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2 ENCE 461 Foundation Analysis and Design Bearing Capacity of Shallow Foundations 1

3 Bearing Capacity Failure a) General Shear Failure Most common type of shear failure; occurs in strong soils and rocks b) Local Shear Failure Intermediate between general and punching shear failure c) Punching Shear Failure Occurs in very loose sands and weak clays 2

4 General and Punching Shear Failure 3

5 Soil Conditions and Bearing Capacity Failure (after Vesi (1973) 4

6 Load Displacement Curves (after Vesi (1973) a) General Shear Failure b) Local Shear Failure c) Punching Shear Failure Ductile Failure Brittle Failure 5

7 Comments on Shear Failure Usually only necessary to analyse general shear failure Local and punching shear failure can usually be anticipated by settlement analysis Failure in shallow foundations is generally settlement failure; bearing capacity failure must be analysed, but in practical terms is usually secondary to settlement analysis 6

8 Development of Bearing Capacity Theory Application of limit equilibrium methods first done by Prandtl on the punching of thick masses of metal Prandtl's methods adapted by Terzaghi to bearing capacity failure of shallow foundations Vesi and others improved on Terzaghi's original theory and added other factors for a more complete analysis 7

9 Sample Limit Equilibrium Method M a q ult Bb B 2 s ubbb zd Bb B 2 0 Assume: No soil strength due to internal friction, shear strength above foundation base neglected 8

10 Sample Limit Equilibrium Method q ult 2 s u zd N c q ult N c s u zd 9

11 Assumptions for Terzaghi's Method Depth of foundation is less than or equal to its width No sliding occurs between foundation and soil (rough foundation) Soil beneath foundation is homogeneous semiinfinite mass Mohr-Coulomb model for soil General shear failure mode is the governing mode (but not the only mode) 10

12 Assumptions for Terzaghi's Method No soil consolidation occurs Foundation is very rigid relative to the soil Soil above bottom of foundation has no shear strength; is only a surcharge load against the overturning load Applied load is compressive and applied vertically to the centroid of the foundation No applied moments present 11

13 Failure Geometry for Terzaghi's Method 12

14 Log-Spiral Geometry 13

15 Application of Log-Spiral Geometry 14

16 Bearing Capacity Factors q ult AN c BN q CN N c = factor of soil cohesion N q = factor of overburden pressure N = factor for unit weight of soil A, B, C = constants depending upon geometry, soil properties, etc. q ult = ultimate bearing capacity of the soil 15

17 Terzaghi Equations and Factors 16

18 Terzaghi Bearing Factors Equations for N c, N q shown at left Values for N originally derived from Cullman's graphical method; can be approximated by equations All strictly a function of 17

19 Tabulated Terzaghi Bearing Factors 18

20 Application to Square and Circular Foundations q 1.3c' N ' N 0.4 'BN ult c zd q Square q 1.3c' N ' N 0.3 'DN ult c zd q Circular 19

21 Example of Terzaghi's Method Given Square Foundation as Shown Grounwater table is 50' below surface Find Ignore slab-on-grade flooring Use Terzaghi's Method Ultimate bearing capacity and column load to produce same 20

22 Example of Terzaghi's Method q ult 1.3c' N c ' zd N q 0.4 'BN Square Solve for q ult Obtain Bearing Capacity Factors 21

23 Example of Terzaghi's Method q ult 1.3c' N c ' zd N q 0.4 'BN q ult q ult q ult 15,780 psf Compute weight of foundation W f = (3.25) 2 (2)(150) = 3169 lbs. Compute design load = (P )/(3.25) 2 0 P = 163,507 lbs. = kips q PW f A u D 22

24 Notes on Terzaghi's Method Since soil cohesion can be difficult to quantify, conservative values of c (cohesion) should be used Frictional strength is more reliable and does not need to be as conservative as cohesion Terzaghi's method is simple and and familiar to many geotechnical engineers; however, it does not take into account many factors, nor does it consider cases such as rectangular foundations 23

25 Vesi's Method Similar in basic format to Terzaghi's Method, but takes into account a large number of factors Some variations in the way it is implemented 24

26 Factors in Vesi Method Bearing Capacity Factor (N) Shape Factor (s) Depth Factor (d) Inclination Factor (i) Ground Factor (g) Base Factor (b) 25

27 Notations for Vesi's Method 26

28 Vesi's Bearing Capacity Factors

29 Tabulated Vesi N Factors 28

30 Shape Factor (s) N q s c 1 B L N c s q 1 B L tan ' s 10.4 B L For continuous footings, s = 1 29

31 Depth Factor (d) d 10.4 k c d 12k tan ' 1sin ' 2 q d 1 Values of k D/B < 1, k = D/B D/B > 1, k = arctan (D/B), result in radians Discontinuity when D = B 30

32 Load Inclination Factor (i) i c 1 mv Ac'N c 0 i q 1 P V Ac' tan ' m 0 i 1 P V Ac' tan ' m1 0 31

33 Load Inclination Factor (i) Variables V = applied shear load P = applied normal load A = base area of footing c' = effective cohesion (use c = s u for undrained analyses) = effective friction angle (use = 0 for undrained analyses) B = foundation width L = foundation length 32

34 Load Inclination Factor (i) Values of m Loads inclined in the B direction: m 2B L 1B L Loads inclined in the L direction: m 2L B 1L B i = 1 if either loads act perpendicular to footing or soil is purely cohesive ( = 0) Applies to loads that are not perpendicular to the base of the foundation; does not apply to eccentric loads 33

35 Ground Inclination Factor (g) g 1 c 147º g g 1tan 2 q For level ground surface, g = 1 34

36 Base Factor (b) b 1 c 147º b b 1tan 2 q For footings with angled foundation bases When footing is level, b = 1 35

37 Allowable Bearing Capacity Factors when considering selection of a factor of safety Most foundations designed by ASD for geotechnical strength q a q ult F Foundation is then designed so that the allowable bearing pressure is not exceeded 36

38 Given General Bearing Capacity Example Continuous Foundation Width of foundation = 1100 mm (B) Depth of Foundation = 1500 mm (D) Soil cohesion c = 15 kpa Soil internal friction angle = 28º, = 19 kn/m 3 Find Water table even with depth of foundation Design loading, FS = 3 Use depth factor term but ignore foundation weight 37

39 Bearing Example For continuous footing, s = 1 For perpendicular load, i = 1 For level foundation, b =1 For level ground, g = 1 Need to compute factors N, d 38

40 N Factors for Example 39

41 Depth Factor (d) Values of k D/B > 1.5/1.1 = k = arctan (1.363) = radians d c 10.4 k1.375 d q 12k tan ' 1sin ' d 1 40

42 Solution of Bearing Capacity Equation q ult c' N c d c ' zd N q d q 0.5 ' BN d q ult kpa q a kpa Allowable wall loading per lineal metre (385 kpa) (1.1 m wide) (1 m long) = 423 kn/m 41

43 Given Example Using Square Foundation Square foundation, load of 1500 kn On Soil Surface Soil Conditions Sand, No Cohesion, = 20 kn/m 3, = 36º Find Acceptable foundation size for square foundation Neglect effect of foundation weight 42

44 Bearing Capacity Equation for Sand Surcharge Overburden Pressure q factor zero if foundation is at the surface =1 0.4 = 0.6 in this case 43

45 Example Using Square Foundation q ult 1 2 'BN s For = 36º, N = s = 0.6 q ult = (0.5)(20)(B)(56.31)(0.6) = B kpa q a = q ult /FS = /3 = B kpa q a = Q/A = 1500 kn/a = 1500 kn/b 2 44

46 Example Using Square Foundation q a = q ult /FS = /3 = B kpa q a = Q/A = 1500 kn/a = 1500 kn/b /B 2 = B B 3 = 1500/ = B = 2.37 m 45

47 Bearing Capacity in Clay 46

48 Questions? 47