VERTICAL STRESS INCREASES IN SOILS TYPES OF LOADING

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1 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASES IN SOILS TYPES OF LOADING Point Loads (P) Line Loads (q/unit length) Figure Das FGE (005). Examples: -Posts Figure 6.1. Das FGE (005). Examples: - Railroad track Revised 1/013 Slide 1 of 3

2 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASES IN SOILS TYPES OF LOADING Strip Loads (q) Area Loads (q) Examples: - Exterior Wall Foundations Examples: - Column Footings Revised 1/013 Slide of 3

3 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE IN SOILS ANALYSIS METHODS - BOUSSINESQ (1883) Based on homogeneous, weightless, elastic, isotropic infinitely large halfspace 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 1/013 Slide 3 of 3

4 Revised 1/ THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE IN SOILS 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 INCREASE IN SOILS 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 3

5 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS POINT LOADING (Boussinesq, 1883) z 3 3P z L 5 3P ( r z 3 z ) 5/ z P z 3 r / z 1 1 5/ P z I 1 Where: 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 1/013 I r / z 1 5/ Slide 5 of 3

6 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS POINT LOADING (Boussinesq, 1883) Table 6.1 Variation of I 1 (Das, FGE 006). Revised 1/013 Slide 6 of 3

7 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS LINE LOADING (Boussinesq, 1883) ( x qz 3 z ) Revised 1/013 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 ( q / z) or x z Where: = Change in Vertical Stress q = Load per Unit Length z = Depth x = Distance from Line Load 1 Slide 7 of 3

8 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS LINE LOADING (Boussinesq, 1883) Table 6.3 Variation of /(q/z) with x/z (Das, FGE 006). Revised 1/013 Slide 8 of 3

9 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS STRIP LOADING (Boussinesq, 1883) q sin cos( ) Where: = Change in Vertical Stress q = Load per Unit Area z = Depth x = Distance from Line Load Revised 1/013 Flexible Strip Load over the Surface of a Semi-infinite Soil Mass Figure Das FGE (005). Angles measured in counterclockwise direction are taken as positive Slide 9 of 3

10 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS STRIP LOADING (Boussinesq, 1883) Table 6.4 Variation of /q with z/b and x/b (Das, FGE 006). Revised 1/013 Slide 10 of 3

11 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS CIRCULAR LOADING (Boussinesq, 1883) q1 1 (R / z) 1 3/ Where: = Change in Vertical Stress q = Load per Unit Area z = Depth R = Radius Revised 1/013 Vertical Stress Below Center of Uniformly Loaded Flexible Circular Area Figure Das FGE (005). Slide 11 of 3

12 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS CIRCULAR LOADING (Boussinesq, 1883) Table 6.5 Variation of /q with z/r (Das, FGE 006). Revised 1/013 Slide 1 of 3

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

14 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS RECTANGULAR LOADING (Boussinesq, 1883) Variation of I with m and n. Figure Das FGE (005). Revised 1/013 Slide 14 of 3

15 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS RECTANGULAR LOADING (Westergaard) Revised 1/013 Figure 1. NAVFAC DM7.01. Slide 15 of 3

16 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS RECTANGULARLY LOADED AREA Within a Rectangularly Loaded Area: qi (1) I () I (3) I (4) Under Center of Footing: c qi c I c f (m 1, n 1 ) Figure Das FGE (005). m 1 L B ;n 1 z B Revised 1/013 Slide 16 of 3

17 14.53 THEORETICAL SOIL MECHANICS VERTICAL STRESS INCREASE ( z ) IN SOILS CENTER OF RECTANGULARLY LOADED AREA (Boussinesq) Table 6.6 Variation of I c with m 1 and n 1 (Das, FGE 006). Revised 1/013 Slide 17 of 3

18 14.53 THEORETICAL SOIL MECHANICS BOUSSINESQ SOLUTIONS SUMMARY (EM Table C-1) Revised 1/013 Slide 18 of 3

19 14.53 THEORETICAL SOIL MECHANICS BOUSSINESQ SOLUTIONS SUMMARY (EM Table C-1) Revised 1/013 Slide 19 of 3

20 14.53 THEORETICAL SOIL MECHANICS BOUSSINESQ SOLUTIONS SUMMARY (EM Table C-1) Revised 1/013 Slide 0 of 3

21 14.53 THEORETICAL SOIL MECHANICS BOUSSINESQ GRAPHICAL SOLUTION (EM Figure 1-) STRIP FOOTING SQUARE FOOTING Revised 1/013 Slide 1 of 3

22 14.53 THEORETICAL SOIL MECHANICS WESTERGAARD GRAPHICAL SOLUTION (NAVFAC DM7.01 Figure 11) Revised 1/013 Slide of 3

23 14.53 THEORETICAL SOIL MECHANICS WESTERGAARD GRAPHICAL SOLUTION (NAVFAC DM7.01 Figure 11) Revised 1/013 Slide 3 of 3

24 14.53 THEORETICAL SOIL MECHANICS V:1H DISTRIBUTION METHOD z Q ( B z)( L z) Where: z = Change in Total Vertical Stress Q = Applied Foundation Load B = Foundation Width L = Foundation Length Figure C-1. USACE EM Revised 1/013 Slide 4 of 3

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