THE PROBLEM OF CENTER CONSTRUCTION. The framing, setting up, and removing of the center is an important feature of the construction of an arch. Since the center is a temporary structure, it should be made with the least possible expenditure for materials and labor, and with the greatest salvage of useful material after the arch is completed. On the other hand, the center must remain as nearly as possible immovable in position and invariable in form, for any change in the position or in the shape of the center, due to in sufficient strength or improper bracing, will be followed by a change in the curve of the intrados and consequently of the line of resistance, which may endanger the safety of the arch itself; and when the time comes to remove the center, it must move altogether and without shock. The problem then is to build a structure that shall be im movable until movement is desired, and that shall then move at will.
Some allowance must be made for the ac cumulation of material on the center and for the effect of jarring during erection. The following analysis of the problem will show roughly what the forces are and why great accu racy is not possible.
To determine the pressure on the center, consider the voussoir DEFG, Fig. 204, and let a = the angle which the joint DE makes with the horizontal; p = the coefficient of friction (see Table 74, page 464), i.e., p is the tangent of the angle of repose; 0 = the angular distance of any point from the crown; W = the weight of the voussoir DEFG; N = the radial pressure on the center due to the weight of If there were no friction, the stone DEFG would be supported by the normal resistance of the surface DE and the radial reaction of the center. The pressure on the surface DE would then be
W cos a, and the pressure in the direction of the radius W sin a.
Friction causes a slight indetermination, since part of the weight of the voussoirs may pass to the abutment either through the arch ring or through the back-pieces (perimeter) of the center. Owing to friction, both of these surfaces will offer, in addition to the above, a resistance equal to the product of the perpendicular pressure and the coefficient of friction. If the normal pressure on the joint DE is W cos a, then the frictional resistance is ft W cos a. Any frictional resistance in the joint D E will decrease the pressure on the center by that amount; and consequently, with friction on the joint DE, the radial pressure on the center is On the other hand, if there is friction between the arch stone and the center, the frictional resistance between these surfaces will decrease the pressure upon the joins DE, as computed above; and consequently the value of N will be greater than in equation 26.
Notice that in passing from the springing toward the crown the pressure of one arch stone on the other decreases. Near the crown this decrease is rapid, and consequently the friction between the voussoirs may be neglected. Under this condition, the radial pressure on the center is The value of the coefficient to be employed in equation 26 is somewhat uncertain. Disregarding the adhesion of the mortar, the coefficient varies from about 0.4 to 0.8 (see Table 74, page 464); and, including the adhesion of good cement mortar, it may be nearly, or even more than, 1. (It is 1 if an arch stone remains at rest, with out other support, when placed upon another one in such a position that the joint between them makes an angle of 45° with the hori zontal.) If the arch is small, and consequently laid up before the mortar has time to harden, probably the smaller value of the co efficient should be used; but if the arch is laid up so slowly that the mortar has time to harden, a larger value could, with equal safety, be employed. As a general average, we will assume that the coef ficient is 0.58, i.e., that the angle of repose is 30°.