-
Axi-symmetric Research Problem (Lower bound Method)
Table 1. Comparison of the values of for smooth and rough
footings from various approaches
Table 2. Comparison of the values of for smooth and rough
footings from various approaches
Table 3. Comparison of the values of for smooth and rough
footings from various approaches
Note: Values within and outside parentheses correspond to rough
and smooth foundations, respectively. a Lower-bound limit analysis
with FEs and linear programming obtained by using the proposed
formulation.
b Lower-bound limit analysis with FEs and linear programming
obtained by using an extended version of the Turgeman and
Pastor
(1982) formulation. c Stress characteristics method.
d Lower-bound limit analysis with FEs and linear
programming.
e Obtained by using FLAC 4.0.
Proposed formulation
a Turgeman and
Pastor(1982)b
Martin
(2004,2005)c
Kumar and
Khatri(2011)d
Erickson and
Drescher(2002)e
0 5.72(6.22) - 5.69(6.05) 5.61(6.01) -
5 7.52(8.40) 7.23(7.89) 7.43(8.06) 7.31(8.00) -
10 10.06(11.66) 9.63(10.80) 9.99(11.09) 9.78(10.99) -
15 14.17(16.81) 13.29(15.36) 13.87(15.84) 13.51(15.66) -
20 20.48(25.34) 18.99(22.68) 20.07(23.67) 19.38(23.22)
19.50(22.30)
25 31.07(40.21) 28.32(35.26) 30.52(36.17) 29.06(36.17) -
30 49.81(67.73) 44.65(58.49) 49.29(61.48) 47.10(61.48) -
35 85.34(122.85) - 85.88(112.47) 81.47(112.47) 84.00(108.00)
40 161.41(244.79) - 164.82(224.27) 153.94(224.27)
161.00(186.00)
45 331.10(537.32) - 358.81(501.74) 324.85(501.74)
320.00(380.00)
Proposed formulation
a Turgeman and
Pastor(1982)b
Martin
(2004,2005)c
Kumar and
Khatri(2011)d
Erickson and
Drescher(2002)e
5 1.66(1.73) 1.63(1.69) 1.65(1.71) 1.64(1.70) -
10 2.78(3.00) 2.70(2.90) 2.76(2.96) 2.72(2.94) -
15 4.78(5.49) 4.56(5.12) 4.72(5.25) 4.62(5.20) -
20 8.45(10.21) 7.91(9.26) 8.31(9.62) 8.05(9.45) -
25 15.49(19.75) 14.23(17.44) 15.23(18.40) 14.55(17.87) -
30 29.50(40.10) 26.78(34.77) 29.46(37.20) 28.20(36.50) -
35 61.21(86.69) - 61.13(80.81) 58.04(79.75) -
40 136.30(206.43) - 139.30(192.83) 130.17(189.19) -
45 332.52(538.35) - 359.81(521.31) 325.85(502.74) -
Proposed formulation
a Turgeman and
Pastor(1982)b
Martin
(2004,2005)c
Kumar and
Khatri(2011)d
Erickson and
Drescher(2002)e
5 0.06(0.09) 0.06(0.08) 0.06(0.08) 0.06(0.08) -
10 0.23(0.35) 0.20(0.30) 0.21(0.32) 0.20(0.30) -
15 0.57(0.94) 0.50(0.86) 0.53(0.93) 0.52(0.88) -
20 1.33(2.42) 1.14(2.21) 1.27(2.41) 1.23(2.27) 1.70(2.80)
25 3.06(6.39) 2.70(5.54) 2.97(6.07) 2.84(5.68) -
30 7.24(15.68) 6.28(13.81) 7.10(15.54) 6.72(14.65) -
35 18.14(41.98) - 18.02(41.97) 16.73(39.97) 21.00(45.00)
40 48.36(126.36) - 50.17(124.10) 45.36(116.20) 58.00(130.00)
45 144.38(392.33) - 160.01(419.47) 138.42(379.79)
186.00(456.00)
-
Figure 1. Mesh used in the analysis
Ni=16
N=4131
E=1377
Dc=2033
-
Figure 2. Plastic zones obtained from analysis with: (a) , ; (b)
, ; (c)
, ; (d) ,
/
/ /
/
(a) (b)
(c) (d)
-
Figure 3. Variation of in soil domain for : (a) , ; (b) , ;
(c)
, ; (d) ,
(a) (b)
)
(c) (d)
-
Shear stress and normal stress distribution below footing
Figure 4: The variation of (a) ; (b) along the footing-soil
interface
-14
-12
-10
-8
-6
-4
-2
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0r/B
=0.1 =0.5 =1.0
-300
-200
-100
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r/B
Present results (Smooth footing)
Present results (Rough footing)
(a)
(b)
-
Axi-symmetric Research Problem (Upper bound Method)
Table 4. Comparison of the values of for smooth and rough
footings from various approaches
Table 5. Comparison of the values of for smooth and rough
footings from various approaches
Table 6. Comparison of the values of for smooth and rough
footings from various approaches
Note: Values within and outside parentheses correspond to rough
and smooth foundations, respectively. a Upper-bound limit analysis
with FEs and linear programming obtained by using the proposed
formulation.
b Upper-bound limit analysis with FEs and linear programming
obtained by using Haar & Von Karman(1909) hypothesis
c Upper-bound limit analysis with FEs and linear programming
obtained by using an extended version of the Turgeman and
Pastor
(1982) formulation. d Stress characteristics method.
e Lower-bound limit analysis with FEs and linear
programming.
f Obtained by using FLAC 4.0.
Proposed formulation
a Kumar and
Chakrabortyb
Turgeman and
Pastor(1982)c
Martin
(2004,2005)d
Kumar and
Khatri(2011)e
Erickson and
Drescher(2002)f
0 5.94(6.36) 5.78(6.16) 5.74(6.07) 5.69(6.05) 5.61(6.01) -
5 8.33(8.41) 7.58(8.11) 7.52(8.09) 7.43(8.06) 7.31(8.00) -
10 11.10(11.60) 10.19(11.18) 10.11(11.18) 9.99(11.09)
9.78(10.99) -
15 15.95(16.63) 14.19(16.10) 14.06(16.05) 13.87(15.84)
13.51(15.66) -
20 21.96(24.82) 20.65(24.24) 20.52(24.16) 20.07(23.67)
19.38(23.22) 19.50(22.30)
25 32.78(39.68) 31.68(38.62) 31.25(37.97) 30.52(37.31)
29.06(36.17) -
30 52.15(67.25) 51.77(65.65) 50.15(63.44) 49.29(62.70)
47.10(61.48) -
35 92.85(128.30) 91.24(120.40) 89.55(118.52) 85.88(113.99)
81.47(112.47) 84.00(108.00)
40 191.89(258.35) 176.77(245.80) 185.80(246.50) 164.82(228.62)
153.94(224.27) 161.00(186.00)
45 424.30(620.55) 397.38(596.02) 408.23(609.42) 358.81(520.30)
324.85(501.74) 320.00(380.00)
Proposed formulation
a Kumar and
Chakrabortyb
Turgeman and
Pastor(1982)c
Martin
(2004,2005)d
Kumar and
Khatri(2011)e
Erickson and
Drescher(2002)f
5 1.73(1.77) 1.68(1.74) 1.65(1.73) 1.65(1.71) 1.64(1.70) -
10 2.95(3.12) 2.84(3.03) 2.80(3.00) 2.76(2.96) 2.72(2.94) -
15 5.27(5.46) 4.89(5.44) 4.81(5.39) 4.72(5.25) 4.62(5.20) -
20 8.99(10.05) 8.66(9.99) 8.62(9.89) 8.31(9.62) 8.05(9.45) -
25 17.28(20.50) 16.08(19.25) 15.98(19.04) 15.23(18.40)
14.55(17.87) -
30 34.10(41.82) 31.31(39.36) 31.15(38.98) 29.46(37.20)
28.20(36.50) -
35 69.01(90.83) 66.13(85.27) 66.16(85.19) 61.13(80.81)
58.04(79.75) -
40 169.33(223.94) 147.90(210.98) 164.56(219.00) 139.30(192.83)
130.17(189.19) -
45 423.62(643.55) 405.65(610.74) 415.23(621.46) 359.81(521.31)
325.85(502.74) -
Proposed Formulation
a Kumar and
Chakrabortyb
Turgeman and
Pastor(1982)c
Martin
(2004,2005)d
Kumar and
Khatri(2011)e
Erickson and
Drescher(2002)f
5 0.09(0.13) 0.08(0.12) 0.07(0.10) 0.06(0.08) 0.06(0.08) -
10 0.37(0.43) 0.33(0.40) 0.28(0.38) 0.21(0.32) 0.20(0.30) -
15 0.73(1.15) 0.69(1.08) 0.61(1.01) 0.53(0.93) 0.52(0.88) -
20 1.65(2.86) 1.51(2.72) 1.47(2.71) 1.27(2.41) 1.23(2.27)
1.70(2.80)
25 3.98(6.94) 3.42(6.78) 3.22(7.04) 2.97(6.07) 2.84(5.68) -
30 8.86(18.86) 8.32(17.54) 8.29(18.19) 7.10(15.54) 6.72(14.65)
-
35 23.69(54.41) 21.60(48.24) 21.78(51.52) 18.02(41.97)
16.73(39.97) 21.00(45.00)
40 69.36(168.30) 62.73(147.85) 66.78(158.64) 50.17(124.10)
45.36(116.20) 58.00(130.00)
45 241.27(559.68) 226.50(525.42) 237.97(543.42) 160.01(419.47)
138.42(379.79) 186.00(456.00)
-
Figure 5: Nodal velocity patterns, along with a zoomed-in view
around the footing edge, for computing : (a) ; (b)
Scale:--=100V0
Scale:--=100V0
u=v=0 u
=0
Zoomed View
Zoomed View
u=v=0
u=
0
(a)
(b)
-
Figure 6: Nodal velocity patterns, along with a zoomed-in view
around the footing edge, for computing : (a) ; (b)
Scale:--=100V0
Zoomed View
Zoomed View
Scale:--=100V0
u=
0
u=v=0
u=v=0 u
=0
(b)
(a)
