In the strict framework of limit analysis, an analytical approach is derived to obtain the upper bound solutions for threedimensional inhomogeneous slopes in clays under undrained conditions. Undrained strength profiles increasing linearly with depth below the crest of the slope and below the outline surface of the slope are assumed representative of cut slopes and natural slopes, respectively. Stability charts are produced for the cut slopes and the natural slopes under both static and pseudostatic seismic loading conditions. The presented charts are convenient to assess the preliminary and shortterm stability for 3D slopes in practical applications, such as rapid excavation or buildup of embankments and slopes subjected to earthquakes. Compared with the available results from the finite element limit analysis method, a better estimate of the slope safety is obtained from the analytical approach.
Stability charts for slopes provide an efficient tool for the preliminary assessment of slope safety and for the calibration of any sophisticated numerical models that are ultimately used for solving more complex slope stability problems. The development of stability charts has been the subject of many investigations since the pioneering work of Taylor [
For homogeneous slopes, stability charts have been derived for twodimensional (2D) slope failures [
Depth contours and undrained strength profile assumed representative of (a) a cut slope and (b) a natural slope (adopted from the study of Li et al., 2010).
In the strict framework of limit analysis, Michalowski and Drescher [
In the present paper, the analytical approach originated by Michalowski and Drescher [
The 3D rotational failure mechanism proposed by Michalowski and Drescher [
3D failure mechanisms for undrained slopes: (a) torustype failure surface; (b) failure surface with limited width
Based on the abovementioned 3D mechanism, an upper bound to the stability number
Since the work rate done by soil weight is independent of the undrained strength of soil, the expressions of
Unlike the work rates
For a cut slope, the energy dissipation rate
For a natural slope, it becomes
According to the balance equation (
Figures
Static stability charts for cut slopes. (a)
Static stability charts for natural slopes. (a)
It can be seen from Figures
In the numerical limit analysis of Li et al. [
Comparisons between upper bounds to the stability number derived from this study and those presented by Li et al. (2009, 2010) for undrained slopes with
Comparisons between analytical upper bound results obtained from this study and the numerical upper and lower bound solutions presented by Li et al. (2009, 2010) for (a) cut slopes and (b) natural slopes with
For cut slopes and natural slopes subjected to seismic excitation, a set of stability charts are presented in Figures
Seismic stability charts for cut slopes with
Seismic stability charts for cut slopes with
Seismic stability charts for cut slopes with
Seismic stability charts for natural slopes with
Seismic stability charts for natural slopes with
Seismic stability charts for natural slopes with
According to the abovederived stability charts (Figures
For comparison purposes, the same example slope as analyzed numerically by Li et al. [
Comparisons between calculated results from the analytical approach of this study and the numerical method of Li et al. (2009, 2010) for the example slope.



2D  

This study  Numerical result  This study  Numerical result  This study  Numerical result  This study  Numerical result  
Cut slopes 

9.33  10.25 
8.62  9.3 
7.92  9.0 
7.19  7.8 

1.68  1.85 
1.55  1.68 
1.43  1.62 
1.30  1.41 



Natural slopes 

8.21  9.1^{†}  7.59  8.25^{†}  7.16  7.75^{†}  6.51  6.3^{†}(6.6^{‡}) 

1.48  1.64^{†}  1.37  1.49^{†}  1.29  1.40^{†}  1.17  1.14^{†}(1.19^{‡}) 
Alternatively, an analytical approximation of the curves in the stability charts can be made as
Coefficients





 










15°  1.5  14.3103  11.5429  10.7045  8.0734  8.5028  6.0669  6.7166  4.1510 
2.0  11.9647  10.6815  8.6907  7.4181  6.8300  5.5594  5.6032  4.3982  
3.0  9.9572  9.9457  6.9418  6.9535  5.3955  5.1557  4.3198  4.0343  
5.0  8.6752  9.4006  5.8925  6.5406  4.4115  4.8650  3.4263  3.7688  
10.0  7.8768  9.0236  5.2417  6.2664  3.7832  4.6572  2.8651  3.5592  

7.2769  8.7267  4.6914  6.0842  3.2662  4.5086  2.2893  3.3425  


30°  1.5  9.6768  6.6362  8.0970  5.3987  6.8890  4.4735  6.0354  3.7967 
2.0  8.7703  6.3038  7.0969  5.1453  5.9931  4.2410  5.2151  3.5279  
3.0  7.7745  6.0632  6.2268  4.9168  5.1412  4.0708  4.3496  3.3635  
5.0  7.1536  5.8459  6.2268  4.9168  4.4777  3.9215  3.4970  3.2897  
10.0  6.7290  5.7188  5.5593  4.7623  4.0284  3.8400  2.9978  3.2242  

6.4288  5.5573  4.7424  4.5944  3.5700  3.8015  2.3854  3.2131  


45°  1.5  8.1155  4.8325  7.1212  4.1478  6.3609  3.5965  5.7601  3.0794 
2.0  7.5187  4.6165  6.5129  3.9617  5.7087  3.4071  5.0744  2.9303  
3.0  6.9890  4.4338  5.8712  3.8236  5.0805  3.2883  4.3923  2.8372  
5.0  6.5002  4.3138  5.3344  3.7402  4.4343  3.2333  3.5826  2.8273  
10.0  6.1923  4.2501  4.9194  3.7136  4.0087  3.2056  3.1355  2.8042  

5.9501  4.1510  4.6053  3.6623  3.6720  3.1776  2.6632  2.8153  


60°  1.5  7.0929  3.7568  6.4416  3.3219  6.0455  2.8950  5.5240  2.5619 
2.0  6.5673  3.6151  5.8267  3.2500  5.3624  2.7979  4.8504  2.4610  
3.0  6.1872  3.4545  5.4323  3.0604  4.9564  2.6698  4.1890  2.4129  
5.0  5.7905  3.3836  5.0814  2.9724  4.2369  2.6748  3.6481  2.3730  
10.0  5.5322  3.3267  4.7003  2.9604  3.9028  2.6499  3.2809  2.3613  

5.3239  3.2561  4.3973  2.9348  3.5675  2.6394  2.8129  2.3864  


75°  1.5  6.1642  3.0140  5.7583  2.6924  5.4741  2.3745  4.9336  2.1758 
2.0  5.7688  2.8644  5.2573  2.5894  4.7992  2.3368  4.4029  2.0869  
3.0  5.3404  2.7583  4.8569  2.4881  4.3137  2.2430  3.9827  2.0093  
5.0  5.0461  2.6674  4.5502  2.3975  4.0368  2.1564  3.7437  1.9028  
10.0  4.8551  2.6220  4.3280  2.3516  3.7027  2.1488  3.4788  1.8779  

4.5987  2.5899  4.1375  2.3088  3.3854  2.1423  3.2900  1.8375  


90°  1.5  5.4109  2.4119  5.1349  2.1369  4.7327  2.0014  4.4387  1.8664 
2.0  4.9769  2.2792  4.6642  2.0652  4.3330  1.8852  4.0007  1.7250  
3.0  4.5868  2.1895  4.1953  2.0087  3.8523  1.8380  3.5886  1.6332  
5.0  4.2956  2.1136  3.9367  1.9290  3.6329  1.7508  3.2815  1.6085  
10.0  4.0946  2.0605  3.7224  1.8910  3.3992  1.7088  3.0884  1.5669  

3.8672  2.0165  3.5075  1.8475  3.1401  1.6933  2.8899  1.5300 
Coefficients





 










15°  1.5  14.0417  5.5213  10.7765  3.9044  8.3917  3.0683  6.8037  2.5367 
2.0  12.1295  5.5304  8.8251  3.9860  6.8320  3.1272  5.5596  2.5655  
3.0  10.2958  5.4513  7.1932  4.0178  5.5081  3.1519  4.3663  2.6045  
5.0  8.9213  5.4354  6.1444  3.9802  4.5560  3.1486  3.5573  2.5891  
10.0  8.1138  5.3815  5.4981  3.9474  3.9846  3.1224  3.0156  2.5757  

7.4668  5.3353  4.9707  3.9254  3.5103  3.1140  2.5252  2.5857  


30°  1.5  9.8257  3.3207  8.2197  2.7972  6.9813  2.4186  6.0906  2.1291 
2.0  9.0010  3.2809  7.2876  2.8136  6.0977  2.4466  5.2739  2.1343  
3.0  7.9860  3.2971  6.4485  2.8280  5.2963  2.4598  4.3928  2.1959  
5.0  7.3384  3.2848  5.7426  2.8412  4.5894  2.5033  3.6805  2.2373  
10.0  6.8809  3.2821  5.3106  2.8429  4.1576  2.4996  3.2194  2.2420  

6.5250  3.2611  4.9572  2.8385  3.7456  2.5202  2.7498  2.2779  


45°  1.5  8.2142  2.6961  7.1962  2.4184  6.3768  2.1509  5.7165  1.9296 
2.0  7.6745  2.6371  6.6418  2.3650  5.7948  2.1230  5.1580  1.8964  
3.0  7.0647  2.6149  6.0183  2.3402  5.1551  2.1331  4.4334  1.9262  
5.0  6.5959  2.6024  5.4339  2.3723  4.5574  2.1465  3.7592  1.9745  
10.0  6.2847  2.5785  5.0647  2.3742  4.1360  2.1727  3.3140  2.0039  

6.0276  2.5560  4.7538  2.3714  3.8115  2.1781  2.8561  2.0462  


60°  1.5  7.1388  2.4670  6.4687  2.2474  5.9017  2.0296  5.4226  1.8260 
2.0  6.6483  2.4103  5.9534  2.1969  5.3860  1.9818  4.8393  1.7918  
3.0  6.1799  2.3493  5.5046  2.1364  4.9641  1.9215  4.2803  1.7737  
5.0  5.8480  2.2981  5.1576  2.0883  4.3155  1.9593  3.6896  1.8075  
10.0  5.6342  2.2414  4.7757  2.0956  3.9675  1.9572  3.3408  1.8128  

5.3836  2.2417  4.4778  2.0925  3.6433  1.9643  2.8754  1.8553  


75°  1.5  6.2502  2.4269  5.7615  2.2425  5.4851  1.9719  4.9201  1.8396 
2.0  5.7408  2.3540  5.2850  2.1560  4.8077  1.9557  4.3905  1.7873  
3.0  5.3571  2.2359  4.8675  2.0673  4.3504  1.8950  3.9983  1.6894  
5.0  5.0488  2.1735  4.5430  1.9894  4.0446  1.8197  3.7401  1.6327  
10.0  4.8306  2.1313  4.3531  1.9368  3.7432  1.8107  3.4934  1.5955  

4.6297  2.0899  4.1505  1.8962  3.4157  1.7986  3.3170  1.5582  


90°  1.5  5.4109  2.4119  5.1349  2.1369  4.7327  2.0014  4.4387  1.8664 
2.0  4.9769  2.2792  4.6642  2.0652  4.3330  1.8852  4.0007  1.7250  
3.0  4.5868  2.1895  4.1953  2.0087  3.8523  1.8380  3.5886  1.6332  
5.0  4.2956  2.1136  3.9367  1.9290  3.6329  1.7508  3.2815  1.6085  
10.0  4.0946  2.0605  3.7224  1.8910  3.3992  1.7088  3.0884  1.5669  

3.8672  2.0165  3.5075  1.8475  3.1401  1.6933  2.8899  1.5300 
For the above slope example, the factor of safety
Factors of safety for the example slope.





2D  



1.84  1.67  1.55  1.44  1.36  1.30 

1.61  1.49  1.38  1.29  1.24  1.18  




1.59  1.43  1.30  1.21  1.11  1.04 

1.43  1.31  1.20  1.12  1.04  0.97  




1.42  1.25  1.14  0.97  0.90  0.82 

1.27  1.16  1.06  0.92  0.85  0.78  




1.26  1.09  0.94  0.81  0.73  0.63 

1.15  1.02  0.90  0.78  0.71  0.61 
Based on the 3D kinematically admissible rotational failure mechanism, an analytical approach is derived for the upper bound limit analysis of the stability of cut slopes and natural slopes under shortterm undrained conditions. Compared with the finite element limit analysis method adopted by Li et al. [
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.