Rectangular Diagrams Research Articles

Effect of the cross-sectional shape of tunnels on the distribution of contact pressures

1. When determining the earth pressure on timbering of tunnels constructed by the cut-and-cover method it is necessary to distinguish two cases of the joint action of the structure and enclosing soil: in a trench and in made ground. Structures for which the condition Hf/Bt≥0.5 can be referred to the first case and Hf/Bt 0.5 B it is necessary to take into account the concentration of pressure caused by the different deformability of the structure and enclosing soil. In calculations of vertical pressure from clay fill it is necessary to take into account cohesion which is restored as the soil is compacted. 2. The distribution of stresses along the foot of the tunnel in the case of weak soils can be taken in calculations in the form of a rectangular diagram. For bases having high values of shear resistance it is necessary to take into account the concentration of pressure on the edge portions of the bottom and take the distribution of pressure on the foot to be saddle-shape in accordance with the recommendations given in [3]. 3. The horizontal pressure on the walls of tunnels in a trench can be assumed uniformly distributed over the height with an intensity equal to 0.25–0.3 of the acting vertical pressure on the ceiling in the case of layerwise compaction of the fill in the pockets and 0.2 of the acting vertical pressure in the case of loose fill; for structures in made ground it is taken as the active pressure according to Coulomb's theory. 4. The indices of the physical and me chanical properties used in calculations of the pressure of soil fills should be designated on the basis of engineering-geologic investigations of the disturbed soil carried out with consideration of the conditions and methods of placing the fill.

HRPA contributions to phonon-phonon interaction in the adiabatic limit

Abstract In the particle-vibration coupling model the leading order phonon-phonon interaction is given by rectangular and two-triangle diagrams. The contributions from diagrams which appear in the adiabatic limit are calculated for the quadrupole vibrator. In this way, the strengths of quartic and cubic anharmonicities are expressed in terms of an underlying shell-model structure.