M, H, V moment, thrust, and shear at the point where deflections are desired. These values are computed from equations 110 to 112, inclusive, for the left part of the arch and equations 113 to 115 for the right part of the arch. In these equations, x, y, and & are measured from the crown to the point where the deflections are desired, not from the point as defined below, which refers to equations 116 to 124, inclusive. s = length along center line of arch from point where deflections are required. angle measured from arch point where deflections are desired to any point in the arch. = x, y coordinates of any arch point with origin at the point at which deflections are desired. angle from arch point at which deflections are desired to Arch deflections at any point are obtained by considering the portion of the arch between the point and the abutment as a curved cantilever beam. The desired arch movements are the sum of the movements due to the applied load between the point considered and the abutment, and the movements due to the moment, thrust, and shear acting at the point. It is evident that the moment, thrust, and shear at the point at which deflections are desired must be determined before deflections can be evaluated. These are easily determined after crown values have been computed. Formulas for the left part of the arch are Equations for arch deflections at any point are derived in the same way as equations 87, 88, and 89 for the crown. However, in this case total forces and moments acting on the abutments can be calculated by equations 110 to 115 after the crown values are known, which makes it possible to consider effects of resulting abutment movements separately. Equations for deflections at any point in the left part of the arch are in which MA, HA, and VA are the moment, thrust, and shear at the abutment. By substituting symbols for integrals or groups of integrals in the brackets, the following equations may be written : Le=A, M+B, H+ C, V-D, + MA α + VA α2 (119) LA r = C, M + B2 H + C2 V-D2 + ( MA Q + VA α2) XA -HA B sin & A + (VAT +MA α2) COS PA (120) As=-B, M-B3 H-B2 V + D3 - (MA α + VA α2) YA - (VA 7 + MA α2) sin PA - HA B COS PA (121) Re-A, M-B, HC, V+ D,- MA α- VA αz (122) RAr=C, M+ B2 H + C2 V-D2+ (MA a +VA α2) XA - HA B sin PA +(VA T +MA α2)COS PA (123) RAS = B, M+B3 H + B2 V-D3 + (MA α + V1 α2) YA +(VA T + MA α 2) sin &A+HA B COS & A (124) 2 The quantities A1, B1, B2, B3, C1, and C2 consist of integrals or groups of integrals which are functions of the arch and are therefore designated "arch constants." The quantities D1, D2, and D, consist of integrals or 3 combinations of integrals which are functions of both the arch and the load and are designated "load constants." Integrals for these constants also appear in quantities required for solution of crown forces. For this reason arch and load constants may be used for the solution of crown forces as well as for calculating arch deflections. The evaluation of constants for various types of arches, together with a general discussion concerning their use, is given in subsequent sections. UNIFORM THICKNESS CIRCULAR ARCH 91. General Statement.-At this point in the development of arch theory it is convenient to consider the arch as circular and of uniform thickness, because arch and load constants can be evaluated for this special case. In derivations of arch and load constants in the following sections, the arch center line is used instead of the neutral axis. This introduces a small error, but simplifies the analysis. Using the notation in section 90, T3/12, s ro, and ds rdo. Since T and Ec remain constant throughout the arch, and since xr sin & and yr vers op, it is evident that the arch and load constants can be integrated. I 92. Arch Constants.-Arch constants are deflections at an arch point due to a unit force or couple at the point. A, is the angular movement at a point due to a unit moment at the point. 1 B1 is the angular movement at a point due to a unit thrust at the point; or, it is the tangential deflection at a point due to a unit. moment at the point. 1 C1 is the angular movement at a point due to a unit shear at the point; or, it is the radial deflection at a point due to a unit moment at the point. 2 B2 is the radial deflection at a point due to a unit thrust at the point; or, it is the tangential deflection at a point due to a unit shear at the point. C2 is the radial deflection at a point due to a unit shear at the point. 3 B, is the tangential deflection at a point due to a unit thrust at the point. The following formulas for arch constants are for any arch point and may be used for either the left or right parts of the arch. Inspection of formulas 125 to 130 shows that quantities in brackets depend only on the arch angle. These quantities have been evaluated for angles between zero and 90 degrees and are given in chapter IX. 93. Load Constants.-Load constants are deflections at a point due to all loads between the point and the abutment. 1 D, is the angular movement at a point due to loads between the point and the abutment. 2 D2 is the radial deflection at a point due to loads between the point and the abutment. 3 D. is the tangential deflection at a point due to loads between the point and the abutment. Integrals for the D-terms for the left side of the arch are Ꭱ These cannot be conveniently expressed until the terms ML, HL, and VL for the left, or MR, HR, and VR for right side of arch are evaluated. This is described in the following sections which deal with the various unit loads separately. Formulas for moment, thrust, shear, and D-terms are derived for the left part of the arch but may be used for the right part also. It should be remembered that ML, HL, and VL are the moment, thrust, and shear in the left part of the arch, due to load at the right of the point. For the purpose of evaluating D-term integrals and deriving equations. for moment, thrust, and shear, ML, HL, and VL or MR, HR, and VR, at any arch point due to unit loads, the following additional notations are used, see figures 33-a and b. A-For uniform loads applied on the arch, 41 is the angle from the point where deflections are desired to the abutment. For triangular loads, 49, is the angle from the point where the loading begins to the abutment. фо = angle from point where deflections are desired, for arch points not under load, to beginning of external load. angle from point where loading begins to any differential element of the arch under load. Φι angle from point where loading begins to any arch point under load. section 83. intensity of applied load acting on the arch, as stated in |