Introduction

For the design of a shallow foundation, the subject of the carrying capacity of shallow foundations must be taken into account. The majority of research focus on studying the behavior of a vertical load that is applied to the foundation at the center. The bearing capacity however, differs from centrally loaded footings when loads are applied eccentrically to the foundation. Due to load eccentricity, the foundation's overall stability declines along with foundation settlement and tilting, lowering its bearing capacity. There are many structures resting on circular footing such as elevated shaft tanks, surface storage tanks, high silos, refinery towers and chimneys. These structures subject mainly to two load cases, main case of loading due to vertical concentric load whether from dead and live loads and secondary case of loading due to wind and earthquake loads. Always circular footing is used to transfer superstructures load to supporting soil to satisfy mainly two conditions which are settlement and bearing capacity requirements. Due to uncertainty of direction of secondary load which produce load eccentricity which always occurs in one way circular footing is conceded as the most economical choice as stated by Hussein et al. [13] Badakhshan and Noorzad [9] and Sharma and Kumar [24].

Different studies have been conducted to look into the impact of eccentric loads on the behavior of foundations, and many researchers have indicated that they have an impact on the carrying capacity of shallow footings. Patra et al. [19, 20], Abbas and Abd [1], Al-Tirkity and Al-Taay [5], Krabbenhoft et al. [15] and Turker et al. (2014) [27] investigated the bearing capacity behavior of eccentrically loaded strip footing on sand reinforced with geogrids. It was found from these studies that; the ultimate bearing capacity of the foundation is reduced by the load eccentricity. Saleh et al. [21], Kaya et al. [14] and Ornek et al. [18]. Studied the performance of skirted strip footing subjected to Eccentric Load. From these studies they concluded that, the use of vertical skirts has improved the load-settlement behavior of strip footing subjected to eccentric loads. El-Sawwaf and Nazer [10], and El-Sawwaf, [11] performed several model tests on a strip footing that was eccentrically loaded and resting on geogrid reinforced sand, and they discovered that the influence of reinforcement on the bearing capacity ratio is stronger at lower values of eccentricity and higher relative densities. Nagaraj and Ullagaddi, [17] studied the effect of shape and size of footing on load settlement behavior of foundation. From this study it was founded that, the increase in size of footing will improve the bearing capacity or load settlement behavior of the supporting soil.

Al-Aghbari and zein, [2,3,4] investigated the use of skirts to improve the bearing capacity and to reduce the settlement of a circular footing resting on dune sand. The skirts' increased bearing capacity and decreased footing settlement are both confirmed by the test findings. Sethy et al. [23] and Kulkarni et al. [16] studied numerically the Behavior of circular footing on sand soil subjected to eccentric loads with and without reinforcement. From these studies it was founded that, soil confinement has a significant effect on improving the performance of circular footing on sand soil.

To improve foundation soil capacity and reduce the expected settlement there are many modification method such as using soil nailing which considered as vertical soil reinforcement or using skirts as stated by Azzam and Basha [7] and Azzam and Basha [8].

Based on the aforementioned studies, it can be seen that the majority of these studies are focused on researching the behavior of eccentrically loaded strip footing on reinforced / unreinforced soil, while eccentrically loaded circular footing has received relatively little attention. As a result, the aim of this research is to investigate experimentally how an eccentrically loaded circular footing behaves on sand with and without skirts under different parameters such as different load eccentricities, the impact of changing the skirts’ depth on the ultimate bearing capacity, settlement of the circular footing and presenting a modified equation to predict the ultimate capacity of circular footing with skirt under different load eccentricities.

Material properties

Sand

In this study, dry, clean silica sand was employed (ASTM D422-63, 2007) [6]. The employed soil had an effective particle size (D10) of 0.26 mm, a uniformity coefficient (Cu) of 2.46, and a coefficient of curvature (Cc) of 1.02. A relative density of Dr of 85% was used to experimentally evaluate the shear parameters (ASTM D3080/D3080M-11). Table 1 lists the characteristics of utilized sand.

Table 1 Geotechnical Properties of used sand in this study
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Footing

Steel plates with a 20 mm thickness are used to represent the circular footing model in order to provide a rigid footing condition. A diameter of 60 mm has been chosen for the footing. As shown in Fig. 1, various models of footing with various eccentricities are utilized to prepare distinct eccentricities on the footing. The load eccentricity for the footing is chosen according to the core of circular footing, which is a portion of the footing where the entire footing experiences compressive pressure when load is applied anywhere other than the center, and where the edge of the footing experiences zero pressure when load is applied to the core boundary. According to Badakhshan and Noorzad [9], the core of a circular footing is calculated to be R/4, which is equivalent to 7.50 mm, as shown below:

$$q = 0 \to q = \frac{P}{A} - \frac{My}{I} = 0 \to \frac{P}{{\pi R^{2} }} = \frac{PeR}{{\frac{{\pi R^{4} }}{4}}} \to e = \frac{R}{4}$$
(1)

where (I) is the moment of inertia, (q) is the pressure at the edge of the footing, (P) is the centric load, (A) is the area of the footing, (M) is the moment, (y) is the greatest distance from the center, (e) is the load eccentricity, and (R) is the radius of the circular footing.

Fig. 1
figure 1

Different models of footing with different eccentricities used in this study

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Nine load eccentricities, including central (e = 0.00), inside the core (e = 2.50 mm and 5.00 mm), on the core border (e = 7.50 mm), outside the core (e = 10.00 mm, 12.50 mm, and 15.00 mm), and farther away from the core (e = 17.5 mm and 20.00 mm), are included in this study. Figure 2 depicts the footing's core and loaded regions, while Fig. 3 displays the various load eccentricity values at the footing’s core.

Fig. 2
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The core of circular foundation and loaded locations used in this study

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Fig. 3
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Schematic Diagram for different values of load eccentricity around the center of the footing

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The ratio between the modeled footing diameter to its thickness was 3.00, which lead to rigid footing The ratio between the inner dimensions of the testing tank to the model footing diameter was about 5.00 which provide a clearance distance between footing perimeter and test tank boundary about 2D in the lateral direction, so the effect of boundary conditions can be discarded as stated by Garnier et al, [12]. Scale effects must be taken into account in all small-scale model tests, but especially those conducted in sand. The use of small-scale models that were built in sand and tested at 1 g is invalidated by a number of significant factors. The research described here was carried out on miniature 1 g physical models. The soil particle size, construction methods, boundary conditions, and dilatancy at low stress are the crucial elements that must be taken into account for such small-scale models. In his summary of test results, Kusakabe noted that for a (D50/B) ratio smaller than 1/100 as stated by Kusakabe [28], the effect of particle size on footing bearing capacity becomes less pronounced. The model layout that was built in the lab was constructed using methods that were comparable to those needed in the field.

Experimental test program and setup

The test set-up consists of a cylindrical container made of stainless steel with inner diameter of 315 mm and a height equal to 850 mm. The cylindrical cell was supported on a base plate. A base plate supports the cylindrical container. Four steel rods form a loading frame and tie the base and top plate together as illustrated in Fig. 4a which shows the details of test apparatus. Also, Fig. 4b shows schematic Diagram for the Test Setup used in this research. Thirty-six tests are carried out to study the effect of eccentric loading on a circular footing, resting on both reinforced and unreinforced sands with vertical skirts. These 36 tests consist of 4 groups of tests on a circular footing on reinforced and unreinforced sand to study the effect of different load eccentricities (e= zero, 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 and 20.00mm and different skirts depths (h = 60, 150 and 240) mm on load-settlement response of foundation. Table 2 presents the different parameters used in this research.

Fig. 4
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Test Setup; a Test Tank b Schematic Diagram for the Test Setup

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Table 2 Constant and variable parameters in experimental testing for this study
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Results and discussions

The load-settlement behavior of circular footing on unreinforced and reinforced sand with vertical skirts with different depths has been investigated in order to show the effect of eccentric loads and the effect of using such soil reinforcement technique to enhance the settlement behavior of circular footing. The results of thirty-six experimental tests are presented and discussed in this part of research as illustrated below.

Effect of load eccentricity on the settlement behavior of circular footing without skirt

The behavior of circular footing subjected to vertical load with different eccentricities (e = zero, 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 and 20.00 mm) were investigated. At the same load value equals (2119 N) the results indicated that, the footing settlement is increased with increasing the load eccentricity. In the first in case of using eccentricity points inside the core of the footing (e = 2.50 mm and 5.00 mm) the settlement is noticed to be increased by 86% and 147% respectively, while in case of using eccentricity point at the boundary of the footing core (e = 7.50 mm) the increasing in footing settlement reached to 207% compared to the case of using uneccentric load (e = 0.00). Moreover, in case of using eccentricity points outside the core (e = 10.00 mm, 15 mm and 20.00 mm) the settlement is found to be increased with large ratio reached to 348% compared to the case of using uneccentric load (e = 0.00) as shows in Fig. 5.

Fig. 5
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Load-settlement behavior of circular footing without skirts (L/D) = 0.00 and subjected to vertical load with different eccentricities

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Effect of load eccentricity on the settlement behavior of circular footing with skirt

The behavior of circular footing subjected to vertical load with different eccentricities (e = zero, 2.50, 5.00, 7.50, 10.00, 12.50, 15.00, 17.50 and 20.00 mm) were investigated. At the same load value equals (2119 N) the results indicated that, the footing settlement is increased with increasing the load eccentricity. In the first in case of using eccentricity points inside the core of the footing (e = 2.50 mm and 5.00 mm) the settlement is noticed to be increased by 86% and 147% respectively, while in case of using eccentricity point at the boundary of the footing core (e = 7.50 mm) the increasing in footing settlement reached to 207% compared to the case of using uneccentric load (e = 0.00). Moreover, in case of using eccentricity points outside the core (e = 10.00 mm, 15 mm and 20.00 mm) the settlement is found to be increased with large ratio reached to 348% compared to the case of using uneccentric load (e = 0.00) as shows in Fig. 5.

Effect of load eccentricity on the settlement behavior of circular footing on reinforced sand

In this part of research, the effect of using skirts with different depths (L/D) = 1, 2.5 and 4 on the load-settlement behavior of circular footing subjected to vertical load with different eccentricities has been discussed. At the same load value equals (1413 N) it was clear that, in case of using skirts with (L/D) = 1 when using eccentricity points inside the core of the footing (e/D) = 0.04 and 0.08 the settlement is noticed to be increased by 20% and 47% respectively, while in case of using eccentricity point at the boundary of the footing core (e/D) = 0.125 the increasing in footing settlement reached to 60% compared to the case of using uneccentric load (e/D) = 0. Moreover, in case of using eccentricity points outside the core (e/D) = 0.167, 0.25 and 0.33) at the same (L/D) ratio equals 1 and loading value equals (1413 N) the settlement is found to be increased with large ratio reached to 110%, 160% and 324% respectively compared to the case of using uneccentric load (e/D) = 0 as shows in Fig. 6. Moreover, in case of using skirts with (L/D) = 2.5 at loading value equals (706 N) when using eccentricity points inside the core of the footing (e/D) = 0.04 and 0.08 the settlement is noticed to be increased by 28% and 30% respectively, while in case of using eccentricity point at the boundary of the footing core (e/D) = 0.125 the increasing in footing settlement reached to 43% compared to the case of using uneccentric load (e/D) = 0. In addition to, in case of using eccentricity points outside the core (e/D) = 0.167, 0.25 and 0.33 the settlement is found to be increased with ratio reached to 71%, 128% and 300% respectively compared to the case of using uneccentric load (e/D) = 0 as shows in Fig. 7. Also, in case of using skirts with (L/D) = 4 at the same loading value equals (2119N) when using eccentricity points inside the core of the footing (e/D) = 0.04 and 0.08 the settlement is noticed to be increased by 40% and 58% respectively, while in case of using eccentricity point at the boundary of the footing core (e/D) = 0.125 the increasing in footing settlement reached to 73% compared to the case of using uneccentric load (e/D) = 0. Moreover, in case of using eccentricity points outside the core (e/D) = 0.167, 0.25 and 0.33 the settlement is found to be increased with ratio reached to 96%, 196% and 287% respectively compared to the case of using uneccentric load (e/D) = 0 as shows in Fig. 8. These results indicate that, the settlement increases when there is an increase in the eccentricity of the loading at the same loading value.

Fig. 6
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Load-settlement behavior of circular footing supported with skirts at (L/D) = 1 and subjected to vertical load with different eccentricities

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Fig. 7
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Load-settlement behavior of circular footing supported with skirts at (L/D) = 2.5 and subjected to vertical load with different eccentricities

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Fig. 8
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Load-settlement behavior of circular footing supported with skirts at (L/D) = 4 and subjected to vertical load with different eccentricities

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Moreover, from the results of experimental tests it is noticed that, using vertical skirts in sand soil under the footing base decreases and improves the settlement behavior of circular footing subjected to eccentric loads. The effect of using different skirts depth (L/D = 1, 2.5 and 4) is investigated at different eccentricity points inside the footing core, at the core boundary and outside the core. It is found that, for the eccentricity point inside the core (e/D) = 0.04 the footing settlement is decreased with increasing the skirt depth. As can be seen in Figs. 6, 7 and 8 the settlement is decreased by 20% and 53% when using skirts with (L/D) = 1 and 2.5 respectively compared to the unreinforced case at constant loading value equals (2119 N). While, in case of using skirts with (L/D = 4) the settlement decreased by 82% compared to the unreinforced case at the same loading value. Moreover, in case of using eccentricity point at the core boundary (e/D) = 0.125 at loading value of (2119N) the footing settlement is found to be decreased by 32% and 91% when using a skirt with (L/D) = 1and 2.5 respectively compared to the unreinforced case. The settlement reduction is increased with increasing the skirt depth it reduced by 92% in case of using skirts with (L/D = 4) compared to the unreinforced case at the same loading value as illustrated in Fig. 8. In addition to, when using eccentricity point outside the footing core (e/D) = 0.25) at loading value of (2119N) the settlement is decreased with increasing the skirt depth, it is found to be reduced by 10%, 92% and 93% when using a skirt with (L/D) equals 1, 2.5 and 4 respectively compared to the unreinforced case. Table 3 illustrates and summarizes the variation of settlement values with different (L/D) and (e/D) ratio during all experimental testing groups. Finally, from these results it can be concluded that, embedment a vertical skirt under the footing provided a resistance to sliding and overturning. All test results indicated the same trend as the settlement decreases with the increase of skirt length. The resistance of the extra passive zone formed in this method can be credited with preventing overturning. The experimental findings show that settlement can be reduced and avoided by increasing the skirts' insertion depth.

Table 3 Variation of footing ultimate capacity and settlement ratio at different studied parameters
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Effect of load eccentricity on the ultimate capacity of circular footing with and without skirts

Laboratory tests were performed to investigate the eccentricity effect on the ultimate capacity of circular footing with and without skirts. According to test results, loads reduce when load eccentricity rises, as illustrated in Fig. 9. In contrast to centric loading (e = 0) at a constant (L/D) ratio of 1, for instance, the final loads reduce by 2.3 percent, 13 percent, and 31 percent when the load is applied at e/D = 0.04, 0.08, and 0.125. When employing an e/D ratio of 0.33 as compared to centric loading (e = 0) at (L/D) = 1, the maximum decrease in ultimate load is obtained, which is a 59 percent reduction. Additionally, it has been discovered through test results that eccentricity effect is reduced or completely removed by introducing rigid skirts vertically into the soil, as shown in Fig. 10 which illustrates how ultimate load values change as the (L/D) ratio varies at various load eccentricities. This graphic makes it clear that as the (L/D) ratio rises, so does the ultimate load. In comparison to the case of an unsupported footing (L/D = 0), the ultimate load increases by 31%, 139%, and 322% when (L/D) = 1, 2.5, and 4 and (e/D) = 0. While in case of using (e/D) = 0.25 the ultimate load is increased by 108% and 269% in case of using (L/D) = 2.5 and 4 and (e/D) = 0 while in case of using (L/D) = 1 the ultimate load is found to be almost constant compared to the case of unsupported footing (L/D) = 0. The maximum capacity of a circular footing is observed to rise with increasing (L/D) at the same eccentricity value. This leads to the conclusion that the additional footing component provides enough more resistance to prevent overturning and sliding without compromising settlement to reclaim the maximum capacity reduction. Additionally, it is established that the inserted skirt's depth (L) significantly affects the maximum loads. For similar eccentricity ratios, the load rises as the insertion depth rises. For various (L/D) and (e/D) ratios, the variation in ultimate load values is shown in Table 3.

Fig. 9
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Variation of footing ultimate capacity with increasing (e/D) ratio at different skirts length (L/D)

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Fig. 10
figure 10

Variation of footing ultimate capacity with increasing (L/D) ratio at different eccentricity points

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Failure mechanism effect of skirted on general shear failure

Understanding the behavior of skirted foundations begins with a review of the failure pattern and bearing capacity of traditional shallow foundations. It is understood that soil shear failure is the most common failure mechanism in shallow foundations. The most essential elements that affect the modes of failure are compressibility and depth to width ratio (d/B) [25]. Generally, general shear failure occurs in dense cohesion soils (Dr > 70%) or stiff cohesive soils. Figure 11 depicts the shear failure mode in general. The carrying capacity of shallow foundation on sand has been carefully resolved through theoretical and experimental investigation. With a combination of lower and upper bound theorems and empiricism Terzaghi [25] created a general bearing capacity formulation. The bearing capacity qult of a footing with width B and length L (A = BL) on a soil with angle of friction (φ), cohesion (c), and density (γ) can be stated as:

$$qult = \frac{Pult}{A} = cNc + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {2\gamma BN\gamma + \gamma DfNq}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\gamma BN\gamma + \gamma DfNq}$}}$$
(2)

where qult = ultimate bearing capacity factors, γ = unit weight of soil, Df = foundation depth, B = foundation width and Nq, Nγ are the bearing capacity factors.

Fig. 11
figure 11

Failure of a general shear foundation due to dense or hard soil (Modified after, [26])

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According to Tripathy [26], the standard equation for ultimate bearing capacity should be modified as follows for shallow foundations with structural skirts resting on dense sand:

  • The soil above the lower edges of the skirts should always be viewed as a surcharge, much like Terzaghi suggested for shallow foundations (1943).

  • For the purpose of calculating the ultimate bearing capacity of a shallow foundation with structural skirts, a skirt factor (Fγ) should be introduced into the second part of the general equation, to account for all the characteristics of the structural skirts, the soil, the foundation and the loading, which influence the ultimate bearing capacity of the foundation. No factor is included in the first part of the general equation because the effect of the skirt can be accounted for by the skirt depth. Thus, the modified ultimate bearing capacity equation may be written as

$$qult = \gamma \left( {Dfs + Ds} \right)Nq + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} B ^{\prime}\gamma N\gamma F\gamma$$
(3)

Where Fγ = Skirt factor, Dfs = Depth to foundation base below ground level, Ds = Depth to the lower edge of the skirt below the foundation base and B ҅ = Total foundation width with skirts (B+2Bs) where, Bs is the skirt thickness.

Figure 12 shows the Bearing capacity failure mechanism in soil under shallow foundation with and without skirts.

Fig. 12
figure 12

Mechanism of bearing capacity failure in soil beneath foundations with and without skirts proposed by Terzaghi [25]

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The similar equation was proposed by Al-Aghbari and Zein in 2004, who claimed that the skirt factor could be calculated using the following formula:

$$F\gamma = 1.15 \left[ {0.4 + 0.6 \left( {\frac{tan\emptyset^\circ }{{tan\delta f}}} \right)} \right]\left[ {0.75 + 0.1\left( {DsB^{\prime}} \right) + 0.37 \left( {\frac{tan\delta s}{{\tan \delta f}}} \right)} \right]\left[ {1.2 - 0.002Dr} \right]$$
(4)

where Ø′= Effective angle of internal friction, δf = Friction angle at foundation base, δs = Friction angle at sides of skirt and Dr = Relative density in percent.

Additionally, Schneider and Senders [22] highlighted three ways that skirting foundations can fail. These three modes of failure are depicted in Fig. 13. The shallow failure mechanism is similar to the first mode. However, as demonstrated in Fig. 13a, the inclusion of skirts shifts the mechanism for shallow foundation collapse to deeper soil layers and potentially stronger soils. For deep skirts, another failure mechanism was seen. A flow round mechanism was shown in Fig. 13b at the skirt foundation's base. The base resistance (Qb=qb*Ab) and the resistance resulting from friction along the skirt's side (Qs=τfπDL). Where (τf) is the unit shaft friction along the skirted foundation's side, are the two sources for the bearing capacity of the skirted foundation in this failure type. When the internal soil and top plate of the skirted foundation are separated, as in Fig. 13c, a third potential mechanism was hypothesized. In this case, in addition to internal and external friction along the skin of the skirt, the end bearing is only produced on the skirt annulus.

Fig. 13
figure 13

Different mechanisms of failure for skirted foundations [22]

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Impact of load eccentricity on the maximum bearing capacity of circular footings with and without skirts

The ultimate bearing capacity ratio (BCRu), a non-dimensional measure, is the best way to describe how skirted and non-skirted footings differ in terms of their ultimate bearing capacity (BCRu). Figure 14 illustrates how using a skirt for various (L/D) ratios increased the ultimate bearing capacity ratio. For various amounts of load eccentricity (e/B) the BCRu is displayed against the corresponding (L/D) ratio. With an increase in skirt length, the BCRu was observed to rise. This is due to the horizontal soil reaction that was brought about on the outer skirt side, which improved footing stability. In general, the BCRu rose as the skirt length grew and peaked at a length. Where the (L/D) ratio was equal to 4. The rate of improvement then approached 4.22 times that of footing without a skirt.

Fig. 14
figure 14

Variation of bearing capacity ratio with increasing (L/D) ratio at different eccentricity points

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Referring to Saleh et al., [21]'s test results, they provided the empirical formula shown below to describe the impact of load inclination angle (Ө), load eccentricity (e/B), and skirt length on ultimate bearing capacity (qult) for strip footing:

$$qult = qult^\circ R \left( {1 - 0.03 \theta } \right) + X \left( \frac{L}{D} \right)$$
(5)

where qult: is the ultimate bearing capacity of the skirted footing, qulto: is the ultimate bearing capacity of the footing without skirt and subjected to vertical concentric load. R: is the eccentricity reduction factor for the ultimate bearing capacity and can be obtained from the relation between (e/B) ratio and the reduction ratio in bearing capacity ratio. And L/D is the ratio of skirts depth to footing width.

In this study the behavior of circular footing subjected to eccentric vertical load is studied, the load inclination is equal to zero. By using [21] equation with load inclination equals zero the equation will be in the following form:

$$qult = qult^\circ R + X \left( \frac{L}{D} \right)$$
(6)

where qult: is the ultimate bearing capacity of the skirted footing, qulto: is the ultimate bearing capacity of the footing without skirt and subjected to vertical concentric load. R: is the eccentricity reduction factor and L/D: is the ratio of skirts length to footing diameter.

At load inclination equals zero (Ө = 0) the factor (X) will be equal 0.3 as stated by Saleh et al., [21] and the reduction factor (R) can be obtained from Fig. 15.

Fig. 15
figure 15

Load eccentricity reduction factor for ultimate bearing capacity

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Substituting the results of the current study into equation No.6 the ultimate bearing capacity values (qult) were obtained at different eccentricities and different (L/D) ratios as presented in Table 4. The results obtained from the equation were then compared with the values of ultimate bearing capacity (qult*) from the experimental test results. A good agreement is found between the results of the experimental work and the results of the calculations using equation No.6 as illustrated in Table 4 especially for concentric footing. It is found that, in case of using a circular footing without skirts (L/D) ratio equals 0 the value of (qult) was equal 813.58, 691.5, 650.9 and 593.9 kN/m2 at (e/D) ratios equal 0, 0.08, 0.125 and 0.25 while, the values of (qult*) was found to be equal 813.58, 707.4, 707.4 and 459.8 kN/m2 at the same (e/D) ratios. Moreover, in case of using a skirted circular footing with (L/D) ratio equals 1 the value of (qult) was equal 816.6, 694.5, 653.9 and 596.9 kN/m2 at (e/D) ratios equal 0, 0.08, 0.125 and 0.25 while, the values of (qult*) was found to be equal 1061.2, 831.3, 831.3 and 459.9 kN/m2 at the same (e/D) ratios. In addition, in case of using a skirted circular footing with (L/D) ratio equals 4 there was also an agreement between the values of (qult) and (qult*) at large (e/D) ratios, the value of (qult) was equal 605.9, 597.8 and 586.9 kN/m2 at (e/D) ratios equal 0.25, 0.29 and 0.33 and the values of (qult*) was found to be equal 1045.5, 972.8 and 672.1 kN/m2 at the same (e/D) ratios. In the case of using centric vertical load or eccentric load with small eccentricity (e/D) equals 0.04, 0.08 and 0.125 the difference between (qult) and (qult*) turned out to be rather large, the value of (qult) was found to equal 826, 760.1, 703.5 and 662.9 kN/m2 at (e/D) ratios equal 0, 0.04, 0.08 and 0.125 and the values of (qult*) was found to be equal 2431 kN/m2 2, 2183.5, 1900.6 and 1175.4 kN/m2 at the same (e/D) ratios as can be seen in Table 4.

Table 4 Variation of ultimate bearing capacity value from mathematical equations and experimental results at different studied parameters
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From these results it can be concluded that, [21] equation can be modified to become the form mentioned in equation No. 6 in this research. Where it can be used to calculate the ultimate bearing capacity (qult) for the circular footing without and with skirts at different (L/D) and (e/D) values. But it was noted that, with the increase in the skirts length at small eccentricity values the difference between the results from the equation calculations and the experimental work was relatively high. So, it can be said that this equation is appropriate in the case of using skirted circular footing with high (L/D) ratios subjected to vertical eccentric load with high (e/D) values. While, in case of using circular footing without skirts or using skirted footing with small (L/D) ratios the equation was found to be appropriate for all cases of load eccentricity.

Modified equation to predict the ultimate capacity of circular footing with skirt

Based on statistical analysis of the experimental data, a modification factors are presented to modify [21] equation to predict the ultimate bearing capacity of circular footing with skirt under eccentric load. The modification factors taking effect of the skirt depth ratio (L/D) and the load eccentricity ratio (e/D). The equation ultimate bearing capacity of circular footing with skirt under eccentric load will be in the following form:

$$qult = \left( { qult^\circ R + X \left( \frac{L}{D} \right)} \right)F_{1} F_{2}$$
(7)

where \({F}_{1}\): is the magnification factor of the ultimate bearing capacity corresponding to the skirt depth ratio (L/D). \({F}_{2}\): is the eccentricity reduction factor for the ultimate bearing capacity corresponding to the load eccentricity ratio (e/D).

These factors can be obtained from the correlation between (L/B) ratio, (e/D) ratio and the modification factors of the ultimate bearing capacity \({F}_{1}\):and \({F}_{2}\):respectively as shown in Figs. 16 and 17. The eccentricity reduction factor has arranged about 0.60 to 1.00 corresponding to the load eccentricity ratio range from 0.00 to 0.33 respectively.

Fig. 16
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Correlation between magnification factor and the skirt depth ratio

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Fig. 17
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Correlation between reduction factor and load eccentricity ratio (e/D)

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Figure 18 represent correlation between the ultimate measured bearing capacity and the ultimate predicted bearing capacity of circular footing with/without skirt. The determination coefficient (R2) equal 0.99. There are a good agreement between and the ultimate predicted bearing capacity of circular footing with/without skirt. From these results it can be concluded that the proposed modified equation can be used to calculate the ultimate bearing capacity (qult) for the circular footing without and with skirts at different skirts length and different load eccentricity values.

Fig. 18
figure 18

Correlation between measured and predicted bearing capacities

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Conclusions

To understand the impact of skirts on footing settlement for load eccentricity inside and outside the footing core boundary, the behavior of eccentrically loaded circular footings supported on both unreinforced and reinforced sand with vertical skirts is investigated experimentally. The experimental investigation on the impact of skirts on footing settlement for load eccentricity inside and outside the footing core boundary has yielded significant findings, providing valuable insights into the behavior of eccentrically loaded circular footings on both unreinforced and reinforced sand with vertical skirts. The study reveals key conclusions that can greatly contribute to the understanding of foundation behavior under eccentric loading conditions.

Firstly, the findings unequivocally demonstrate that footing settlement increases with load eccentricity (e/D) compared to the case of no eccentricity (e = 0). This observation emphasizes the importance of considering eccentric loading scenarios in foundation design and analysis. Moreover, the use of vertical skirts proves to be a highly effective technique in mitigating settlement under eccentric loading conditions. This practical approach can significantly enhance the overall performance of foundations subjected to such loading conditions.

Secondly, the investigation highlights the influence of vertical skirts on reducing footing settlement. When compared to unsupported footings, the presence of skirts leads to substantial improvements in load-settlement behavior. Furthermore, the study reveals that deeper skirts result in even more favorable outcomes, with the maximum improvement observed at a skirt length (L/D) of 4. These findings underscore the potential benefits of using longer skirts to achieve more stable and reliable foundations.

Thirdly, the study delves into the ultimate load capacity of circular footings with and without skirts. As load eccentricity increases, footings without skirts experience a decrease in ultimate load capacity. However, the incorporation of vertical skirts with varying lengths connected to the footing bottom significantly enhances the ultimate load capacity under eccentric loads. This insight sheds light on the critical role of skirts in improving the load-carrying capacity of footings, even in challenging loading scenarios.

Additionally, the examination of different skirt lengths (L/D) provides valuable information on how to optimize footing designs for enhanced performance. Skirts with lengths of 1, 2.5, and 4 show considerable increases in the ultimate capacity of the footing when compared to unsupported footings at (e/D) = 0. Remarkably, higher eccentricity values yield even greater enhancements in ultimate capacity with longer skirts. These findings offer valuable guidelines for engineers in choosing suitable skirt configurations for specific loading conditions.

Finally, the study proposes a modified equation to predict the ultimate capacity of circular footings with skirts, which exhibits good agreement with the experimental results. This predictive model holds great promise for practical applications in estimating the behavior of skirt-supported footings under various loading scenarios. Engineers can benefit from this valuable tool in designing more robust and efficient foundations.