Induced stresses

Induced stresses

1. Hooke's Law (basic formula of elasticity)

1.1 Hooke's law in three dimensional 

       (1)

       (2)

       (3)

1.2 Definition of Elastic modulus

Figure 1. Definitions of Elastic Modulus in Terms of Loading & Applied Boundary Conditions (after Mayne et al. 2001)

       (4)

       (5)

       (6)

1.3 Correlations between Elastic modulus

Shear modulus vs. Young's modulus:

       (7)

Constrained modulus:

       (8)

2. Induced Stresses

2.1 Boussinesq's equations

Induced stresses in the ground due to external loads are calculated utilizing Boussinesq's method (Boussinesq, 1885).  

Figure 2. Element of soil in three dimensional x-y-z space (after Coduto 1998)

       (9)

       (10)

         (Ishihara)

       (11)

       (12)

       (13)

       (14)

       (15)

       (16)

For the infinitely long condition, Flamant (1892) derived the equations for induced normal and shear stresses due to an applied external line load (F, pounds per foot) as (Verruijt, 2001):

Figure 3. Line load on half space (after Verruijt (2001))

       (17)

       (18)

       (19)

Because of the plane strain condition of the line load, the strain in the y-direction should be equal to 0. From Eq. (2), we obtain

       (20)