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Homework #3 By Rith Makara Stress Invariant, and Resilient Modulus in Kenlayer Stress Invariant, : Stress Invariant is every stress that are independent of the coordinate system. The principle stresses can be combined to form the stress invariants, I 1 , I 2 and I 3 Resilient Modulus: In Boussines’s solutions, the material that constitutes the half-space is linear elastic which means the elastic constant (Ex. Modulus of Elasticity) is not vary with state of stresses. However, it is not true for soils because their axial deformation depends strongly on the magnitude of confining pressures. Therefore, the soil should be considered as nonlinear material and the Resilient Modulus should be used. To show the effect of nonlinearity of granular materials on vertical stresses and deflection, in 1968, Huang divided the half-space into seven layers and applied Burmister’s layered theory to determine the stresses at the mid-height of each layer by Iterative Method and lowest layer is a rigid base with a very large elastic modulus. By Rith Makara 1

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Homework #3By Rith MakaraStress Invariant, and Resilient Modulus in KenlayerStress Invariant, : Stress Invariant is every stress that are independent of the coordinate system. The principle stresses can be combined to form the stress invariants, I1, I2 and I3

Resilient Modulus: In Boussiness solutions, the material that constitutes the half-space is linear elastic which means the elastic constant (Ex. Modulus of Elasticity) is not vary with state of stresses. However, it is not true for soils because their axial deformation depends strongly on the magnitude of confining pressures. Therefore, the soil should be considered as nonlinear material and the Resilient Modulus should be used. To show the effect of nonlinearity of granular materials on vertical stresses and deflection, in 1968, Huang divided the half-space into seven layers and applied Burmisters layered theory to determine the stresses at the mid-height of each layer by Iterative Method and lowest layer is a rigid base with a very large elastic modulus.

Iterative Method in solving this problem: First, an elastic modulus is assumed for each layer and the stresses can be obtained from the layered theory. Then new set of modulus is determined from Eq. 2.11 and then the new set of stress is computed. The process is repeated until the modulus between two consecutive iterations converge to a specified tolerance. After obtaining the stress at each layer, the elastic modulus of each layer is determined from the equation below:

Where: E0 Initial elastic modulus - Soil constant indicating the increase in elastic modulus per unit increase in stress invariant. - Stress Invariant (It should include both effect of applied load and the geostatic load.

Where: z, r, and t Vertical, radial and tangential stresses due to loading - Unit weight of soilz Distance below ground surface at which the stress invariant is computedK0 Coefficient of earth pressure at rest Conclusion of Burmisters test result: Depending on the depth of the point in question, the vertical stresses based on nonlinear theory may be greater or small than those based on linear theory, and at a certain depth, both theories could yield the same stresses. Thats why Boussinesqs solutions for vertical stress based on linear theory have been applied with success even though the soil themselves are basically nonlinear. In Layer Elastic Analysis of Kenlayer, the Resilient modulus, E can be computed from a more popular relationship as shown below: Granular Materials

Where: K1 and K2 can be determined from the table below:

Fine-Grained Soil:

Summary: Chapter 3, Huang (Layer Elastic Part) The basis component of KENLAYER is the elastic multilayer system under a circular loaded area. Each layer is linear elastic, homogeneous, isotropic and infinite in areal extent. The problem is axisymmetric and the solutions are in terms of cylindrical coordinates r and z. For axisymmetric problem in elasticity, a convenient method is to assume a stress function. After the stress function is found, the stresses and displacements can be determined.

Summary: Chapter 5 (How can we consider the curling stress in rigid pavement in Kenslabs?) Generally, the solution of Kenslabs can be determine from the equation below:

Where: [K] The overall stiffness matrix (Combination of slab, Joint and foundation stiffness){} Nodal displacements vector{f} Externally applied nodal forces vector In analyzing the temperature curling, the general formulation of curling is similar to that for loading. And to solve the nodal displacements, the equation below can be used:Applied Force by Curling

Where: [Kp] is the stiffness matrix of slab including joint{} Nodal displacements vector of slab{} Nodal displacements vector of foundation[Kf] is the stiffness matrix of foundation{f} Externally applied nodal forces vector

Where:w Vertical deflection (downward positive) - Rotation (about x and y axis)Fw Applied vertical force c Initial curling (upward positive) due to weightless and unrestrained slab. And where c =0 then Eq. 5.28a will be the same as Eq. 5.15

Where: t Temperature differentialt Coefficient of thermal expansion

By Rith Makara6