VERTICAL STRESS INCREASES IN SOIL TYPES OF LOADING. Point Loads (P) Line Loads (q/unit length) Examples: - Posts. Examples: - Railroad track

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1 VERTICAL STRESS INCREASES IN SOIL Point Loads (P) TYPES OF LOADING Line Loads (q/unit length) Revised 0/015 Figure Das FGE (005). Examples: - Posts Figure 6.1. Das FGE (005). Examples: - Railroad track Slide 1 of 4

2 VERTICAL STRESS INCREASES IN SOIL Strip Loads (q) TYPES OF LOADING Area Loads (q) Examples: - Exterior Wall Foundations Examples: - Column Footings Revised 0/015 Slide of 4

3 VERTICAL STRESS INCREASES IN SOIL ANALYSIS METHODS: BOUSSINESQ (1993) Based on homogeneous, weightless, elastic, isotropic infinitely large half-space free of initial stress and deformation. The modulus of elasticity is assumed constant and the principle of linear superposition is assumed valid (EM , 1990). Not accurate for layered soil stratigraphy with substantial thickness (NAVFAC DM7.01, 1986). Rigid Surface Layer Over Weaker Underlying Layer: If the surface layer is the more rigid, it acts as a distributing mat and the vertical stresses in the underlying soil layer are less than Boussinesq values. Weaker Surface Layer Over Stronger Underlying Layers: If the surface layer is less rigid than the underlying layer, then vertical stresses in both layers exceed the Boussinesq values. Revised 0/015 Slide 3 of 4

4 Revised 0/ SOIL MECHANICS VERTICAL STRESS INCREASES IN SOIL ANALYSIS METHODS: WESTERGAARD Based on the assumption that the soil on which load is applied is reinforced by closely spaced horizontal layers which prevent horizontal displacement. The effect of the Westergaard assumption is to reduce the stresses substantially below those obtained by the Boussinesq equations. VERTICAL STRESS INCREASES IN SOIL ANALYSIS METHODS: V:1H METHOD An approximate stress distribution assumes that the total applied load on the surface of the soil is distributed over an area of the same shape as the loaded area on the surface, but with dimensions that increase by an amount equal to the depth below the surface. Vertical stresses calculated V:1H method agree reasonably well with the Boussinesq method for depths between B and 4B below the foundation. Slide 4 of 4

5 VERTICAL STRESS INCREASE ( σ σ Z ) IN SOIL POINT LOADING (BOUSSINESQ 1883) σ z = 3 3P z π L 5 = 3P π ( r z 3 z ) 5/ σ = P 3 π [( r / z) 1] z = 5/ z 1 P z I 1 Where: σ z = Change in Vertical Stress P = Point Load Stresses in an Elastic Medium Caused by Point Loading Figure Das FGE (005). *Based on homogeneous, elastic, isotropic infinitely large half-space Revised 0/015 I 1 = 3 1 π ( r / z) 1 5/ Slide 5 of 4

6 Table 6.1 Variation of I 1 (Das, FGE 006) SOIL MECHANICS VERTICAL STRESS INCREASE ( σ σ Z ) IN SOIL POINT LOADING (BOUSSINESQ 1883) Revised 0/015 Slide 6 of 4

7 VERTICAL STRESS INCREASE ( σ σ = σ Z ) IN SOIL LINE LOADING (BOUSSINESQ 1883) π ( x qz 3 z ) Revised 0/015 Line Load over the Surface of a Semi-infinite Soil Mass Figure 6.1. Das FGE (005). *Based on flexible line load of infinite length on a homogeneous, elastic, isotropic semi-infinite half-space Dimensionless Form σ ( q / z) = or π x z 1 Where: σ = Change in Vertical Stress q = Load per Unit Length z = Depth x = Distance from Line Load Slide 7 of 4

8 VERTICAL STRESS INCREASE ( σ Table 6.3 Variation of σ/(q/z) with x/z (Das, FGE 006). σ Z ) IN SOIL LINE LOADING (BOUSSINESQ 1883) Revised 0/015 Slide 8 of 4

9 VERTICAL STRESS INCREASE ( σ σ Z ) IN SOIL STRIP LOADING (BOUSSINESQ 1883) q σ = π [ β sin β cos( β δ )] Where: σ = Change in Vertical Stress q = Load per Unit Area z = Depth x = Distance from Line Load Flexible Strip Load over the Surface of a Semi-infinite Soil Mass Figure Das FGE (005). Revised 0/015 Angles measured in counterclockwise direction are taken as positive Slide 9 of 4

10 VERTICAL STRESS INCREASE ( σ Table 6.4 Variation of σ/q with z/b and x/b (Das, FGE 006). σ Z ) IN SOIL STRIP LOADING (BOUSSINESQ 1883) Revised 0/015 Slide 10 of 4

11 VERTICAL STRESS INCREASE ( σ σ Z ) IN SOIL CIRCULAR LOADING (BOUSSINESQ 1883) σ = q 1 1 (R / z) 1 3/ Where: σ = Change in Vertical Stress q = Load per Unit Area z = Depth R = Radius Revised 0/015 Vertical Stress Below Center of Uniformly Loaded Flexible Circular Area Figure Das FGE (005). Slide 11 of 4

12 VERTICAL STRESS INCREASE ( σ Table 6.5 Variation of σ/q with z/r (Das, FGE 006). σ Z ) IN SOIL CIRCULAR LOADING (BOUSSINESQ 1883) Revised 0/015 Slide 1 of 4

13 Slide 13 of 4 Revised 0/ SOIL MECHANICS = 1 1 tan n m n m n m mn n m n m n m n m n m mn I π = = = = = B y L x qi z y x dxdy qz d 0 0 5/ 3 ) ( ) ( 3 π σ σ Vertical Stress Below Corner of Uniformly Loaded Flexible Rectangular Area Figure Das FGE (005). Where: σ = Change in Vertical Stress q = Load per Unit Area z = Depth z L n z B m = = ; VERTICAL STRESS INCREASE ( σ σ σ σ Z ) IN SOIL RECTANGULAR LOADING (BOUSSINESQ 1883)

14 VERTICAL STRESS INCREASE ( σ SOIL Variation of I with m and n. Figure Das FGE (005) SOIL MECHANICS σ Z ) IN RECTANGULAR LOADING (BOUSSINESQ 1883) Revised 0/015 Slide 14 of 4

15 VERTICAL STRESS INCREASE ( σ SOIL SOIL MECHANICS σ Z ) IN RECTANGULAR LOADING (WESTERGAARD) Figure 1. NAVFAC DM7.01. Revised 0/015 Slide 15 of 4

16 VERTICAL STRESS INCREASE ( σ RECTANGULAR LOADED AREA Within a Rectangular Loaded Area: σ Z ) IN SOIL σ = q I (1) I () I (3) I (4) Under Center of Footing: σ c = qi c I c = f (m 1, n 1 ) Figure Das FGE (005). m 1 = L B ;n 1 = z B Revised 0/015 Slide 16 of 4

17 VERTICAL STRESS INCREASE ( σ Table 6.6 Variation of I c with m 1 and n 1 (Das, FGE 006). σ Z ) IN SOIL CENTER OF RECTANGULAR LOADED AREA Revised 0/015 Slide 17 of 4

18 BOUSSINESQ SOLUTIONS SUMMARY (EM TABLE C-1) Revised 0/015 Slide 18 of 4

19 BOUSSINESQ SOLUTIONS SUMMARY (EM TABLE C-1) Revised 0/015 Slide 19 of 4

20 BOUSSINESQ SOLUTIONS SUMMARY (EM TABLE C-1) Revised 0/015 Slide 0 of 4

21 BOUSSINESQ GRAPHICAL SOLUTION (EM FIGURE 1-) STRIP FOOTING SQUARE FOOTING Revised 0/015 Slide 1 of 4

22 WESTERGAARD GRAPHICAL SOLUTION (NAVFAC DM7.01 FIGURE 11) Revised 0/015 Slide of 4

23 WESTERGAARD GRAPHICAL SOLUTION (NAVFAC DM7.01 FIGURE 11) Revised 0/015 Slide 3 of 4

24 VERTICAL STRESS INCREASES IN SOIL ANALYSIS METHODS: V:1H METHOD σ z = Q ( B z)( L z) Where: σ z = Change in Total Vertical Stress Q = Applied Foundation Load B = Foundation Width L = Foundation Length Figure C-1. USACE EM Revised 0/015 Slide 4 of 4