3-BEARING CAPACITY OF SOILS

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1 3-BEARING CAPACITY OF SOILS INTRODUCTION The soil must be capable of carrying the loads from any engineered structure placed upon it without a shear failure and with the resulting settlements being tolerable for that structure. The evaluation, of the limiting shear resistance or ultimate bearing capacity qult of the soil under a foundation load, depends on the soil strength perimeters. The recommendation for the allowable bearing capacity qa to be used for design is based on the minimum of either 1. Limiting the settlement to a tolerable amount.. The ultimate bearing capacity, which considers soil strength, where qa qult / F. Where F is the factor of safety, which typically is taken as 3 for the bearing capacity of shallow foundations. Probably, one of two potential mode of failure may occur when a footing loaded to produce the ultimate bearing capacity of the soil. a. Rotate about some center of rotation (probably along the vertical line oa) with shear resistance developed along the perimeter of the slip zone shown as a circle b. Punch into the ground as the wedge agb. a. Footing on 0 Soil Upper Bound Solution 0 0 Lower Bound Solution At point o, left side element 1 and adjacent right side element, the horizontal stresses should be equal 3, 1 1, 45 For 0, tan 45 1 then For the case of 0, i.e. foundation on ground surface, the average value is 45. pg. 19

2 b. Footing on φ-c Soil pg. 0

3 BEARING CAPACITY EQUATIONS Terzaghi Bearing-Capacity Equation One of the early sets of bearing-capacity equations was proposed by Terzaghi (1943) for Shallow (D ) strip footing. Terzaghi's equations were produced from a slightly modified bearing-capacity theory developed by Prandtl (190) but Terzaghi used shape factors.. Where C cohesion of soil angle of internal friction of soil B least lateral dimension of footing (diameter for round footing). unit weight of soil (use buoyant (submerged) unit weight for soil below water table. D a. cos (45 /) Shape Factor ( 1) Strip 1 Round 1.3 Square 1.3 Rectangle 1 0.3B/L B/L Conclusions 1. The ultimate bearing capacity increases with the depth of footing.. The ultimate baring capacity of cohesive soil ( 0)is independent of footing size; i.e., at ground level qult 5.7 c 3. The ultimate bearing capacity of a cohesionless soil(c0) is directly dependent on the footing size, but the depth of footing is more important than size. Meyerhof s Bearing-Capacity Equation Meyerhof (1951, 1963) proposed a bearing-capacity equation similar to that of Terzaghi but included a shape factor Sq with the depth term Nq. He also included depth factors di, and inclination factors ii, for cases where the footing load is inclined from the vertical. Up to a depth of D «B the Meyerhof qult is not greatly different from the Terzaghi value. The difference becomes more pronounced at larger D/B ratios... Vertical load Inclined load pg. 1

4 ( ) tan(1.4) Shape, depth and inclination factors Hansen's Bearing-Capacity Method Hansen (1970) proposed the general bearing-capacity case and N factor equations. This equation is readily seen to be a further extension of the earlier Meyerhof work. The extensions include base factors for situations in which the footing is tilted from the horizontal bi and for the possibility of a slope of the ground supporting the footing to give ground factors gi. The Hansen equation implicitly allows any D/B and thus can be used for both shallow and deep foundation. When 0 (. ) ( tan() su undrained 0 ultimate shear strength. ) Vesic's Bearing-Capacity Equations The Vesic (1973, 1975) procedure is essentially the same as the method of Hansen (1961) with select changes. The Nc and Nq terms are those of Hansen but N is slightly different. There are also differences in the ii, bi, gi terms. The Vesic equation is somewhat easier to use than Hansen's because Hansen uses the / terms in computing shape factors si whereas Vesic does not.. ( ) tan() pg.

5 pg. 3

6 Which Equations to Use The bearing capacity can be obtained often using empirical SPT or CPT data directly to a sufficient precision for most projects. The Terzaghi equations, being the first proposed, have been very widely used because of their greater ease of use. They are only suitable for a concentrically loaded footing on horizontal ground. Both the Meyerhof and Hansen methods are widely used. The Vesic method has not been much used Use Terzaghi Hansen, Meyerhof, Vesic Hansen, Vesic Best for Very cohesive soils where D/B < 1 or for a quick estimate of quit to compare with other methods. Don t use for footings with moments and/or horizontal forces or for tilted bases and/or sloping ground. Any situation that applies, depending on preference or familiarity with a particular method. When base is tilted; when footing is on a slope or when D/B > 1. pg. 4

ADDITIONAL CONSIDERTIONS 1. A reduction factor is applicable for the third term of B effect as follows; r 1-0.5 ( ) for B.

7 ADDITIONAL CONSIDERTIONS 1. A reduction factor is applicable for the third term of B effect as follows; r ( ) for B. Which φ The soil element beneath the centerline of a strip footing is subjected to plane strain loading, and therefore, the plain strain friction angle must be used in calculating its bearing capacity, the plane strain friction angle can be obtained from a plane strain compression test. The loading condition of a soil element along the vertical centerline of a square or circular footing more closely resembles axisymmetric loading than plane strain loading, thus requiring a triaxial friction loading, which can be determined from a consolidated drained or undrained triaxial compression test. Meyerhof proposed the corrected friction angle for use with rectangular footing as: ( ) Example: Compute the allowable bearing pressure using the Terzaghi equation for the footing and soil parameters shown. Use a safety factor of 3 to obtain qa? Critical Thinking: F3 is recommended for cohesive soil Solution: Find the bearing capacity. (B not known but D is) From equations and tables Nc 17.7 Nq7.4 Nϒ5.0 Sc 1.3 Sϒ x 17.7 x x 1. x x B x 17.3 x 5.0 x B kpa qa qult / F. qa B)/ B Likely B ranges from 1.5 to 3m, with B3m qa x kpa qa x 3 x kpa r log( ) 0.95 consider qa 0 kpa pg. 5

EFFECT OF WATER TABLE ON BEARING CAPACITY The effective unit weight of the soil is used in the bearing-capacity equations for computing the ultimate capacity, in calculation for q in the qnq term and

. When the water table lies within the wedge zone, if B is known, computing the average effective unit weight to be used in the 0.5 BN term will be as follows ( ) ( ) Where H 0.

8 EFFECT OF WATER TABLE ON BEARING CAPACITY The effective unit weight of the soil is used in the bearing-capacity equations for computing the ultimate capacity, in calculation for q in the qnq term and in the term 0.5yBNy. 1. When the water table is below the wedge zone [depth approximately 0.5B tan (45 /)], the water table effects can be ignored for computing the bearing capacity.. When the water table lies within the wedge zone, if B is known, computing the average effective unit weight to be used in the 0.5 BN term will be as follows ( ) ( ) Where H 0.5 B tan (45 /) dw depth of water table below base of footing wet unit weight of soil in depth dw submerged unit weight of soil below water table Example: for the square footing shown, what is the allowable bearing capacity? use F, Assume the soil above water table is wet with normal water content wn 10% and Gs.68 Critical Thinking: F is recommended for cohesionless soil Effective unit weight should be calculated Solution: ( B/L) tri 35 H 0.5 B tan (45 /) 0.5x.5xtan (4535/).4 m dw Water table lies within the wedge zone / (9.81).68(9.81) The saturated unit weight is the dry weight weight of water in voids (9.81) 0.1 / ( ) ( ) ( ) ( ) / pg. 6

9 To calculate q in the qnq term use q / In the 0.5 BN term use / 1. Bearing Capacity by Terzaghi s equations 35 Nc 57.8 Nq 41.4 r log(.. Nϒ 4.4 ) x x.5x14.85x4.4x0.8x kn/m qa qult / 1438/719 kn/m. Bearing Capacity by Meyerhof s equations 35 Nc 46.1 Nq 33.3 Nϒ Sq s 10.1Kp B/L 10.1x3.7x11.37 d q d x qult 19.91x33.3x1.37x x.5x14.85x37.1x1.37x1.08x kn/m qa qult / 000/1000 kn/m 3. Bearing Capacity by Hansen s equations qa qult / 154/76 kn/m BEARING CAPACITY FOR FOOTINGS ON LAYERED SOILS There are three general cases of the footing on a layered soil as follows: Case 1. Footing on layered clays (all 0). a. Top layer weaker than lower layer (c1 < c) b. Top layer stronger than lower layer (c1> c) CR strength ratio If CR When 0.7 CR 1 IF CR Reduce the aforesaid 0.5 & by 10% pg. 7

improvement method Case. Footing on layered -c soils with a, b same as case 1.

10 & Notice: When the top layer is very soft with a small d 1/B ratio, one should give consideration either to placing the footing deeper onto the stiff clay or to using some kind of soil improvement method Case. Footing on layered -c soils with a, b same as case 1. A modified angle of friction and soil cohesion may be used to determine the bearing capacity when the depth of the top layer( ) under the base of foundation is less than(h). Calculate H ; H 0.5 B tan (45 /) IF H > d1 then compute ( ) ( ) For case 1 and,if the top layer is soft the calculated bearing capacity should not be more than qult 4c1 q Case3. Footing on layered sand and clay soils. a. Sand overlying clay b. Clay overlying sand IF H > d1 then 1. Find qult based on top-stratum soil parameters using one of bearing capacity equations.. Assume a punching failure bounded by the base perimeter of dimensions B x L. Here include the q contribution from d1, and compute q ult of the lower stratum using the base dimension B. 3. Compare qult to q'ult and use the smaller value. Example: A footing of B 3, L 6 m is to be placed on a two-layer clay deposit as shown. Estimate the ultimate bearing Capacity using Hansen s equation? Critical Thinking: Footing on layered caly soil 0 pg. 8

11 Solution: Check H ; H 0.5 B tan (45 /) H0.5 x 3 tan m > (d11. m) > 1 77 Nc x3/ N c x3/ ( ) qult 6 x 77 (1 (0.x3/6) (0.4x1.83/3) ) 17.6 x1.83 x1x1 65 kpa BEARING CAPACITY FOR FOOTINGS WITH ECCENRIC LOADING Two methods can be followed to solve this case Method 1. Use either the Hansen or Vesic bearing-capacity equation with the following adjustments: a. Use B' in the ybny term. b. Use B' and L' in computing the shape factors. c. Use actual B and L for all depth factors. qa qult/f and Pa qab'l' Method. Use the Meyerhof general bearing-capacity equation and a reduction factor Re used as quit, des quit, comp x Re Re 1 - e/b (cohesive soil) Re 1 - e/b (cohesionless soil and for 0 < e/b < 0.3) Alternatively, one may directly use the Meyerhof equation with B' and L used to compute the shape and depth factors and B used in the 0.5 B'N term. pg. 9

12 Example: A square footing is 1.8 X 1.8 m with a 0.4 X 0.4 m square column. It is loaded with an axial load of 1800 kn and Mx 450 kn m; My 360 kn m. Undrained triaxial tests (soil not saturated) give 36 and c 0 kpa. The footing depth D 1.8 m; the soil unit weight y kn/m3; the water table is at a depth of 6.1 m from the ground surface. What is the allowable soil pressure, if F 3.0, using the Hansen bearing-capacity equation with B', L', Meyerhof's equation; and the reduction factor Re? Solution: ex My/V 360/ < B/6 ey Mx/V 450/ < L/6 Bmin 4ex wx 4x <1.8 Lmin 4ey wy 4x <1.8 B B - ex 1.8-x0. 1.4m L L - ey 1.8-x m Hansen s bearing-capacity equation Nc 51 Nq 38 N 40 Compute shape factors using B, L sc 1 (Nq/Nc) (B /L ) 1.69 sq 1 ( B /L ) sin 1.55 s 1-0.4B /L 0.6 Compute depth factors using B, L dc 10.4D/B 1.4 dq 1tan (1-sin )D/B1.5 d 1. 0x51x1.69x1.4 18x1.8x38x1.55x x1.4x18x40x0.6x kpa qa 5100/3 1700kpa The actual soil pressure is 1800/ (1.4x1.3) 989 kpa pg. 30

Meyerhof s equation Kp tan (45 /) 3.85 Nc 51 Nq38 N 44 sc 1 (0.Kp B/L) 1.77 sq s 1 (0.1Kp B/L) 1.39 dc 1 (0. Kp D/B) 1.39 dq d 1 (0.1 Kp D/B) 1.0 qult 0x51x1.77x1.39 18x1.8x38x1.39x1. 0.5x18x1.8x44x1.

8) 555 kpa Re 1- (e/b) cohesionless soil (This can be used since the cohesion of the soil is low (0) Rex 1- (e/b) 1-0./1.8 0.67 Rey 1- (e/b) 1-0.5/1.8 0.63 qult 575x0.67x0.

13 Meyerhof s equation Kp tan (45 /) 3.85 Nc 51 Nq38 N 44 sc 1 (0.Kp B/L) 1.77 sq s 1 (0.1Kp B/L) 1.39 dc 1 (0. Kp D/B) 1.39 dq d 1 (0.1 Kp D/B) 1.0 qult 0x51x1.77x x1.8x38x1.39x1. 0.5x18x1.8x44x1.39x kpa Re1-(e/B) cohesive soil Rex1-(x0./1.8) 0.78 Rey 1-(x0.5/1.8) 0.7 qult 575x0.78x kpa qa 330/ kpa The actual soil pressure is 1800/ (1.8x1.8) 555 kpa Re 1- (e/b) cohesionless soil (This can be used since the cohesion of the soil is low (0) Rex 1- (e/b) 1-0./ Rey 1- (e/b) 1-0.5/ qult 575x0.67x kpa qa48/3 809 kpa BEARING CAPACITY FOR FOOTINGS WITH INCLINED LOAD Inclined loads are produced when the footing is loaded with both a vertical V and a horizontal component(s) Hi of loading (refer to Table 4-5 and its figure). This loading is common for many industrial process footings where horizontal wind loads are in combination with the gravity loads. In any case the load inclination results in a bearing capacity reduction over that of a foundation subjected to a vertical load only. Safety Factor Against Sliding The footing must be stable against both sliding and bearing in case it subjected to a horizontal load component, for sliding the general equation for the factor of safety is; Example: The data obtained by the Load test is Vult 1060 kn HL ult 38 kn Find the ultimate bearing capacity by Hansen and Vesic s equations? Critical Thinking: Change triaxial to plane strain pg. 31

14 Solution: plane strain ( B/L) triaxial ( x 0.5/)43 47 Hansen s Bearing Capacity Equation From Equations ; Nq 187 N 300. dqb 1 tan (1-sin ) D/B dql 1 tan (1-sin ) D/L 1.04 d B d L 1 iqb (1-(0.5HB / (VAf Ca Cot ))).5 1 i B (1-(0.7HB / (V Af Ca Cot )))3.5 1 Horizontal force parallel to B 0 iql (1-(0.5HL / (VAf Ca Cot ))).5 (1 0.5x 38/1060) i L (1-(0.7HL / (VAf Ca Cot ))) SqB 1 sin B iqb / L 1.18 SqL 1 sin L iql /B.78 S B B i B /( L i L) 0.73 S L L i L /( B i B) 0.4 Use 0.6 both should be equal or more than 0.6 Substitute the factors obtained for each direction in Hansen s equation and take the smaller ultimate bearing capacity qult 9.43 x 0.5x187x1.18x1.16x1 0.5x0.5x9.43x300x0.73x1x kpa qult 9.43 x 0.5x187x.78x1.04x x.0x9.43x300x0.6x1x kpa Vesic s Bearing Capacity Equation. From Equations Nq 187 N 404 Sq 1 B/L tan 1 (0.5/) tan S B/L (0.5/) OK dq 1 tan (1 sin ) k 1.16 d 1 m( (L/B)) / (1(L/B)) iql (1-(HL / (VAf Ca Cot )))m (1 38/1060) i L (1-(HL / (VAf Ca Cot )))m qult 9.43 x 0.5x187x1.7x1.16x x0.5x9.43x404x0.9x1x kpa pg. 3

15 BEARING CAPACITY OF FOOTINGS ON SLOPES Example: A x m square footing has a ground slope of 0 and base slope of 10. Are the footing dimensions adequate for the given load with a factor of safety 3? Use Hansen s and Vesic s equations Critical Thinking: Assume angle of friction between concrete and soil ( ) 5 Base adhesion Ca 0.6 to 1 x C 5 kpa Neglect passive earth pressure for D 0.3 m Solution: Factor of safety against sliding must be checked 600 tan Hansen s Bearing Capacity Equation From table Nc 0.7 Nq 10.7 N 6.8. dc D/B 1.06 dq 1 tan (1-sin ) D/B 1.05 d 1 iqb (1-(0.5HB / (VAf Ca Cot ))) i B (1-((0.7- /450)HB / (VAf Ca Cot ))) i L 1 ic iq (1-iq)/(Nq-1) ScB 1 (Nq/Nc)( B iqb / L) 1.39 SqB 1 sin B iqb / L 1.85 S B B i B / L i L rad bcb 1 / bqb exp( - tan ) b B exp( -.7 tan ) 0.80 Ground factors gi 1 for 0 qult 5x0.7x1.39x1.06x0.641x x0.3x10.7x1.85x1.05x0.675x x17.5xx6.8x0.808x1x0.481x kpa Pa A x qa x x 515/3 686 kn > 600 OK pg. 33

BEARING CAPACITY FROM IN SITUE SPT (1 0.33 ) ( (1 0.33 ) 0.3 1. F1 F ) (1 0.33 ) N55 0.05 0.08 N 70 0.04 0.06 > 1. qa allowable bearing pressure for Ho 5-mm or 1-in.

16 BEARING CAPACITY FROM IN SITUE SPT ( ) ( ( ) F1 F ) ( ) N N > 1. qa allowable bearing pressure for Ho 5-mm or 1-in. settlement, kpa,for any other value of settlement Hi, qai Hi/ Ho x qa Kd D/B 1.33 N is the statistical average value of blows for the footing influence zone of about 0.5B above footing base to at least B below. To convert N 70 x 70 N55 x 55 Example: Find the allowable bearing pressure for a soil of N70 4, if the foundation depth 1m and width3m for a tolerable settlement of 15 mm? Solution: ( ) ( ) qa for 15mm settlement 537 x 15/5 3 kpa BEARING CAPACITY FROM IN SITUE CPT An approximation for the bearing capacity should be applicable for D/B < 1.5 from an averaged value of qc (cone point resistance in kg/ cm) over the depth interval from about B/ above to 1.1B below the footing base. For cohesionless soils one may use ( (300 For cohesive soils one may use ) ) / / / / pg. 34

Foundation Engineering Prof. Dr N.K. Samadhiya Department of Civil Engineering Indian Institute of Technology Roorkee

Foundation Engineering Prof. Dr N.K. Samadhiya Department of Civil Engineering Indian Institute of Technology Roorkee Module 01 Lecture - 03 Shallow Foundation So, in the last lecture, we discussed the