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G-1 Appendix G. Determination of Stress State in the Faulting Model The faulting model that incorporates the effects of a load spectrum on the permanent deformation of the base course requires a simple consistent and reasonably accurate method of estimating the state of stress in the middle of the base course. The method adopted for this faulting model is from Timoshenko (95) for estimating the stress state in a half space in which the only material properties that is needed is Poissonâs ratio. The tire load for both single and dual tires is represented as a point load on the concrete pavement surface. The use of these equations permits the superposition of the stresses caused by both dual tires. CALCULATION OF STRESS IN THE BASE COURSE FOR SINGLE TIRE The stress calculation for a single tire is based on one point load applied on the surface of a half space. According to Figure G.1, the analytical solutions (96) for normal stresses and shear stresses at any point caused by the point load P are expressed as: 2 1 3 1 2 2 (G.1) 2 1 3 1 2 2 (G.2) 3 2 (G.3) 2 1 3 1 2 2 (G.4) 3 2 (G.5) 3 2 (G.6) where is the point load; is the distance between an arbitrary point (x, y, z) and the x-axis; is the distance between an arbitrary point (x, y, z) and the y-axis; is the vertical depth of the point P from the surface; is the distance between an arbitrary point (x, y, z) and the point P; and is Poissonâs ratio.
G-2 Figure G.1. Stress Caused by a Point Load (96). The normal stresses immediately beneath the point load P is considered for stress calculations. Accordingly, both of and are equal to zero and the value of is equal to . The analytical Eqs. G.1âG.6 with input values of , , and yield: 2 1 0 1 2 1 12 4 2 1 (G.7) 2 1 0 1 2 1 12 4 2 1 (G.8) 3 2 (G.9) 0 (G.10) The average stresses in the base course immediately beneath the point load P are required to reflect the stress effects on the development of permanent deformation in the base course and thus on faulting. Thus, the average term of the vertical depth ( ) in the analytical Eqs. G.7âG.9 of normal stresses is determined by integration from the bottom of the concrete slab to the bottom of the base course and expressed as: 1 1 1 1 (G.11) (G.12) where is the depth of mean vertical stress; is the thickness of slab; and is the total thickness of slab and base course. In this way, Eqs. G.7âG.9 are expressed as:
G-3 4 2 1 (G.13) 3 2 (G.14) CALCULATION OF STRESS IN THE BASE COURSE FOR DUAL TIRES The stress calculation for dual tires depends on two point loads applied on the surface of a concrete slab as shown in Figure G.2. The point loads and are the same for dual tires. Figure G.2. Stress Caused by Two Point Loads. The final normal stresses are the sum of the normal stress produced by the point loads and . The normal stresses produced by the point load are determined by Eqs. G.18âG.20 and substituting 0, 0, and ( ) and expressed as: 4 2 1 (G.15) 3 2 (G.16) 0 (G.17) where , , and are normal stresses caused by the point load . The normal stresses caused by the point load are determined by Eqs. G.1âG.3 and substituting 0, , and â ( ) and generated as: 2 1 2 1 (G.18)
G-4 2 1 3 1 2 2 (G.19) 3 2 (G.20) 0 (G.21) 3 2 (G.22) where is the distance between the two point load and also the tire spacing, which is typically 12 in. Given the normal stresses from single and dual tires, the first invariant of stress tensor and the second invariant of the deviatoric stress tensor in the center of the base course at the axle load level can be determined by: (G.23) 1 6 (G.24)