1 Introduction

With the rapid process of urbanization, more superstructures would be built on slopes owing to the shortage of land and the complexity of terrain [19]. Generally, slopes always negatively affect the strip footings. In view of this, for deeper insight into how slopes affecting the footing stability, a number of methods including the semi-empirical method [22], limit equilibrium method [2, 13, 15], limit analysis method [5, 9, 29,30,31] and slip line method [7, 20] were used to predict the stability about the slope–footing system. In recent years, more researchers adopted numerical method to investigate this issue by dint of the rapid development of computational power [5, 6]. Many researchers investigated different types of slope–footing issues numerically, including cases with layered slope [11, 23, 24, 28, 34], seismic action [1, 3, 16, 17, 33] and spatially variable of soils [8, 35].

Abovementioned efforts offered deeper insight into this kind of topic; however, none of them took rock slopes into consideration. It is because the core theory of the most traditional numerical method is obtaining the P–S curve through the completed simulation process, by which the ultimate bearing capacity can be got. However, the rock is with nonlinear characteristics due to the uncertainty of rock in situ and natural discontinuities including the complicate joints and fractures. When considering rock nonlinearity in traditional numerical method, the intact P–S curve is hard to acquire, and the computing speed also would be very slow. Finite element limit analysis, as an advanced numerical method which collaborates the advantages of finite element method and limit analysis theory, can compute the ultimate bearing capacity efficiently without obtaining P–S curves. In view of this, the authors embedded GHB yield criterion into self-developed FELA code to achieve the nonlinearity of rock masses. Furthermore, footings are always subjected to wind force, earthquake action and the uneven load-bearing of granary, surface parking lots [4], which may cause eccentric load applying on the footing. The eccentric load not only impairs the footing stability, but also converts the failure mechanisms, which is necessary to be considered in stability studies. To the authors’ knowledge, there is no work discussing this topic.

In order to address this special problem, this study employed self-developed FELA code to investigate the stability of eccentrically loaded strip footing on rock slopes with nonlinearity. Numerous influential factors, especially the load eccentricity, were considered in the parametric study to investigate the combined influence of the slope and the eccentric load on the footing stability. Strict UB and LB results were acquired by the FELA code. Visualized results were presented to reveal how the potential failure patterns evolve with different parameters, and detailed design tables were presented to facilitate engineering practice. In general, the research method (FELA code with nonlinear yield criterion) is innovative, and the results can further ensure the safety of buildings near slopes, which has important research value.

2 Problem definition

Figure 1 graphically depicts the issues to be investigated under plane-strain condition. A strip footing of width B subjected to an eccentric load q locates on the rock slope. The load eccentricity is e, considering that the eccentric direction of the load is of great importance in this study. To accurately distinguish load with different eccentric direction, eccentricity toward the slope face is defined as negative, whereas the eccentricity to the opposite direction is defined as positive. The footing setback distance is L. The height of the slope is H which is set as 6B and the slope angle is β. Following previous investigations [3, 10, 23, 25, 26, 32], the side boundaries are fixed in horizontal, and the bottom boundary is fixed both vertically and horizontally, whereas the ground of the slope is totally free. The influence of strength σc and unit weight γ of the rock is incorporated in a widely used dimensionless factor σc/γB [12, 14, 27], which is set as 100, 500 for normal rock and + ∞ for weightless rock. The value of GSI (geological strength index is set as 10–80; the material constant of rock masses mi is set as 7, 10, 15, 17, 25 for carbonate rocks, lithified argillaceous rock, fine-grained polyminerallic igneous crystalline rocks and coarse-grained polyminerallic igneous rocks, respectively [29]. By previous researches [23, 27], the cases e/B = 0, 0.05, 0.1, 0.15 and cases of β = 15°, 30°, 45°, 60° are considered in this study for a comprehensive investigation of the slope–footing system. In addition, following previous studies [14, 21], a dimensionless bearing capacity factor N is used to assess the stability of footings more conveniently:

$$ N = \frac{q}{{\sigma_{c} }} = f\left( {\frac{{\sigma_{ci} }}{\gamma B},\frac{e}{B},{\text{GSI}},m_{i} ,\beta ,\frac{L}{B}} \right) $$
(1)

where the q is ultimate bearing capacity of the footing; e/B is dimensionless load eccentricity.

Fig. 1
figure 1

Problem definition

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3 Verifications of proposed models

The proposed numerical model with detailed mesh arrangement and model size is shown in Fig. 2. In this study, four iteration step number is adopted for optimal calculation precision and efficiency. Triangular element is adopted to establish numerical models. Initial and final number of elements is set as 1000 and 5000, respectively, which can guarantee the relative errors between UB and LB results less than 6%. The footing is set as rigid, and the interface between the footing and the ground cannot bear the tension force.

Fig. 2
figure 2

The sketch of the proposed FELA model

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Figure 3 shows the ultimate bearing capacity of footings on slopes obtained from UB solutions, LB solutions and previous study [18]. It can be seen that the errors between the previous study and the present study are small. The maximum relative error is 5.3% at the case of GSI = 15 and mi = 15, which is calculated by Eq. (2). Furthermore, the UB and LB results bracket the results of Saada et al. [18] tightly. In view of this, the average value of UB and LB results would be utilized in the results and discussions section to achieve higher precision.

$$ {\text{Relative error}} = \left| {\frac{{\left( {A - B} \right)}}{A + B}} \right| \cdot 100\% $$
(2)
Fig. 3
figure 3

Comparisons of ultimate bearing capacity of footings with previous study

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Figure 4 presents comparisons between the self-developed FELA code and existing commercial software (Plaxis for FE method and OptumG2 for FELA method) for cases of footings on rock slopes under eccentric load. Obviously, for cases with any GSI, mi and slope angles, the results of different method are pretty close to each other. The small relative errors between them are mainly caused by different mesh arrangement strategies. And considering the aforementioned strictly close results, the FELA model can be considered as reliable.

Fig. 4
figure 4

Comparisons of ultimate bearing capacity factor of eccentrically loaded footings on slopes with existing methods

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4 Results and discussions

The effects of main parameters (including the σc/γB, β, mi, GSI, and e/B) on the bearing capacity factor are shown in this section. The variation trends of bearing capacity factor are presented in Fig. 5 to investigate the differences between opposite load eccentricity on the bearing capacity factor. It should be noted that previous studies [11, 23] pointed out that for some cases of cohesive-friction soil slopes, the bearing capacity with slight load eccentricity can be greater than those with centric load due to the influence of soil slopes. However, it can be seen from Fig. 5 that for all cases, the maximum bearing capacity is at the e/B = 0, indicating that the strip footing obtains its optimal bearing capacity under the centric load. And the bearing capacity of footing under eccentric load toward the slope (negative load eccentricity) is smaller than the bearing capacity of footing under load with same eccentric distance and opposite direction (positive load eccentricity). The reason for this inequality phenomenon is that the slope is relatively more instable than the flat ground, and the load closer to the slope crest is more likely to induce the collapse of slopes. Furthermore, it also can be observed by comparing Fig. 5a and b; (c) and (d) that the above phenomenon would be more obvious with a greater slope angle, and the differences between the symmetrical cases become greater with the increasing absolute value of e/B. It indicates that the steeper slopes and greater absolute values of load eccentricity would intensify the inequality phenomenon. Figure 5e shows the effect of unit weight of rock masses, and the weightless rock is represented by σc/γB = + ∞. It is obvious that the greater the rock unit weight is, the greater the bearing capacity factor of the footing becomes. And the smaller σc/γB would make the inequality phenomenon more obvious.

Fig. 5
figure 5

Variation of N with different e/B and GSI for a σc/γB = 100, β = 15°, mi = 7; b σc/γB = 100, β = 45°, mi = 7; c σc/γB = 100, β = 15°, mi = 17; d σc/γB = 100, β = 45°, mi = 17; e β = 45°, mi = 7

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Figure 6 presents the effects of mi and load eccentricity on the bearing capacity. It can be seen that the hollow icons (representing the positive load eccentricity) are always higher than the solid icons (representing the negative load eccentricity), which verifies the aforementioned conclusion. And it can be seen that the bearing capacity factor increases with the increasing mi almost linearly. Hence, detailed design tables are presented in Tables 1, 2 and 3. By using the design tables, researchers can reckon the accurate bearing capacity of any mi from two known bearing capacity factors with different mi. And by comparing Fig. 6a and b, it can be seen that the greater GSI leads the bearing capacity factor increasing faster with an increasing mi, whereas Fig. 6b and c illustrate that the steeper slope would decrease the rising amplitude of the N with the increase in mi. Furthermore, the gaps between cases with the load eccentricity of same absolute values (depicted as the same shape icons) would be larger with the increase in load eccentricity, indicating that the inequality phenomenon is more obvious with a load of greater eccentricity.

Fig. 6
figure 6

Variation of N with different e/B and mi for a σc/γB = 100, GSI = 40, β = 30°; b σc/γB = 100, GSI = 70, β = 30°; c σc/γB = 100, GSI = 70, β = 60°

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Table 1 The bearing capacity factors for β = 15° and β = 30°
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Table 2 The bearing capacity factors for β = 45° and β = 60°
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Table 3 The critical distance L/B of centric loaded strip footings lying on rock slopes
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Figure 7 shows the variation trends of bearing capacity factors with different GSI. As expected, the greater GSI leads an increase in bearing capacity. And all the curves would be steeper with the increase in GSI, and the steeper angle means a faster increment speed of the bearing capacity. In addition, the differences between the same shape icons would be more distinct with a greater GSI, indicating that the greater GSI would intensify the inequality phenomenon. To investigate the effect of different mi on the inequality phenomenon, the cases with e/B = ± 0.15 and GSI = 80 in Fig. 7a and b are compared. The relative error of cases in Fig. 7a is 2.36%, whereas the relative difference of the case in Fig. 7b is 2.06%. It means that the increase in mi would slightly weaken the inequality phenomenon. By comparing the same cases of Fig. 7b and c, the relative difference is 2.06% for β = 30° and 2.81% for β = 60°, which illustrates that the steeper the rock slope is, the more obvious the inequality phenomenon becomes.

Fig. 7
figure 7

Variation of N with different e/B and GSI for a σc/γB = 100, mi = 10, β = 30°; b σc/γB = 100, mi = 25, β = 30°; c σc/γB = 100, mi = 25, β = 60°

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5 Failure mechanisms

In this section, the variation trends of failure patterns with different parameters are presented by the distribution of shear dissipation. Figure 8 presents the failure pattern of GSI = 50, mi = 17, σc/γB = 100, β = 30° with different load eccentricity. The failure pattern of centric load is shown in Fig. 8a. It can be seen that there is a failure curve extends to the slope face, which would lead an overall landslide of the slope. And another failure curve means the strip footing would contrarotate due to the slope. Figure 8b and c show the failure pattern of e/B = 0.05 and 0.15, respectively. It can be seen that the left failure curve in Fig. 8b becomes lighter, which means the anticlockwise rotation becomes unapparent. And with the further shift of the load, there only exists a single failure curve. This is because the rotation directions induced by the eccentric load and the slope are same. This failure pattern is the typical face failure, indicating that the footing would not be contrarotated. The failure patterns of e/B = − 0.05 and − 0.15 are presented in Fig. 8d and e. It can be seen that the failure pattern of Fig. 8d is similar to the failure pattern of centric load. However, the color at the slope crest in Fig. 8d is thicker than the color in Fig. 8a. It reasons that the collapse of the slope crest would be more dominant due to the load near the crest. Then, the slope crest would be further destructed with e/B = − 0.15. And the right failure curve of Fig. 8e becomes lighter, which means the greater load eccentricity would weaken the landslide of the rock slope. It should be noted that the numerous stress concentration at the slope crest would reduce the ultimate bearing capacity, which may be the main reason for the inequality phenomenon mentioned above. Furthermore, by comparison between Fig. 8a, c and e, it can be observed that the failure zone would move up with the increase in load eccentricity. It is because the greater load eccentricity would cause a more drastic rotation of the footing, which would induce shallow collapse of the rock masses.

Fig. 8
figure 8

UB failure patterns with GSI = 50, mi = 17, σc/γB = 100, β = 30° for a e/B = 0; b e/B = 0.05; c e/B = 0.15; d e/B = − 0.05; e e/B = − 0.15

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Figure 9 shows the effect of the slope angle on the failure patterns; thereinto, the failure patterns of different β with e/B = 0.15 are presented in Fig. 9a–d. The failure pattern for all cases with e/B = 0.15 are similar; namely, there is only single failure curve extending to the slope face from the footing. And it can be seen from Fig. 9a–d that the failure curve would move downward along the slope face with the increase in the slope angle until the failure pattern transforms from face failure (Fig. 9a–c) to toe failure (Fig. 9d). Furthermore, Fig. 9e and f illustrate that the form of failure curves of e/B = − 0.15 with β = 30° and β = 60° are alike, and the failure zone of steeper slope angle also extends more deeply. Based on abovementioned observation, it can be concluded that the variation of slope angle nearly has no effect on the form of failure curves, and the steeper slope angle would cause a deeper collapse.

Fig. 9
figure 9

UB failure patterns with GSI = 50, mi = 17, σc/γB = 100 for a e/B = 0.15, β = 15°; b e/B = 0.15, β = 30°; c e/B = 0.15, β = 45°; d e/B = 0.15, β = 60°; e e/B = − 0.15, β = 30°; f e/B = − 0.15, β = 60°

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The failure mechanisms of different GSI and mi are presented in Figs. 10 and 11 for different GSI and Figs. 12 and 13 for different mi. Figures 10 and 12 show that the failure patterns of e/B = 0.15 with different GSI and mi are all typical face failure, whereas all the failure patterns of e/B = − 0.15 are face failure with anticlockwise rotation (depicted in Figs. 11 and 13). This phenomenon illustrates that the variation of GSI and mi nearly has no effect on the failure pattern. Besides, it can be seen that the greater the GSI and mi is, the deeper the failure curve extends, indicating that the increase in GSI and mi would induce a deeper collapse of rock masses.

Fig. 10
figure 10

UB failure patterns with e/B = 0.15, mi = 15, σc/γB = 100, β = 30°for a GSI = 20; b GSI = 40; c GSI = 60; d GSI = 80

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Fig. 11
figure 11

UB failure patterns with e/B = − 0.15, mi = 15, σc/γB = 100, β = 30°for a GSI = 20; b GSI = 40; c GSI = 60; d GSI = 80

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Fig. 12
figure 12

UB failure patterns with e/B = 0.15, GSI = 50, σc/γB = 100, β = 30°for a mi = 7; b mi = 15; c mi = 17; d mi = 25

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Fig. 13
figure 13

UB failure patterns with e/B = − 0.15, GSI = 50, σc/γB = 100, β = 30°for a mi = 7; b mi = 15; c mi = 17; d mi = 25

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Figure 14 shows the failure mechanisms of different σc/γB with β = 45°, e/B = 0.15, GSI = 50 and mi = 15. It can be seen from Fig. 14a that the landslide only occurs above the middle slope. Then, with the increase in σc/γB, the slip line would extend more deeply, which is similar to aforementioned variation trends of GSI and mi. Based on it, a conclusion can be drawn is that the increase of the rock property parameters (GSI = 50, mi and σc/γB) can lead a deeper collapse of the rock slope, and the general form of failure curve wouldn’t change a lot with different parameters. Furthermore, the difference between failure pattern of σc/γB = 500 and σc/γB = + ∞ (weightless rock) is smsall, and Fig. 5e also indicates that the bearing capacity factors with σc/γB = 500 and σc/γB = + ∞ are approximate. Taking advantage of this feature, in engineering design, cases with relatively great σc/γB can be regarded as weightless rock to simplify the calculation.

Fig. 14
figure 14

UB failure patterns with e/B = 0.15, GSI = 50, mi = 15, σc/γB = 100, β = 45°for a σc/γB = 50; b σc/γB = 100; c σc/γB = 500; d σc/γB = + ∞

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In general, the farther footing setback distance would weaken the effect of the slope. And the failure pattern would become Prandtl-type failure once the slope no longer affects the footing. Figures 15 and 16 show the effect of footing setback distance of β = 15°and β = 45°with mi = 15, GSI = 50 and σc/γB = 100. It can be seen that the failure pattern of L/B = 5, e/B = 0 is face failure with anticlockwise rotation, which means the slope still has influence on the strip footing. Then, with the further increase in L/B, the failure pattern transforms to typical Prandtl-type failure (depicted in Fig. 15b). Hence, it can be predicted that the L/B = 5 is a critical state, beyond which the slope no longer affects the footing. Figure 15c shows the failure mechanism of L/B = 5 and e/B = − 0.15. It can be seen that the footing would contrarotate due to the negative eccentric load, and the failure curve would not extend to the slope face. By comparison between Fig. 15a and c, it can be predicted that the negative eccentric load would weaken the influence of the slope, corresponding to previous studies about the soil slope. Furthermore, previous study indicated that the positive load eccentricity would strengthen the connection between the footing and the slope. However, it can be seen in Fig. 15d that the strip footing would rotation clockwise, and the collapse would not spread to the slope face. And Fig. 16 also shows the similar variation trend of failure patterns, which illustrates that any eccentric load would weaken the influence of rock slopes on the footing. It reasons that the strength of rock masses is much better than the soil; therefore, the collapse induced by the clockwise rotation of the footing cannot extend to the slope surface.

Fig. 15
figure 15

UB failure patterns with σc/γB = 100, GSI = 50, mi = 15, β = 15°for a L/B = 5, e/B = 0; b L/B = 6, e/B = 0; c L/B = 5, e/B = − 0.15; d L/B = 5, e/B = 0.15

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Fig. 16
figure 16

UB failure patterns with σc/γB = 100, GSI = 50, mi = 15, σc/γB = 100, β = 45°for a L/B = 8, e/B = 0; b L/B = 9, e/B = 0; c L/B = 8, e/B = − 0.15; d L/B = 8, e/B = 0.15

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Figure 17a shows the failure mechanism of σc/γB = 100, GSI = 50, mi = 15, β = 45°, e/B = 0 and L/B = 8. As mentioned above, this Prandtl-type failure would transform to toe failure once the footing moves a little away from the slope. Hence, this critical condition can be taken as a benchmark to further investigate the effect of the rock property. Figure 17b–d show that the increases in σc/γB, GSI and mi make the failure pattern becomes toe failure from Prandtl-type failure, which illustrates that the increase in rock property parameter (σc/γB, GSI and mi) can lead a broader influence sphere of the slope.

Fig. 17
figure 17

UB failure patterns with β = 45°, e/B = 0 and L/B = 8 for a σc/γB = 100, GSI = 50, mi = 15; b σc/γB = + ∞, GSI = 50, mi = 15; c σc/γB = 100, GSI = 90, mi = 15; d σc/γB = 100, GSI = 50, mi = 30

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The critical distances of footings lying on rock slopes are summarized in Table 3 to facilitate the engineering practice. It can be seen that the greater the σc/γB, GSI and mi is, the greater the critical distance becomes, corresponding to the conclusion drawn above. If the footing setback distance is greater than the critical distance in Table 3, the failure pattern would be Prandtl-type failure, indicating that the slope no longer affects the footing. And if the footing setback distance is less than or equal to the critical distance, the failure pattern would be footing-slope combination failure, indicating that the slope has an effect on the footing bearing capacity.

6 Conclusions

This study embedded GHB nonlinear yield criterion into a self-developed FELA code to investigate a special topic, the stability of eccentrically loaded strip footing on rock slopes. The emphasis of this study is to comprehensively investigate the inequality phenomenon induced by the slope and eccentric load. Effects of different influential factors on the ultimate bearing capacity of footings were analyzed. Variation trends of potential failure patterns were summarized for revealing the internal failure mechanisms. Furthermore, detailed design tables were presented for engineering use. Several conclusions can be drawn after analysis:

  1. 1.

    In slope–footing system, any offset of load would weaken the footing stability. And the bearing capacity of footing under eccentric load toward the slope is smaller than the bearing capacity of footing under load with same eccentric distance and opposite direction. This inequality phenomenon would be more obvious with the decrease in σc/γB, mi, and increase in GSI load eccentricity and slope angle.

  2. 2.

    For cases with any load eccentricity, the bearing capacity of footings would increase with the increasing mi linearly.

  3. 3.

    The greater the σci/γB is, the smaller the bearing capacity factor becomes. And the effect of σci/γB would be negligible if the σci/γB greater than 500.

  4. 4.

    There are three typical failure patterns (face failure, toe failure, Prandtl-type failure) in the rock slope–footing system. The failure pattern would transform to toe failure from face failure with the increase in the slope angle. And the failure zone would be deeper with the increase in GSI, mi and σci/γB.

  5. 5.

    The increase in footing setback distance and load eccentricity would weaken the effect of slopes on the strip footing; inversely, the increase in GSI, mi and σci/γB would intensify the interaction between the slope and the footing.