A theoretical method of mechanical analysis for a circular tunnel reinforced by fully grouted rock bolts

Analysis of the mechanical behavior of rock mass reinforced by fully grouted rock bolts is introduced based on the interaction between the rock mass and the bolts. The model is based on the following premises: (1) the elastic behavior of the rock mass and rock bolts; (2) the plane strain condition; (3) a deeply buried circular tunnel; (4) complete contact between the bolts and the surrounding rock, that is, they are bonded together; (5) the loads on the surrounding rock from the fully grouted rock bolts are replaced by innumerable concentrated forces along the longitudinal direction of the bolts. For this, the analytical radial displacement solution for a deeply buried circular tunnel subjected to concentrated forces at arbitrary points in surrounding rock is derived. As long as this displacement solution is integrated along the length direction of the bolt, the effect of the bolt on the surrounding rock can be obtained. According to the complete contact condition at the anchoring interface and the force balance condition of the bolts, under the action of the in situ stress, linear equations made up of shear stresses on the bolts are established, from which the distribution of shear stresses and axial forces along the bolts can be solved. Model simulations confirm the previous findings that each installed bolt has a pick-up length, an anchor length and a neutral point. Besides, the influence of the parameters of the rock bolts and the surrounding rock are discussed. The conclusion is consistent with the results of a practical project without adopting any empirical equations. The results of this method can provide a theoretical basis for the design and layout of rock bolts in underground caverns.

Mechanical Calculation of Flexible Wires Taking into Account Taps to Electrical Apparatuses

A method is proposed for the mechanical calculation of flexible wires of substation in the presence of one tap to an electrical apparatus located in the middle of the span, taking into account its real location. The technique takes into account the presence of tension insulators strings, different heights of the suspension of wires. The loads on the conductor and tap depend on the climatic regime and take into account wind and icy conditions, as well as temperature changes. The transition to another climatic regime is performed by solving the equation of state, taking into account the coefficients of the horizontal and vertical load of the given and initial regimes. Formulas are given for calculating load factors that take into account weight, wind and ice cover loads on wires and insulator strings, as well as the slope of the span. They are calculated for each climatic mode based on the actual location of the tap to the electrical apparatus and the forces from it to the busbar. It is shown that the calculation results are in good agreement with the proposed method and the computer program. Formulas are given for calculating the sag and horizontal deviations of the wires when the tap is located in the middle of the span. A method is proposed for calculating the components of the forces on the busbar from the tap, taking into account its real location in substation. Formulas for calculating these components are given for different variants of the tap orientation. Knowledge of the values of the components of concentrated forces from the tap allows to increase the accuracy of calculating the sag and tension of the substation wires. Formulas are given for calculating the coefficients of increasing the sag, taking into account the components of the forces from the tap.

Mechanical Calculation of Flexible Wires in the Presence of Horizontal Concentrated Loads

The linear wind load on the wires and cables acting perpendicular to the wire depends on the angle between the direction of the wind and the axis of the overhead line. In the methodology of mechanical calculation of wires and cables, it is recommended to take the wind directed at an angle of 90° to the axis of span and it is not specified which side the wind blows from. For spans of air, this is not so much significant as for switchgear spans, where the deviations of the wires depend on the direction of action of the taps to the electrical apparatus. The article discusses various options for the location of taps and their effect on the wire, as well as changing the direction of the wind. An algorithm for calculating the horizontal deviation of a flexible wire and its increase coefficients in the presence of horizontal concentrated loads due to the action of windon spacers, barriers, taps to electrical apparatuses and other structural elements of substations and overhead lines is given. In the absence of wind, horizontal concentrated loads and deviations occur when an arrangement of the taps is non-keel. The formulas for calculating the horizontal component of the load coefficient to solve the equation of state in the presence of horizontal concentrated forces acting in any direction have been derived. The results of the mechanical calculation are obtained for the cases of one and two horizontal concentrated forces, differently oriented with respect to the distributed wind load. In design practice it is recommended to take the wind flow in the direction of the action of horizontal concentrated forces, since in this case the greatest horizontal deviations and load factors are obtained. The reduction in the coefficients of the horizontal load occurs when the current lead is unloaded because of the opposite directions of the wind and horizontal concentrated forces. In the absence of wind, it is proposed to use the formulas for calculating horizontal deviations and load after finding the product of the coefficient of increase in horizontal deviations and the horizontal component of the coefficient of load per linear load.

Antiplane point load solution for a linear slot with rounded ends

An exact linear elastic solution is derived for a pair of opposing point loads, which act in the middle of a straight slot having rounded ends. This problem is analogous to the mode III crack problem where two concentrated forces act in opposite directions to open the surfaces of a finite-length crack. The corresponding path independent J integral for this slot problem is also determined.

Concentrated Force Action on 1/8 Homogeneous Isotropic Space

Using the example of vertical displacements, it is shown that by combining a solution to the problem of determining vertical displacements from the action of four identical concentrated forces symmetrically applied to an elastic half-space and two identical concentrated forces symmetrically applied to an elastic quarter-space, one can obtain a solution about the action of one force on 1/8 of the elastic space with free edges. To find vertical displacements in an elastic half-space, the Boussinesq solution is used, and vertical displacements in an elastic quarter-space – an integral equation obtained by Ya. S. Uflyand to determine vertical displacements in the face of a homogeneous elastic isotropic quarter-space, for which a deformation modulus and Poisson’s ratio are constant. However, an integral equation of Ya. S. Uflyand is very inconvenient for practical use, therefore, in the paper, an approximate expression written in terms of elementary functions is proposed to find vertical displacements in the face of an elastic quarter-space from the action of a concentrated force. To obtain the latter, a special approximation method is used. The desired solution is also expressed in terms of elementary functions. In this case, an accurate calculation is obtained for an incompressible material with Poisson’s ratio 1/8 of the space n = 0.5. Since the solution is obtained in the case of a concentrated force acting on 1/8 of the elastic space, it is easy to find an expression for determining the vertical displacements of the edge of 1/8 of the elastic space from the action of any distributed load by integrating over the area of action of this load from the influence function, which is taken as required decision. Recommendations for improving the accuracy of calculations are offered. The described approach can also be used to determine the stress-strain of 1/8 of the space with both hingedly supported and free edges.