FORM AND DIMENSIONS OF CROSS-SECTION.
In deciding upon the sectional profile of a tunnel two factors have to be taken into consideration: (1) The form of section best suited to the conditions, and (2) the interior dimensions of this section.
Form of Section.
—The form of the sectional profile of a tunnel should be such that the lining is of the best form to resist the pressures exerted by the unsupported walls of the tunnel excavation, and these vary with the character of the material penetrated. These pressures are both vertical and lateral in direction; the roof, deprived of support by the excavation, tends to fall, and the opposite sides for the same reason tend to slide inward along a plane more or less inclined, depending upon the friction and cohesion of the material. In some rocks the cohesion is so great that they will stand vertically, while it may be very small in loose earth which slides along a plane whose inclination is directly proportional to the cohesion.
From the theory of resistance of profiles we know that the resistance of a line to exterior normal forces is directly proportional to its degree of curvature, and consequently inversely proportional to the radius of the curve. Hence the sectional profile of a tunnel excavated through hard rock, where there are no lateral pressures owing to the great cohesion of the material, and having to resist only the vertical pressure, should be designed to offer the greatest resistance at its highest point, and the curve must, therefore, be sharper there, and may decrease toward the base. In quicksand, mud, or other material practically without cohesion, the pressures will all be normal to the line of the profile, and a circular section is the one best suited to resist them. These theoretical considerations have been proved correct by actual experience, and they may be employed to determine in a general way the form of section to be adopted. Applying them to very hard rock, they give us a section with an arched roof and vertical side walls. In softer materials they give us an elliptical section with its major axis vertical, and in very soft quicksands and mud they give us the circular section. These three forms of cross-section and their modifications are the ones commonly employed for tunnels. An important exception to this general practice, however, is met with in some of the city underground rapid-transit railways built of late years, where a rectangular or box section is employed. These tunnels are usually of small depth, so that the vertical pressures are comparatively light, and the bending strains, which they exert upon the flat roof, are provided for by employing steel girders to form the roof lining.
Fig. 7.—Diagram of Polycentric Sectional Profile.
From what has been said it will be seen that it is impossible to establish a standard sectional profile to suit all conditions. The best one for the majority of conditions, and the one most commonly employed, is a polycentric figure in which the number of centers and the length of the radii are fixed by the engineer to meet the particular conditions which exist. In a general way this form of center may be considered as composed of two parts symmetrical in respect to the vertical axis. [Fig. 7] shows such a profile, in which DH is the vertical axis. The section is unsymmetrical in respect to the horizontal axis GE. The upper part forming the roof arch is usually a semi-circle or semi-oval, while the lower part, comprising the side walls and invert of floor, varies greatly in outline. Sometimes the side walls are vertical and the invert is omitted, as shown by [Fig. 8]; and sometimes the side walls are inclined, with their bottoms braced apart by the invert, as shown by [Fig. 9]. In more treacherous soils the side walls are curved, and are connected by small curved sections to the invert, as shown by [Fig. 10]. In the last example the side walls are commonly called skewbacks, and the lower part of the section is a polycentric figure like the upper part, but dissimilar in form.
In a tunnel section whose profile is composed entirely of arcs the following conditions are essential: The centers of the springer arcs Ga and Ea′, [Fig. 7], must be located on the line GE; the center of the roof arc bDb′ must be located on the axis HD; the total number of centers must be an odd number; the radii of the succeeding arcs from G toward D and E toward D must decrease in length, and finally the sum of the angles subtended by the several arcs must equal 180°.
