- Email: [email protected]
The influence of horizontal confinement on the bearing capacity factor Nγ of smooth strip footing
Computers and Geotechnics 61 (2014) 127–131
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
The influence of horizontal confinement on the bearing capacity factor Nc of smooth strip footing Lianheng Zhao, Feng Yang ⇑, Hancheng Dan School of Civil Engineering, Central South University, Changsha, Hunan 410075, China
a r t i c l e
i n f o
Article history: Received 3 September 2013 Received in revised form 1 April 2014 Accepted 22 May 2014
Keywords: Bearing capacity Smooth strip footing Horizontal confinement Upper-bound limit analysis Failure mechanism
a b s t r a c t We studied the upper-bound ultimate bearing capacity of smooth strip shallow footings with symmetrical and asymmetrical horizontal confinements on purely frictional sand within the framework of upperbound limit analysis. The subsoil follows the associated flow rule, and no surcharge on the soil surface is assumed. The contact between the soil and the horizontal confinement walls is assumed to be perfectly rough. The upper-bound solutions for the objective functions are obtained using nonlinear sequential quadratic programming. The results for the different internal friction angles u are provided in terms of the variation of two parameters, namely, the bearing capacity factor Nc and the correction factor of bearing capacity Kc, with respect to the change in the clear spacing between the edge of smooth footing and the rigid vertical walls. The values of Nc and Kc increase with u and decrease with the clear spacing between the edge of the smooth footing and the rigid vertical walls. Nc and Kc are more sensitive to this confining effect as u increases. The numerical results, a comparative analysis with the results from previous studies, and design charts are also included. Ó 2014 Published by Elsevier Ltd.
1. Introduction A number of studies on shallow isolated strip footings on either purely cohesive soil or frictional soil (Fig. 1) were recently reported [1–8]. However, shallow foundations are usually not placed on a semi-infinite solid bed. These foundations instead have space circumscription such as isolated footings with rigid walls at a finite distance on either side. This problem has recently received considerable attention, and the confinement effect on isolate footings was explored. For instance, Salençon [9–11] and Kumar et al. [12] studied some analytical solutions of the bearing capacity factor for rough footings with horizontal symmetrical confinements. Salençon [9] performed an upper-bound estimate on the bearing capacity for strip rough footing using the yield design theory. In his study, the strip footing acts on a soil foundation with rigid boundaries at a finite distance, where the soil is taken as both purely cohesive soil and frictional soil. In the analysis process, the effect of the horizontal confinement on the rough footing is modeled as a footing on a soil layer with limited thickness [13], which means that the thickness of the collapse zone will be reduced if the rigid walls are close enough to the footing. Salençon [10] revisited the bearing capacity of the strip footing within the ⇑ Corresponding author. Tel.: +86 135 4964 1242. E-mail address: [email protected] (F. Yang). http://dx.doi.org/10.1016/j.compgeo.2014.05.010 0266-352X/Ó 2014 Published by Elsevier Ltd.
scope of the yield design theory. The footing is placed on the purely cohesive soil, which is confined by rigid walls. The correction factors for the classical value of the bearing capacity were determined using upper-bound limit analysis by considering both a perfectly rough and a frictionless contact condition at the wall interfaces. These studies demonstrated that spacing and the interference effects have a significant influence on the bearing capacity of a rough footing. The bearing capacity is negatively correlated with the space between the footings and the horizontal confinements and is positively correlated with the soil friction angle and the friction contact condition at the wall interfaces. However, previous studies are limited regarding the ultimate bearing capacity of a smooth footing with horizontal confinements located at a limited distance from the footing. In this study, we investigate the effects of symmetrical and asymmetrical horizontal confinements on the ultimate bearing capacity of shallow smooth footings placed on purely frictional sand by using upper-bound limit analysis. The contact between the footing and the soil is assumed to be perfectly smooth. The contact between the soil and the asymmetrically horizontal confinement walls is assumed to be perfectly rough. The upper-bound solutions for the bearing capacity factor and the modification factor are obtained using nonlinear sequential quadratic programming. The effect of the friction angle of purely frictional sand on the results is examined. This paper extends the research scope with
128
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
Fig. 1. Rigid-block failure mechanism and the corresponding velocity field for a smooth strip footing with horizontal confinements.
respect to the ultimate bearing capacity of shallow smooth strip footings. Among some previous researches, the confinements are not taken into consideration, while it is incorporated in this paper. The bearing capacity charts are presented as an alternative way for facilitating the research verification. 2. Problem definition When the ultimate load is applied to the footing and the rigid footing moves downward vertically, the slip lines of the failure mechanism spread outward to a critical distance, Dcr, depending on the internal friction angle of the soil and the contact between the footing and the soil. If the horizontal confinement is located at a distance of L < Dcr, then the failure mechanism, the spread of the slip lines at the failure, and the ultimate bearing capacity of the footing are moderately affected. Fig. 1 shows the failure mechanism and its corresponding velocity field, including the variables used to investigate the horizontal confinement effects on the ultimate bearing capacity of a shallow smooth footing. This velocity field (for shallow smooth footing) is improved based on the mesh-like rigid blocks system (for shallow rough footing) (Yang and Yang [14], Yang et al. [15], Zhao [16], Zhao and Yang [17]). Under the assumption that the subsoil is purely sand and obeys the associated flow rule with no surcharge on the soil surface, the bearing capacity factor Nc can be given as:
Nc ¼
2qu cB
ð1Þ
where c = unit weight of sand, B = the width of the strip footing, and qu = the ultimate bearing capacity. The true distribution of the ultimate load is not uniform (for both symmetric and nonsymmetric problems), and the calculated ultimate bearing capacity qu is the upper bound to the average limit load.
Given the effects of the horizontal confinements on the smooth footing:
Nc ðuÞ ¼ K c
d1 d2 ; ; u N 0c ðuÞ B B
ð2Þ
where Nc ðuÞ is the bearing capacity factor of a smooth footing with confinement walls, N0c ðuÞ is the bearing capacity factor of an isolated smooth footing without confines, K c ðd1 =B; d2 =B; uÞ is the correction factor of the footing with confines relating to c (c – 0, q = 0, and c = 0), and all depend on the internal friction angle of the soil u and the ratios d1/B and d2/B. If d1 = d2 < Dcr, the smooth strip footing has a symmetrical horizontal confinement. If d1 < Dcr and d2 = Dcr, the smooth strip footing has a nonsymmetrical horizontal confinement. If d1 – d2, d1 < Dcr, and d2 ? 1, the smooth strip footing has a single-side horizontal confinement.
3. Comparison calculation for an isolated smooth strip footing The obtained values of N0 c for an isolated smooth footing with different values of internal friction angle (u) are compared with the analyses of Meyerhof [18], Booker [19], Bolton and Lau [20], Michalowski [2], Hjiaj et al. [7], and Martin [8]. Table 1 shows that the present N0 c values match those from the published numerical solutions. The accuracy of N0 c is closely related to the number of rigid blocks within the failure mechanism. A larger number of rigid blocks leads to a better value of N0 c. When the number of the rigid sliding blocks reaches to 2 30 20 = 1200, the numerical results of this approach closely approximate the exact solution derived by Martin [8]. The proposed rigid-block failure mechanism with an upper-bound solution is an effective method for approximating the exact solution of N0 c for an isolated smooth footing with more rigid sliding blocks.
Table 1 Comparison of N0 c values for an isolated smooth footing.
u (°) 5 10 15 20 25 30 35 40 45
Present method m = 20 n = 10
m = 30 n = 20
0.09685 0.3172 0.7920 1.7249 3.6933 8.1699 18.5886 45.7610 121.8589
0.09542 0.2925 0.74843 1.6178 3.5804 7.8101 17.9592 44.0799 120.2724
Martin [8]
Hjiaj [7]
Booker [19]
Meyerhof [18]
Bolton and Lau [20]
Michalowski [2]
0.0844649 0.280879 0.699096 1.57862 3.46108 7.65300 17.5771 43.1866 117.576
0.0888 0.2910 0.7187 1.6215 3.5536 7.8504 17.9890 44.0967 119.4376
0.149 0.336 0.757 1.704 3.836 8.636 19.443 43.775 98.557
0.035 0.183 0.565 1.435 3.383 7.834 18.576 46.845 131.371
0.09 0.29 0.71 1.6 3.51 7.74 17.8 44 120
0.127 0.423 1.050 2.332 5.02 10.918 24.749 60.215 164.308
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
4. The influence of symmetrical confinement walls on Nc and Kc Fig. 2 shows the values of Nc and Kc for an isolated smooth footing with symmetrical confinement walls with d1 =B ¼ d2 =B increasing from 0.0 to infinity and u from 20° to 45°. The figure shows that when d1 =B ¼ d2 =B P Dcr =B, Kc = 1.0, the horizontal confinement walls have no influence on Nc. When 0 6 d1 =B ¼ d2 =B < Dcr =B, K c ðd1 =B;d2 =B; uÞ > 1, N c ðuÞ > N 0c ðuÞ, the values of Nc and Kc remarkably increase with a decrease in the space between the footing and the horizontal confinements.
129
5. The influence of asymmetrical confinement walls on Nc and Kc 5.1. Single-side confinement wall Fig. 3 shows Nc and Kc for an isolated smooth footing with unilateral confinement wall with increasing d1 =B from 0.0 to infinity and u from 20° to 45°. The figure also shows that the confinement walls have no effect on the bearing capacity of the footing for a single-side confinement
Fig. 2. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for symmetrical confinement walls (d1/B = d2/B).
Fig. 3. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for single-side confinement wall.
130
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
Fig. 4. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for non-symmetrical confinement walls (d2/B = Dcr/B).
and when d1 =B P Dcr =B, Nc = N0 c, and Kc = 1.0. When 0 < d1 =B < Dcr =B, the values of Nc and Kc increase non-linearly as d1/B decreases. When d1/B = 0.0, the magnitude of Nc becomes exactly equal to 2 N0 c (Kc = 2.0) for any u. Therefore, the ultimate bearing capacity of the footing becomes exactly twice that of an isolated footing without confinement walls. 5.2. Nonsymmetrical confinement walls Fig. 4 shows Nc and Kc for an isolated smooth footing limited by asymmetrical confinement walls with d2 =B ¼ Dcr =B, d1 =B increasing from 0.0 to Dcr =B and u increasing from 20° to 45°. The figure also shows that when d1 =B P Dcr =B; d2 =B ¼ Dcr =B, a smooth footing can be treated as an isolated foundation without any confinement, and Nc = N0 c, Kc = 1.0 for all values of u. When 0 6 d1 =B < Dcr =B; d2 =B ¼ Dcr =B, the values of Nc and Kc increase nonlinearly as d1/B decreases. A constant Kc with d1 =B ¼ 0 for different u is then obtained: (i) Kc = 2.39 for u = 45°, (ii) Kc = 2.34 for u = 40°, (iii) Kc = 2.29 for u = 35°, (iv) Kc = 2.24 for u = 30°, and (v) Kc = 2.18 for u = 20°.
(2) Isolated smooth footing with unilateral confinement walls. When d1/B P Dcr/B, Nc = N0 c, and Kc = 1.0 and if no gap between the confinement and the footing exists, the magnitude of Kc becomes exactly equal to 2.0. The ultimate bearing capacity of the footing becomes exactly twice that of an isolated footing without confinement walls. (3) Isolated smooth footing with asymmetrical confinement walls. When d1 P Dcr, Nc = N0 c, and Kc = 1.0 for all values of u, the confinement walls have no effect on the bearing capacity of the footing. When 0 6 d1 =B < Dcr =B;d2 =B ¼ Dcr =B, the values of Nc and Kc increase nonlinearly with a decrease in d1/B. A constant Kc with d1 =B ¼ 0 for different u is then obtained. Acknowledgments The writers are grateful to X. Liu, T. Zhang and Y.P. Li for their assistance. The present work was sponsored by the National Natural Science Foundation of China (Nos. 51208522, U1134207) and the Postdoctoral Science Foundation of China (No. 2012T50708). The financial support is greatly appreciated.
6. Conclusions References The bearing capacity of a smooth footing, which is confined by symmetrical or nonsymmetrical rigid walls, is found within the framework of the upper-bound limit analysis. The subsoil comprises purely frictional sand and obeys the associated flow rule given that no surcharge is loaded on the soil surface. For a different friction angle of u, the results are provided in terms of the variation between two parameters, namely, the bearing capacity factor Nc and the correction factor of bearing capacity Kc with respect to a change in the clear spacing between the smooth footing edge and the rigid vertical walls. The values of Nc and Kc increase with u and decrease with the clear spacing between the smooth footing and the rigid vertical walls. Nc and Kc also prove to be more sensitive to the confining effect as u increases. (1) Isolated smooth footing with symmetrical confinement walls. The values of Nc and Kc increase continuously with u but decrease with d1/B (= d2/B) until d1 =B ¼ d2 =B P Dcr =B .
[1] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. [2] Michalowski RL. An estimate of the influence of soil weight on bearing capacity using limit analysis. Soils Found 1997;37(4):57–64. [3] Soubra AH. Upper-bound solutions for bearing capacity of foundations. J Geotech Geoenviron Eng (ASCE) 1999;125(1):59–68. [4] Zhu DY. The least upper-bound solution for the bearing capacity factor Nc. Soils Found 2000;40(1):123–9. [5] Kumar J, Bhoi MK. Interference of multiple strip footings on sand using small scale model tests. Geotech Geol Eng 2008;26:469–77. http://dx.doi.org/ 10.1007/s10706-008-9175-6. [6] Ukritchchon B, Whittle AW, Klangvijit C. Calculations of bearing capacity factor Nc using numerical limit analyses. J Geotech Geoenviron Eng (ASCE) 2003;129(7):468–74. [7] Hjiaj M, Lyamin AV, Sloan SW. Numerical limit analysis solutions for the bearing capacity factor Nc. Int J Solids Struct 2005;42:1681–704. [8] Martin CM. Exact bearing capacity calculations using the method of characteristics. In: Proceedings of the 11th International Conference of IACMAG, Turin, vol. 4; 2005. p. 441–50. [9] Salençon J. The influence of confinement on the bearing capacity of strip footings. Comptes Rendus Mecanique 2002;330:319–26. [10] Salençon J. Confining effect on the bearing capacity of circular footings on a purely cohesive soil. Comptes Rendus Mecanique 2002;330:521–5.
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131 [11] Salençon J. Bearing capacity of strip footings with horizontal confinement. Comptes Rendus Mecanique 2003;331:319–24. [12] Kumar J. Nc for rough strip footing using the method of characteristics. Canad Geotech J 2003;40(3):669–74. [13] Mandel J, Salençon J. Force portante d’un sol sur une assise rigide (étude théorique). Géotechnique 1972;22(1):79–93. [14] Yang F, Yang JS. A revised failure mechanism of strip footings for upper bound solution. Electron J Geotech Eng 2008;F:1–17. [15] Yang F, Yang JS, Fu LL. Rigid block upper bound limit analysis method for determination of ultimate bearing capacity factor Nc. In: Proceedings of 17th National Conference on Structural Engineering; 2008.
131
[16] Zhao LH. Energy analysis study on slope stability and reinforcing design. Changsha: PhD thesis of Central South University; 2009. [17] Zhao LH, Yang F. Construction of improved rigid blocks failure mechanism for ultimate bearing capacity caculation based on slip-line field theory. J Central South Univ Technol 2013;20(4):1047–57. [18] Meyerhof GG. Some recent research on the bearing capacity of foundations. Canad Geotech J 1963;1(1):16–26. [19] Booker JR. Applications of theories of plasticity to cohesive frictional soils. Ph D thesis, Sydney University; 1969. [20] Bolton MD, Lau CK. Vertical bearing capacity factors for circular and strip footings on Mohr–Coulomb soil. Canad Geotech J 1993;30(6):1024–33.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
The influence of horizontal confinement on the bearing capacity factor Nc of smooth strip footing Lianheng Zhao, Feng Yang ⇑, Hancheng Dan School of Civil Engineering, Central South University, Changsha, Hunan 410075, China
a r t i c l e
i n f o
Article history: Received 3 September 2013 Received in revised form 1 April 2014 Accepted 22 May 2014
Keywords: Bearing capacity Smooth strip footing Horizontal confinement Upper-bound limit analysis Failure mechanism
a b s t r a c t We studied the upper-bound ultimate bearing capacity of smooth strip shallow footings with symmetrical and asymmetrical horizontal confinements on purely frictional sand within the framework of upperbound limit analysis. The subsoil follows the associated flow rule, and no surcharge on the soil surface is assumed. The contact between the soil and the horizontal confinement walls is assumed to be perfectly rough. The upper-bound solutions for the objective functions are obtained using nonlinear sequential quadratic programming. The results for the different internal friction angles u are provided in terms of the variation of two parameters, namely, the bearing capacity factor Nc and the correction factor of bearing capacity Kc, with respect to the change in the clear spacing between the edge of smooth footing and the rigid vertical walls. The values of Nc and Kc increase with u and decrease with the clear spacing between the edge of the smooth footing and the rigid vertical walls. Nc and Kc are more sensitive to this confining effect as u increases. The numerical results, a comparative analysis with the results from previous studies, and design charts are also included. Ó 2014 Published by Elsevier Ltd.
1. Introduction A number of studies on shallow isolated strip footings on either purely cohesive soil or frictional soil (Fig. 1) were recently reported [1–8]. However, shallow foundations are usually not placed on a semi-infinite solid bed. These foundations instead have space circumscription such as isolated footings with rigid walls at a finite distance on either side. This problem has recently received considerable attention, and the confinement effect on isolate footings was explored. For instance, Salençon [9–11] and Kumar et al. [12] studied some analytical solutions of the bearing capacity factor for rough footings with horizontal symmetrical confinements. Salençon [9] performed an upper-bound estimate on the bearing capacity for strip rough footing using the yield design theory. In his study, the strip footing acts on a soil foundation with rigid boundaries at a finite distance, where the soil is taken as both purely cohesive soil and frictional soil. In the analysis process, the effect of the horizontal confinement on the rough footing is modeled as a footing on a soil layer with limited thickness [13], which means that the thickness of the collapse zone will be reduced if the rigid walls are close enough to the footing. Salençon [10] revisited the bearing capacity of the strip footing within the ⇑ Corresponding author. Tel.: +86 135 4964 1242. E-mail address: [email protected] (F. Yang). http://dx.doi.org/10.1016/j.compgeo.2014.05.010 0266-352X/Ó 2014 Published by Elsevier Ltd.
scope of the yield design theory. The footing is placed on the purely cohesive soil, which is confined by rigid walls. The correction factors for the classical value of the bearing capacity were determined using upper-bound limit analysis by considering both a perfectly rough and a frictionless contact condition at the wall interfaces. These studies demonstrated that spacing and the interference effects have a significant influence on the bearing capacity of a rough footing. The bearing capacity is negatively correlated with the space between the footings and the horizontal confinements and is positively correlated with the soil friction angle and the friction contact condition at the wall interfaces. However, previous studies are limited regarding the ultimate bearing capacity of a smooth footing with horizontal confinements located at a limited distance from the footing. In this study, we investigate the effects of symmetrical and asymmetrical horizontal confinements on the ultimate bearing capacity of shallow smooth footings placed on purely frictional sand by using upper-bound limit analysis. The contact between the footing and the soil is assumed to be perfectly smooth. The contact between the soil and the asymmetrically horizontal confinement walls is assumed to be perfectly rough. The upper-bound solutions for the bearing capacity factor and the modification factor are obtained using nonlinear sequential quadratic programming. The effect of the friction angle of purely frictional sand on the results is examined. This paper extends the research scope with
128
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
Fig. 1. Rigid-block failure mechanism and the corresponding velocity field for a smooth strip footing with horizontal confinements.
respect to the ultimate bearing capacity of shallow smooth strip footings. Among some previous researches, the confinements are not taken into consideration, while it is incorporated in this paper. The bearing capacity charts are presented as an alternative way for facilitating the research verification. 2. Problem definition When the ultimate load is applied to the footing and the rigid footing moves downward vertically, the slip lines of the failure mechanism spread outward to a critical distance, Dcr, depending on the internal friction angle of the soil and the contact between the footing and the soil. If the horizontal confinement is located at a distance of L < Dcr, then the failure mechanism, the spread of the slip lines at the failure, and the ultimate bearing capacity of the footing are moderately affected. Fig. 1 shows the failure mechanism and its corresponding velocity field, including the variables used to investigate the horizontal confinement effects on the ultimate bearing capacity of a shallow smooth footing. This velocity field (for shallow smooth footing) is improved based on the mesh-like rigid blocks system (for shallow rough footing) (Yang and Yang [14], Yang et al. [15], Zhao [16], Zhao and Yang [17]). Under the assumption that the subsoil is purely sand and obeys the associated flow rule with no surcharge on the soil surface, the bearing capacity factor Nc can be given as:
Nc ¼
2qu cB
ð1Þ
where c = unit weight of sand, B = the width of the strip footing, and qu = the ultimate bearing capacity. The true distribution of the ultimate load is not uniform (for both symmetric and nonsymmetric problems), and the calculated ultimate bearing capacity qu is the upper bound to the average limit load.
Given the effects of the horizontal confinements on the smooth footing:
Nc ðuÞ ¼ K c
d1 d2 ; ; u N 0c ðuÞ B B
ð2Þ
where Nc ðuÞ is the bearing capacity factor of a smooth footing with confinement walls, N0c ðuÞ is the bearing capacity factor of an isolated smooth footing without confines, K c ðd1 =B; d2 =B; uÞ is the correction factor of the footing with confines relating to c (c – 0, q = 0, and c = 0), and all depend on the internal friction angle of the soil u and the ratios d1/B and d2/B. If d1 = d2 < Dcr, the smooth strip footing has a symmetrical horizontal confinement. If d1 < Dcr and d2 = Dcr, the smooth strip footing has a nonsymmetrical horizontal confinement. If d1 – d2, d1 < Dcr, and d2 ? 1, the smooth strip footing has a single-side horizontal confinement.
3. Comparison calculation for an isolated smooth strip footing The obtained values of N0 c for an isolated smooth footing with different values of internal friction angle (u) are compared with the analyses of Meyerhof [18], Booker [19], Bolton and Lau [20], Michalowski [2], Hjiaj et al. [7], and Martin [8]. Table 1 shows that the present N0 c values match those from the published numerical solutions. The accuracy of N0 c is closely related to the number of rigid blocks within the failure mechanism. A larger number of rigid blocks leads to a better value of N0 c. When the number of the rigid sliding blocks reaches to 2 30 20 = 1200, the numerical results of this approach closely approximate the exact solution derived by Martin [8]. The proposed rigid-block failure mechanism with an upper-bound solution is an effective method for approximating the exact solution of N0 c for an isolated smooth footing with more rigid sliding blocks.
Table 1 Comparison of N0 c values for an isolated smooth footing.
u (°) 5 10 15 20 25 30 35 40 45
Present method m = 20 n = 10
m = 30 n = 20
0.09685 0.3172 0.7920 1.7249 3.6933 8.1699 18.5886 45.7610 121.8589
0.09542 0.2925 0.74843 1.6178 3.5804 7.8101 17.9592 44.0799 120.2724
Martin [8]
Hjiaj [7]
Booker [19]
Meyerhof [18]
Bolton and Lau [20]
Michalowski [2]
0.0844649 0.280879 0.699096 1.57862 3.46108 7.65300 17.5771 43.1866 117.576
0.0888 0.2910 0.7187 1.6215 3.5536 7.8504 17.9890 44.0967 119.4376
0.149 0.336 0.757 1.704 3.836 8.636 19.443 43.775 98.557
0.035 0.183 0.565 1.435 3.383 7.834 18.576 46.845 131.371
0.09 0.29 0.71 1.6 3.51 7.74 17.8 44 120
0.127 0.423 1.050 2.332 5.02 10.918 24.749 60.215 164.308
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
4. The influence of symmetrical confinement walls on Nc and Kc Fig. 2 shows the values of Nc and Kc for an isolated smooth footing with symmetrical confinement walls with d1 =B ¼ d2 =B increasing from 0.0 to infinity and u from 20° to 45°. The figure shows that when d1 =B ¼ d2 =B P Dcr =B, Kc = 1.0, the horizontal confinement walls have no influence on Nc. When 0 6 d1 =B ¼ d2 =B < Dcr =B, K c ðd1 =B;d2 =B; uÞ > 1, N c ðuÞ > N 0c ðuÞ, the values of Nc and Kc remarkably increase with a decrease in the space between the footing and the horizontal confinements.
129
5. The influence of asymmetrical confinement walls on Nc and Kc 5.1. Single-side confinement wall Fig. 3 shows Nc and Kc for an isolated smooth footing with unilateral confinement wall with increasing d1 =B from 0.0 to infinity and u from 20° to 45°. The figure also shows that the confinement walls have no effect on the bearing capacity of the footing for a single-side confinement
Fig. 2. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for symmetrical confinement walls (d1/B = d2/B).
Fig. 3. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for single-side confinement wall.
130
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131
Fig. 4. Correction bearing capacity factor Kc and bearing capacity factor Nc against d1/B and soil friction angle u for non-symmetrical confinement walls (d2/B = Dcr/B).
and when d1 =B P Dcr =B, Nc = N0 c, and Kc = 1.0. When 0 < d1 =B < Dcr =B, the values of Nc and Kc increase non-linearly as d1/B decreases. When d1/B = 0.0, the magnitude of Nc becomes exactly equal to 2 N0 c (Kc = 2.0) for any u. Therefore, the ultimate bearing capacity of the footing becomes exactly twice that of an isolated footing without confinement walls. 5.2. Nonsymmetrical confinement walls Fig. 4 shows Nc and Kc for an isolated smooth footing limited by asymmetrical confinement walls with d2 =B ¼ Dcr =B, d1 =B increasing from 0.0 to Dcr =B and u increasing from 20° to 45°. The figure also shows that when d1 =B P Dcr =B; d2 =B ¼ Dcr =B, a smooth footing can be treated as an isolated foundation without any confinement, and Nc = N0 c, Kc = 1.0 for all values of u. When 0 6 d1 =B < Dcr =B; d2 =B ¼ Dcr =B, the values of Nc and Kc increase nonlinearly as d1/B decreases. A constant Kc with d1 =B ¼ 0 for different u is then obtained: (i) Kc = 2.39 for u = 45°, (ii) Kc = 2.34 for u = 40°, (iii) Kc = 2.29 for u = 35°, (iv) Kc = 2.24 for u = 30°, and (v) Kc = 2.18 for u = 20°.
(2) Isolated smooth footing with unilateral confinement walls. When d1/B P Dcr/B, Nc = N0 c, and Kc = 1.0 and if no gap between the confinement and the footing exists, the magnitude of Kc becomes exactly equal to 2.0. The ultimate bearing capacity of the footing becomes exactly twice that of an isolated footing without confinement walls. (3) Isolated smooth footing with asymmetrical confinement walls. When d1 P Dcr, Nc = N0 c, and Kc = 1.0 for all values of u, the confinement walls have no effect on the bearing capacity of the footing. When 0 6 d1 =B < Dcr =B;d2 =B ¼ Dcr =B, the values of Nc and Kc increase nonlinearly with a decrease in d1/B. A constant Kc with d1 =B ¼ 0 for different u is then obtained. Acknowledgments The writers are grateful to X. Liu, T. Zhang and Y.P. Li for their assistance. The present work was sponsored by the National Natural Science Foundation of China (Nos. 51208522, U1134207) and the Postdoctoral Science Foundation of China (No. 2012T50708). The financial support is greatly appreciated.
6. Conclusions References The bearing capacity of a smooth footing, which is confined by symmetrical or nonsymmetrical rigid walls, is found within the framework of the upper-bound limit analysis. The subsoil comprises purely frictional sand and obeys the associated flow rule given that no surcharge is loaded on the soil surface. For a different friction angle of u, the results are provided in terms of the variation between two parameters, namely, the bearing capacity factor Nc and the correction factor of bearing capacity Kc with respect to a change in the clear spacing between the smooth footing edge and the rigid vertical walls. The values of Nc and Kc increase with u and decrease with the clear spacing between the smooth footing and the rigid vertical walls. Nc and Kc also prove to be more sensitive to the confining effect as u increases. (1) Isolated smooth footing with symmetrical confinement walls. The values of Nc and Kc increase continuously with u but decrease with d1/B (= d2/B) until d1 =B ¼ d2 =B P Dcr =B .
[1] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier; 1975. [2] Michalowski RL. An estimate of the influence of soil weight on bearing capacity using limit analysis. Soils Found 1997;37(4):57–64. [3] Soubra AH. Upper-bound solutions for bearing capacity of foundations. J Geotech Geoenviron Eng (ASCE) 1999;125(1):59–68. [4] Zhu DY. The least upper-bound solution for the bearing capacity factor Nc. Soils Found 2000;40(1):123–9. [5] Kumar J, Bhoi MK. Interference of multiple strip footings on sand using small scale model tests. Geotech Geol Eng 2008;26:469–77. http://dx.doi.org/ 10.1007/s10706-008-9175-6. [6] Ukritchchon B, Whittle AW, Klangvijit C. Calculations of bearing capacity factor Nc using numerical limit analyses. J Geotech Geoenviron Eng (ASCE) 2003;129(7):468–74. [7] Hjiaj M, Lyamin AV, Sloan SW. Numerical limit analysis solutions for the bearing capacity factor Nc. Int J Solids Struct 2005;42:1681–704. [8] Martin CM. Exact bearing capacity calculations using the method of characteristics. In: Proceedings of the 11th International Conference of IACMAG, Turin, vol. 4; 2005. p. 441–50. [9] Salençon J. The influence of confinement on the bearing capacity of strip footings. Comptes Rendus Mecanique 2002;330:319–26. [10] Salençon J. Confining effect on the bearing capacity of circular footings on a purely cohesive soil. Comptes Rendus Mecanique 2002;330:521–5.
L. Zhao et al. / Computers and Geotechnics 61 (2014) 127–131 [11] Salençon J. Bearing capacity of strip footings with horizontal confinement. Comptes Rendus Mecanique 2003;331:319–24. [12] Kumar J. Nc for rough strip footing using the method of characteristics. Canad Geotech J 2003;40(3):669–74. [13] Mandel J, Salençon J. Force portante d’un sol sur une assise rigide (étude théorique). Géotechnique 1972;22(1):79–93. [14] Yang F, Yang JS. A revised failure mechanism of strip footings for upper bound solution. Electron J Geotech Eng 2008;F:1–17. [15] Yang F, Yang JS, Fu LL. Rigid block upper bound limit analysis method for determination of ultimate bearing capacity factor Nc. In: Proceedings of 17th National Conference on Structural Engineering; 2008.
131
[16] Zhao LH. Energy analysis study on slope stability and reinforcing design. Changsha: PhD thesis of Central South University; 2009. [17] Zhao LH, Yang F. Construction of improved rigid blocks failure mechanism for ultimate bearing capacity caculation based on slip-line field theory. J Central South Univ Technol 2013;20(4):1047–57. [18] Meyerhof GG. Some recent research on the bearing capacity of foundations. Canad Geotech J 1963;1(1):16–26. [19] Booker JR. Applications of theories of plasticity to cohesive frictional soils. Ph D thesis, Sydney University; 1969. [20] Bolton MD, Lau CK. Vertical bearing capacity factors for circular and strip footings on Mohr–Coulomb soil. Canad Geotech J 1993;30(6):1024–33.