	308	4.5	2.03	1.80	0.84
0	0	0.15	138	2.2	1.02	1.00	0.65
0	1	0.15	171	2.3	1.27	1.05	0.71
0	2	0.15	240	2.5	1.78	1.14	0.85
0	3	0.15	255	3.8	1.89	1.73	0.73
0	4	0.15	290	4.2	2.15	1.91	0.79
0.5	0	0	330	1.4	1.00	1.00	1.00
0.5	1	0	370	3.8	1.12	2.71	1.00
0.5	2	0	400	4	1.21	2.86	1.00
0.5	3	0	450	4.2	1.36	3.00	1.00
0.5	4	0	470	5	1.42	3.57	1.00
0.5	0	0.05	310	1.3	1.00	1.00	0.94
0.5	1	0.05	360	2.8	1.16	2.15	0.97
0.5	2	0.05	390	3.7	1.26	2.85	0.98
0.5	3	0.05	420	4	1.35	3.08	0.93
0.5	4	0.05	460	5	1.48	3.85	0.98
0.5	0	0.1	262	1.3	1.00	1.00	0.79
0.5	1	0.1	322	1.5	1.23	1.15	0.87
0.5	2	0.1	352	2.1	1.34	1.62	0.88
0.5	3	0.1	365	3	1.39	2.31	0.81
0.5	4	0.1	400	3.2	1.53	2.46	0.85
0.5	0	0.15	250	1	1.00	1.00	0.76
0.5	1	0.15	302	1.5	1.21	1.50	0.82
0.5	2	0.15	320	2	1.28	2.00	0.80
0.5	3	0.15	332	2.7	1.33	2.70	0.74
0.5	4	0.15	372	2.8	1.49	2.80	0.79
1	0	0	400	3.2	1.00	1.00	1.00
1	1	0	460	6	1.12	1.88	1.00
1	2	0	510	7	1.24	2.19	1.00
1	3	0	540	7.5	1.32	2.34	1.00
1	4	0	580	8.5	1.41	2.66	1.04
1	0	0.05	398	3.1	1.00	1.00	0.97
1	1	0.05	440	5	1.11	1.61	0.96
1	2	0.05	480	5.2	1.21	1.68	0.94
1	3	0.05	500	6	1.26	1.94	0.93
1	4	0.05	560	8	1.41	2.58	0.64
1	0	0.1	360	2.1	1.00	1.00	0.88
1	1	0.1	410	2.8	1.14	1.33	0.89
1	2	0.1	428	4.2	1.19	2.00	0.84
1	3	0.1	470	4.5	1.31	2.14	0.87
1	4	0.1	510	6	1.42	2.86	0.91
1	0	0.15	310	1.5	1.00	1.00	0.76
1	1	0.15	380	2.5	1.23	1.67	0.83
1	2	0.15	400	2.8	1.29	1.87	0.78
1	3	0.15	420	3.5	1.35	2.33	0.78
1	4	0.15	480	4.5	1.55	3.00	0.86

Table 3

Results of eccentrically loaded (e/B) rectangular footing in unreinforced and reinforced sand conditions.

D_f/B	α°	N	q_u	S_u	BCR	SR	$\frac{q_{u (e - i)}}{q_{u (e - i = 0)}}$ {{{q_{u(e - i)}}} \over {{q_{u(e - i = 0)}}}}
0	0	0	212	2.2	1.00	1.00	1.00
0	1	0	240	3.8	1.13	1.73	1.00
0	2	0	282	4.1	1.33	1.86	1.00
0	3	0	348	4.7	1.64	2.14	1.00
0	4	0	368	6	1.74	2.73	1.00
0	0	0.05	180	2.2	1.00	1.00	0.85
0	1	0.05	206	3.2	1.14	1.45	0.86
0	2	0.05	270	3.5	1.50	1.59	0.96
0	3	0.05	312	3.7	1.73	1.68	0.90
0	4	0.05	325	4.5	1.81	2.05	0.88
0	0	0.1	148	1.8	1.00	1.00	0.70
0	1	0.1	160	2	1.08	1.11	0.67
0	2	0.1	248	3	1.68	1.67	0.88
0	3	0.1	270	3.2	1.82	1.78	0.78
0	4	0.1	305	3.8	2.06	2.11	0.83
0	0	0.15	132	1.2	1.00	1.00	0.00
0	1	0.15	140	1.5	1.06	1.25	0.58
0	2	0.15	232	2	1.76	1.67	0.82
0	3	0.15	212	2.8	1.61	2.33	0.61
0	4	0.15	280	3.3	2.12	2.75	0.76
0.5	0	0	330	1.4	1.00	1.00	1.00
0.5	1	0	370	3.8	1.12	2.71	1.00
0.5	2	0	400	4	1.21	2.86	1.00
0.5	3	0	450	4.2	1.36	3.00	1.00
0.5	4	0	470	5	1.42	3.57	1.00
0.5	0	0.05	300	1.5	1.00	1.00	0.91
0.5	1	0.05	338	2.8	1.13	1.87	0.91
0.5	2	0.05	380	3	1.27	2.00	0.95
0.5	3	0.05	400	4	1.33	2.67	0.89
0.5	4	0.05	450	5	1.50	3.33	0.96
0.5	0	0.1	258	1.3	1.00	1.00	0.78
0.5	1	0.1	310	2.7	1.20	2.08	0.84
0.5	2	0.1	342	2.9	1.33	2.23	0.86
0.5	3	0.1	358	3.5	1.39	2.69	0.80
0.5	4	0.1	380	3.8	1.47	2.92	0.81
0.5	0	0.15	240	1	1.00	1.00	0.73
0.5	1	0.15	280	1.5	1.17	1.50	0.76
0.5	2	0.15	300	2.4	1.25	2.40	0.75
0.5	3	0.15	328	2.6	1.37	2.60	0.73
0.5	4	0.15	360	3	1.50	3.00	0.77
1	0	0	400	3.2	1.00	1.00	1.00
1	1	0	460	6	1.12	1.88	1.00
1	2	0	510	7	1.24	2.19	1.00
1	3	0	540	7.5	1.32	2.34	1.00
1	4	0	580	8.5	1.41	2.66	1.04
1	0	0.05	390	2	1.00	1.00	0.95
1	1	0.05	430	3	1.10	1.50	0.93
1	2	0.05	470	4.8	1.21	2.40	0.92
1	3	0.05	490	5.1	1.26	2.55	0.91
1	4	0.05	550	6.2	1.41	3.10	0.98
1	0	0.1	342	1.4	1.00	1.00	0.83
1	1	0.1	400	2.4	1.17	1.71	0.87
1	2	0.1	420	3.1	1.23	2.21	0.82
1	3	0.1	450	3.4	1.32	2.43	0.83
1	4	0.1	500	6	1.46	4.29	0.89
1	0	0.15	302	1.1	1.00	1.00	0.74
1	1	0.15	362	2.4	1.20	2.18	0.79
1	2	0.15	385	2.8	1.27	2.55	0.75
1	3	0.15	400	2.9	1.32	2.64	0.74
1	4	0.15	460	3.5	1.52	3.18	0.82

Table 4

Results of rectangular footing subjected to inclined loading (a°) in unreinforced and reinforced sand conditions.

D_f/B	α°	N	q_u	S_u	BCR	SR	$\frac{q_{u (e - i)}}{q_{u (e - i = 0)}}$ {{{q_{u(e - i)}}} \over {{q_{u(e - i = 0)}}}}
0	0	0	212	2.2	1.00	1.00	1.00
0	0	1	240	3.8	1.13	1.73	1.00
0	0	2	282	4.1	1.33	1.86	1.00
0	0	3	348	4.7	1.64	2.14	1.00
0	0	4	368	6	1.74	2.73	1.00
0	7	0	134	2.8	1.00	1.00	0.63
0	7	1	222	3.4	1.66	1.21	0.93
0	7	2	240	3.8	1.79	1.36	0.85
0	7	3	280	4	2.09	1.43	0.80
0	7	4	310	4.5	2.31	1.61	0.84
0	14	0	110	1.7	1.00	1.00	0.52
0	14	1	154	2.4	1.40	1.41	0.64
0	14	2	224	3	2.04	1.76	0.79
0	14	3	268	4.1	2.44	2.41	0.77
0	14	4	282	4.3	2.56	2.53	0.77
0.5	0	0	330	1.4	1.00	1.00	1.00
0.5	0	1	370	3.8	1.12	2.71	1.00
0.5	0	2	400	4	1.21	2.86	1.00
0.5	0	3	450	4.2	1.36	3.00	1.00
0.5	0	4	470	5	1.42	3.57	1.00
0.5	7	0	270	1.7	1.00	1.00	0.82
0.5	7	1	300	2.5	1.11	1.47	0.81
0.5	7	2	350	3	1.30	1.76	0.88
0.5	7	3	400	3.5	1.48	2.06	0.89
0.5	7	4	422	4	1.56	2.35	0.90
0.5	14	0	195	1.2	1.00	1.00	0.59
0.5	14	1	232	1.5	1.19	1.25	0.63
0.5	14	2	300	2.5	1.54	2.08	0.75
0.5	14	3	340	2.6	1.74	2.17	0.76
0.5	14	4	360	4	1.85	3.33	0.77
1	0	0	400	3.2	1.00	1.00	1.00
1	0	1	460	6	1.12	1.88	1.00
1	0	2	510	7	1.24	2.19	1.00
1	0	3	540	7.5	1.32	2.34	1.00
1	0	4	580	8.5	1.41	2.66	1.04
1	7	0	350	3.2	1.00	1.00	0.85
1	7	1	385	4.1	1.10	1.28	0.84
1	7	2	455	4.4	1.30	1.38	0.89
1	7	3	490	6.12	1.40	1.91	0.91
1	7	4	520	6.98	1.49	2.18	0.93
1	14	0	270	3	1.00	1.00	0.66
1	14	1	322	3.2	1.19	1.07	0.70
1	14	2	390	3.5	1.44	1.17	0.76
1	14	3	420	3.8	1.56	1.27	0.78
1	14	4	430	5	1.59	1.67	0.77

Table 5

Results of eccentrically inclined, loaded (α°, e/L) rectangular footing in unreinforced and reinforced sand conditions.

D_f/B	α°	N	e/B	q_u	S_u	BCR	SR	$\frac{q_{u (e - i)}}{q_{u (e - i = 0)}}$ {{{q_{u(e - i)}}} \over {{q_{u(e - i = 0)}}}}
0	7	0	0.05	112	1.7	1.00	1.00	0.53
0	7	1	0.05	190	3.2	1.70	1.88	0.79
0	7	2	0.05	220	3.5	1.96	2.06	0.78
0	7	3	0.05	242	3.8	2.16	2.24	0.70
0	7	4	0.05	280	4.2	2.50	2.47	0.76
0	7	0	0.1	106	1.6	1.00	1.00	0.50
0	7	1	0.1	170	2.4	1.60	1.50	0.71
0	7	2	0.1	185	3	1.75	1.88	0.66
0	7	3	0.1	225	3.4	2.12	2.13	0.65
0	7	4	0.1	252	3.7	2.38	2.31	0.68
0	7	0	0.15	95	1.6	1.06	1.00	0.45
0	7	1	0.15	155	2	1.72	1.25	0.65
0	7	2	0.15	172	2.2	1.91	1.38	0.61
0	7	3	0.15	202	2.5	2.24	1.56	0.58
0	7	4	0.15	230	2.8	2.56	1.75	0.63
0	14	0	0.05	100	1.4	1.00	1.00	0.47
0	14	1	0.05	135	2	1.35	1.43	0.56
0	14	2	0.05	220	2.5	2.20	1.79	0.78
0	14	3	0.05	232	3	2.32	2.14	0.67
0	14	4	0.05	245	3.7	2.45	2.64	0.67
0	14	0	0.1	84	1.3	0.95	1.00	0.40
0	14	1	0.1	122	1.5	1.39	1.15	0.51
0	14	2	0.1	195	2	2.22	1.54	0.69
0	14	3	0.1	228	2.5	2.59	1.92	0.66
0	14	4	0.1	231	2.7	2.63	2.08	0.63
0	14	0	0.15	76	1.2	1.00	1.00	0.36
0	14	1	0.15	102	1.3	1.34	1.08	0.43
0	14	2	0.15	180	1.8	2.37	1.50	0.64
0	14	3	0.15	200	2.1	2.63	1.75	0.57
0	14	4	0.15	210	2.3	2.76	1.92	0.57
0.5	7	0	0.05	258	1.3	1.00	1.00	0.78
0.5	7	1	0.05	282	2	1.09	1.54	0.76
0.5	7	2	0.05	331	2.5	1.28	1.92	0.83
0.5	7	3	0.05	370	2.6	1.43	2.00	0.82
0.5	7	4	0.05	400	3.8	1.55	2.92	0.85
0.5	7	0	0.1	221	1.2	1.00	1.00	0.67
0.5	7	1	0.1	242	1.5	1.10	1.25	0.65
0.5	7	2	0.1	310	2.4	1.40	2.00	0.78
0.5	7	3	0.1	340	2.5	1.54	2.08	0.76
0.5	7	4	0.1	360	2.8	1.63	2.33	0.77
0.5	7	0	0.15	208	0.98	1.04	1.00	0.63
0.5	7	1	0.15	220	1.1	1.10	1.12	0.59
0.5	7	2	0.15	270	1.4	1.35	1.43	0.68
0.5	7	3	0.15	290	1.6	1.45	1.63	0.64
0.5	7	4	0.15	320	2	1.60	2.04	0.68
0.5	14	0	0.05	188	0.98	1.00	1.00	0.57
0.5	14	1	0.05	225	1.1	1.20	1.12	0.61
0.5	14	2	0.05	285	1.8	1.52	1.84	0.71
0.5	14	3	0.05	300	2.3	1.60	2.35	0.67
0.5	14	4	0.05	320	2.8	1.70	2.86	0.68
0.5	14	0	0.1	178	0.9	1.00	1.00	0.54
0.5	14	1	0.1	200	1	1.12	1.11	0.54
0.5	14	2	0.1	250	1.4	1.40	1.56	0.63
0.5	14	3	0.1	280	1.6	1.57	1.78	0.62
0.5	14	4	0.1	290	2.7	1.63	3.00	0.62
0.5	14	0	0.15	160	0.7	1.00	1.00	0.48
0.5	14	1	0.15	170	0.8	1.06	1.14	0.46
0.5	14	2	0.15	220	1.3	1.38	1.86	0.55
0.5	14	3	0.15	250	1.4	1.56	2.00	0.56
0.5	14	4	0.15	270	1.6	1.69	2.29	0.57
1	7	0	0.05	340	1.6	1.00	1.00	0.83
1	7	1	0.05	360	3.2	1.06	2.00	0.78
1	7	2	0.05	440	4	1.29	2.50	0.86
1	7	3	0.05	460	4.62	1.35	2.89	0.85
1	7	4	0.05	480	5	1.41	3.13	0.86
1	7	0	0.1	300	1.6	1.00	1.00	0.73
1	7	1	0.1	320	3	1.07	1.88	0.70
1	7	2	0.1	390	3.2	1.30	2.00	0.76
1	7	3	0.1	420	3.9	1.40	2.44	0.78
1	7	4	0.1	430	4.2	1.43	2.63	0.77
1	7	0	0.15	270	1	1.00	1.00	0.66
1	7	1	0.15	290	2.5	1.07	2.50	0.63
1	7	2	0.15	340	2.8	1.26	2.80	0.67
1	7	3	0.15	380	3.1	1.41	3.10	0.70
1	7	4	0.15	395	3.3	1.46	3.30	0.71
1	14	1	0.05	290	2.8	1.12	1.12	0.63
1	14	2	0.05	372	3	1.43	1.20	0.73
1	14	3	0.05	390	3.4	1.50	1.36	0.72
1	14	4	0.05	400	3.6	1.54	1.44	0.71
1	14	0	0.1	242	1.5	1.00	1.00	0.59
1	14	1	0.1	270	1.8	1.12	1.20	0.59
1	14	2	0.1	320	2	1.32	1.33	0.63
1	14	3	0.1	360	2.5	1.49	1.67	0.67
1	14	4	0.1	380	3	1.57	2.00	0.68
1	14	0	0.15	198	1.2	1.00	1.00	0.48
1	14	1	0.15	220	1.5	1.11	1.25	0.48
1	14	2	0.15	270	1.8	1.36	1.50	0.53
1	14	3	0.15	320	2	1.62	1.67	0.59
1	14	4	0.15	350	2.1	1.77	1.75	0.63

4.2.1

Effect of load eccentricity

In the case of eccentric loading, the UBC value is calculated from the load–settlement curves that are shown in Fig. 6(a). From the graphs, it can be concluded that the embedment depth ratio D_f/B has a significant effect on the reduction of the load eccentricity, where increasing the depth ratio reduced the effect of the eccentricity ratio. The BCR and SR were calculated using Eqs (2) and (3), and the results are presented in Tables 2 and 3 for both dimensions. It can be inferred from both tables that as the load eccentricity increases BCR increases significantly. The results also show that the BCR significantly increased with the increase in the number of reinforcing layers. Also, it is noticed that there is an optimum value of (N) after which a little increase in the value of BCR is observed. This increase in BCR for eccentrically loaded footing, in comparison with centrically loaded footing, is previously shown in an experiment by Sadoglu et al. [24] for strip footing on geotextile reinforced sand. A similar conclusion is also reported by Sawwaf and Nazir [26] for eccentrically loaded ring footing on reinforced layered soil. It is observed that by increasing the number of the reinforcement layers from one layer to three layers the BCR increases.

Variation of L–S curve for eccentric loading with an embedment depth D_f/B along with (a) larger dimension (e/L) and (b) smaller dimension (e/B)

4.2.2

Effect of load inclination

The load–settlement curve for inclined loading with an embedment depth is shown in Fig. 6(b). The centric loads were applied with different inclination angles changed from 0° to 14°. The ultimate load decreases as the load inclination angle increases and it is expected that q_u reaches its maximum value at vertical loading (i.e. a = 0°). The results of BCR and SR are presented in Table 4. According to the theoretical solutions, the BCR increases with an increase in inclination angle at any embedment depth D_f/B.

Variation of L–S curve for inclined loading with an embedment depth D_f/B.

4.2.3

Effect of eccentrically inclined loading

The load–settlement curve for different eccentric-inclined loading in both dimensions is shown in Fig. 6(c). These results indicate that the value of q_u reduces with an increase in the load eccentricity and inclination of applied load in both dimensions. While the load eccentricity rises from 0 to 0.15 the value of q_u reduces about 33% and 31% in the larger and smaller dimensions of the foundation, and when it applies on inclined load 0° to 14° the value of q_u reduces about 48%. The main reason for decreases in the UBC in the case of eccentric loading is that the contact pressure of the footing increases on the side of eccentricity and reduces on another side, resulting from the model footing tilt on the side of eccentricity. When the vertical pressure reaches the ultimate value, there will be a failure of the supporting soil on the side of eccentricity. Similarly, in the case of incline load, the contact pressure on the model foundation is not equally distributed. As a result, the footing loses contact with supporting soil and the UBC reduces. The results of theoretical solutions are presented in Tables 5 and 6 for both dimensions. It can be concluded that, by increasing the eccentrically inclined load, BCR and SR decreases. Thus, when the reinforcement layers are placed in optimum number (N = 3) for eccentricity and eccentrically inclined load, the BCR reaches the maximum value, and after that if the number of reinforced layers are increased, this factor decreases.

Variation of L–S curve for D_f/B = 0) at eccentric-inclined loading along with (a) larger dimension (e/L, a) and (b) smaller dimension (e/B, a).

Table 6

Results of eccentrically inclined, loaded (α°, e/B) rectangular footing in unreinforced and reinforced sand conditions.

D_f/B	α°	N	e/B	q_u	S_u	BCR	SR	$\frac{q_{u (e - i)}}{q_{u (e - i = 0)}}$ {{{q_{u(e - i)}}} \over {{q_{u(e - i = 0)}}}}
0	7	0	0.05	105	2.2	1.00	1.00	0.50
0	7	1	0.05	180	3	1.71	1.36	0.75
0	7	2	0.05	212	3.5	2.02	1.59	0.75
0	7	3	0.05	232	4	2.21	1.82	0.67
0	7	4	0.05	262	4.2	2.50	1.91	0.71
0	7	0	0.1	99	1.9	1.00	1.00	0.47
0	7	1	0.1	165	2.2	1.67	1.16	0.69
0	7	2	0.1	178	2.5	1.80	1.32	0.63
0	7	3	0.1	199	3	2.01	1.58	0.57
0	7	4	0.1	232	3.2	2.34	1.68	0.63
0	7	0	0.15	84	1.7	1.00	1.00	0.40
0	7	1	0.15	152	1.9	1.81	1.12	0.63
0	7	2	0.15	168	2.2	2.00	1.29	0.60
0	7	3	0.15	175	2.8	2.08	1.65	0.50
0	7	4	0.15	220	3.2	2.62	1.88	0.60
0	14	0	0.05	98	1.2	1.00	1.00	0.46
0	14	1	0.05	132	1.4	1.35	1.17	0.55
0	14	2	0.05	210	2	2.14	1.67	0.74
0	14	3	0.05	220	2.8	2.24	2.33	0.63
0	14	4	0.05	242	2.9	2.47	2.42	0.66
0	14	0	0.1	88	2	0.47	1.00	0.42
0	14	1	0.1	120	1.2	0.64	0.60	0.50
0	14	2	0.1	190	1.7	1.01	0.85	0.67
0	14	3	0.1	201	2.5	1.07	1.25	0.58
0	14	4	0.1	222	2.6	1.18	1.30	0.60
0	14	0	0.15	64	1	1.00	1.00	0.30
0	14	1	0.15	100	1.2	1.56	1.20	0.42
0	14	2	0.15	170	1.5	2.66	1.50	0.60
0	14	3	0.15	180	2	2.81	2.00	0.52
0	14	4	0.15	201	2.1	3.14	2.10	0.55
0.5	7	0	0.05	252	1.1	1.00	1.00	0.76
0.5	7	1	0.05	270	1.4	1.07	1.27	0.73
0.5	7	2	0.05	322	1.6	1.28	1.45	0.81
0.5	7	3	0.05	360	2.5	1.43	2.27	0.80
0.5	7	4	0.05	380	3	1.51	2.73	0.81
0.5	7	0	0.1	210	1.1	1.00	1.00	0.64
0.5	7	1	0.1	232	1.2	1.10	1.09	0.63
0.5	7	2	0.1	300	1.7	1.43	1.55	0.75
0.5	7	3	0.1	332	2.2	1.58	2.00	0.74
0.5	7	4	0.1	320	3	1.52	2.73	0.68
0.5	7	0	0.15	200	1	1.00	1.00	0.61
0.5	7	1	0.15	210	1.1	1.05	1.10	0.64
0.5	7	2	0.15	260	1.6	1.30	1.60	0.65
0.5	7	3	0.15	278	1.8	1.39	1.80	0.62
0.5	7	4	0.15	298	2	1.49	2.00	0.63
0.5	14	0	0.05	185	1	1.00	1.00	0.56
0.5	14	1	0.05	215	1.1	1.16	1.10	0.58
0.5	14	2	0.05	282	1.5	1.52	1.50	0.71
0.5	14	3	0.05	290	1.8	1.57	1.80	0.64
0.5	14	4	0.05	310	2	1.68	2.00	0.66
0.5	14	0	0.1	162	0.8	1.00	1.00	0.49
0.5	14	1	0.1	189	0.9	1.17	1.13	0.51
0.5	14	2	0.1	242	1.2	1.49	1.50	0.61
0.5	14	3	0.1	270	1.4	1.67	1.75	0.60
0.5	14	4	0.1	280	1.5	1.73	1.88	0.60
0.5	14	0	0.15	152	0.6	1.00	1.00	0.46
0.5	14	1	0.15	165	0.7	1.09	1.17	0.45
0.5	14	2	0.15	210	0.9	1.38	1.50	0.53
0.5	14	3	0.15	240	1.2	1.58	2.00	0.53
0.5	14	4	0.15	255	1.4	1.68	2.33	0.54
1	7	0	0.05	328	1.2	1.00	1.00	0.80
1	7	1	0.05	344	2.5	1.05	2.08	0.75
1	7	2	0.05	430	3	1.31	2.50	0.84
1	7	3	0.05	450	5	1.37	4.17	0.83
1	7	4	0.05	470	6.2	1.43	5.17	0.84
1	7	0	0.1	290	1	1.00	1.00	0.71
1	7	1	0.1	310	2.2	1.07	2.20	0.67
1	7	2	0.1	380	2.5	1.31	2.50	0.75
1	7	3	0.1	400	3.1	1.38	3.10	0.74
1	7	4	0.1	420	3.4	1.45	3.40	0.75
1	7	0	0.15	260	0.9	1.00	1.00	0.63
1	7	1	0.15	282	1.5	1.08	1.67	0.61
1	14	1	0.1	260	1.8	1.13	1.20	0.57
1	7	2	0.15	320	1.8	1.23	2.00	0.63
1	7	3	0.15	360	2.5	1.38	2.78	0.67
1	7	4	0.15	370	2.8	1.42	3.11	0.66
1	14	0	0.05	255	1.7	1.00	1.00	0.62
1	14	1	0.05	280	2.8	1.10	1.65	0.61
1	14	2	0.05	362	3	1.42	1.76	0.71
1	14	3	0.05	380	3.4	1.49	2.00	0.70
1	14	4	0.05	390	4.5	1.53	2.65	0.70
1	14	0	0.1	230	1.5	1.00	1.00	0.56
1	14	2	0.1	305	2	1.33	1.33	0.60
1	14	3	0.1	350	2.5	1.52	1.67	0.65
1	14	4	0.1	360	3.4	1.57	2.27	0.64
1	14	0	0.15	190	1.1	1.00	1.00	0.46
1	14	1	0.15	210	1.4	1.11	1.27	0.46
1	14	2	0.15	260	1.8	1.37	1.64	0.51
1	14	3	0.15	310	2	1.63	1.82	0.57
1	14	4	0.15	330	2.4	1.74	2.18	0.59

4.2.4

Effects of the number of reinforcement layers

The load–settlement curve for different embedded model footing with multiple layers of the geogrid reinforcement at centric vertical loading is shown in Fig. 7. From the graphs, it can be seen that the value of UBC of model footing increases with an increase in the depth of embedment and the settlement of the model foundation also increases with the same. These figures also indicate that the value of UBC increases with the increasing numbers of reinforcing layers at any depth of embedment (D_f/B).

Variation of L–S curve with multiple layers of geogrid N for (a) surface footing D_f/B = 0, (b) embedded footing D_f/B = 0.5, and (c) embedded footing D_f/B = 1.0 at axial centric loading (e = 0, α = 0).

These figures also indicate that the value of UBC increases with increasing the numbers of reinforcing layers. With the increase of the number of reinforcing layers in the contact area, the interlocking between sand particles and reinforcing layers are also increasing. Thus, the larger displacement and shear stresses are built up into the soil beneath the model foundation and transferred in reinforcement layers to the larger mass of soil. Therefore, the failure wedge becomes larger and the frictional resistance in failure planes become more [2].

4.2.5

Effect of the number of geogrid layers

This study aims to investigate the optimum number of geogrid layers to take the maximum benefit of soil reinforcement. The four layers of geogrid were adopted in the present study. The first layer of geogrid (u_r/B) was placed at a depth of 0.08 m beneath the model foundation and afterward, other layers of geogrid (h_r/B) were placed at 0.06 m depth. Fig. 8 showed the variation of UBC with several layers of geogrid (N = 0 to 4) in eccentrically loaded modeled footing in both dimensions. These figures clearly showed that the UBC increases at the third layer of reinforcement and after that, the rate of load increment becomes much less. This study concludes that N = 3 is the optimum number of the reinforcement layers, which was also present in prior studies of the strip and square footing over reinforced sand [4, 26]. These studies also showed that the optimum number of the soil reinforcement layer depends on the vertical spacing between geogrid layers and the embedment depth of the first layer. This is since the soil reinforcement would be significant when placed in an effective zone under the footing.

Variation of BCR with N at eccentric loading along the length and width side.

Conclusions

A series of laboratory tests were conducted to know the behaviour of the UBC of shallow rectangular foundation on unreinforced and multi-layered geogrid reinforced sand, under eccentric and inclined load with an embedment ratio varying from 0 to 1. Only one type of sand at one average relative density of compaction was used for all the tests. The test sand was reinforced with multiple layers of the bi-axial geogrid which varied from 0 to 4. The load eccentricity (e/L, e/B) varied from 0 to 0.15, and the inclination of applied load a varied from 0° to 14°. The non-dimensional equations BCR and SR have been developed in Eqs (2) and (3). The following conclusions are drawn from the present study:

The value of UBC reduces about 33% and 31% with an increase in the load eccentricity 0 to 0.15 in both dimensions. Similarly, the reduction is in the range of 21% and 24% if the footing is placed over geogrid reinforced sand.

As the load inclination increases from 0° to 14°, the UBC is decreased by 48%. Similarly, this decrement is in the range of 24% when the footing is placed on reinforced sand.

The value of UBC increases by about 75% with an increasing number of layers of geogrid reinforcement (N = 0 to 4).

The value of UBC increases by about 93% with an increasing the depth of embedded footing (D_f/B = 0 to 1).

The BCR increases with an increase in the numbers of geogrid reinforcing layers in both dimensions of the model footing and it is decreased with increasing the eccentricity and eccentrically inclined loads.

The optimum layer of reinforcement is N = 3 because the UBC increases at the third layer reinforcement and after that, the rate of load increment becomes much less.

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Behaviour of eccentrically inclined loaded rectangular foundation on reinforced sand

Article Category: Original Study

Publié en ligne: 30 juin 2021

Pages: 74 - 89

Reçu: 27 juil. 2020

Accepté: 06 févr. 2021

DOI: https://doi.org/10.2478/sgem-2021-0003

Mots clés
Eccentric loading, load inclination, ultimate bearing capacity (UBC), rectangular footing, geogrid

© 2021 Sujata Gupta et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1(a)

Figure 1(b)

Figure 1(c)

Figure 2(a)

Figure 2(b)

Figure 3

Figure 4(a)

Figure 4(b)

Figure 5

Figure 6(a)

Figure 6(b)

Figure 6(c)

Figure 7

Figure 8

Behaviour of eccentrically inclined loaded rectangular foundation on reinforced sand

Article Category: Original Study

Publié en ligne: 30 juin 2021

Pages: 74 - 89

Reçu: 27 juil. 2020

Accepté: 06 févr. 2021

DOI: https://doi.org/10.2478/sgem-2021-0003

Mots clésEccentric loading, load inclination, ultimate bearing capacity (UBC), rectangular footing, geogrid

© 2021 Sujata Gupta et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1(a)

Figure 1(b)

Figure 1(c)

Figure 2(a)

Figure 2(b)

Figure 3

Figure 4(a)

Figure 4(b)

Figure 5

Figure 6(a)

Figure 6(b)

Figure 6(c)

Figure 7

Figure 8

Mots clés
Eccentric loading, load inclination, ultimate bearing capacity (UBC), rectangular footing, geogrid