Signs of D2 below operate on sign of D-Term as calculated
1/2 Point

Cr. D1

* Use coordinates and trigonometric functions for points (A-), comp. sheet 1 of 4, Figure 40

For multipliers see sheet I of 41

FIGURE 42- COMPUTATION FORM FOR MOMENTS, THRUSTS, SHEARS, AND RADIAL DEFLECTIONS

FIGURE 43-COMPUTATION FORM FOR TANGENTIAL AND ANGULAR MOVEMENTS

usually 1,000. The coefficient of expansion of concrete, c, for 1° F. change of temperature may be found in text books on concrete; and the magnitude of the temperature change, t, is usually taken as + 1° F. 2. The value of 4 and the quarter values thereof, in degrees, are tabulated on the line designated . Trigonometric functions of angles are secured from tables in chapter IX.

A

3. Arch constants, A1, B1, and C1, are each the product of a tabular value secured from table 10, chapter IX, and a multiplier which is evaluated at the bottom of the computation sheet. Arch constants, B2, C2, and B,, are each the sum of two products as indicated. Each product is made up of a tabular value and a multiplier, which are obtained in the same way as A1, B1, and C1. 4. Abutment movement functions are copied from the form given in figure 14, chapter III.

124. Crown Constants and Forces.-Figure 41, sheet 2 of the set, deals with the evaluation of crown constants and their use in obtaining the moment, thrust, and shear at the crown. A comparison of the form with table 3, chapter V, shows that if integrals in the table are replaced by their respective symbols, the constants are the same. Equations for the solution of crown forces are given in chapter V. The procedure is briefly discussed below.

1. Copy arch parts of A1, B1, C1, B2, C2, and B, from the crown point on the preceding form, sheet 1, and compute abutment parts from the values on the same sheet. Total values are the sum of the arch and abutment parts.

3

2. The arch part of D1, D2, or D, is the product of a tabular term and its multiplier, or the sum of two such products. These products are secured from their component parts, for the crown point, listed at the right central portion of sheet 3, see subsequent section for explanation. Abutment parts are composed of products of values found on sheets 1 and 3. The AML, AHL, and AVL, or the AMR, AHR, and AVR values for these terms are from sheet 3. Total values of D-terms are sums of arch and abutment parts. For concentrated loads, total D-terms include only the effects of yielding abutments. Temperature D-terms are evaluated by formulas given on sheet 1. These are total D-terms because temperature changes are assumed to have no direct effect on abutment yielding.

Ꭱ

3. After values of Mo, Ho, and Vo have been computed by the equations on the right side of the sheet, they should be checked by substituting them in the equations of equilibrium.

125. Moments, Thrusts, Shears, and Radial Deflections. Moments, thrusts, and shears at arch points are computed in the upper part of sheet 3, figure 42. Radial deflections are computed in the lower part of the sheet. Equations for moments, thrusts, and shears are the same as equations 110 to 115, inclusive, given in chapter V. Computations of radial deflections are based on equations 120 and 123 in chapter V. The procedure is as follows:

L,

L

1. Before sheet 2 can be completed, it is necessary to enter on sheet 3 tabular values of D-terms for the crown point and values of MÅ, HL, and Vь for the point, Abut. 4. However, at the time these values are determined, it is expedient to fill in values for other points between the crown and abutment. Formulas for ML, HL, and VL for uniform and triangular radial, tangential, and twist loads are given in tables 5, 6, and 7. Radial, tangential, and twist concentrated loads are, respectively, a shear AVL P, a thrust AHL P, and a moment AMLP, P being negative in the case of thrust.

=

=

2. Tabular D-term constants at the right end of the sheet are copied from tables 17 to 21, inclusive, chapter IX. Products of the tabular values and their proper multipliers are D-terms or Dconstants used in computing deflections. Crown D-terms are used on sheet 2 in computing crown forces.

3. With Mo, Ho, and Vo, computed on sheet 2, and coordinates and trigonometric functions for arch points, computed on sheet 1, the remaining columns across the top of the sheet can be computed, giving values of M, H, and V at arch points.

4. All quantities for radial deflections are now calculated. The portion of the deflection at a point due to the elastic movement of the arch is the algebraic sum of the D-terms on sheet 3 and the arch constants, from sheet 1, multiplied by the moments, thrusts, and shears on sheet 3. The portion of the deflection due to elastic deformation of the yielding abutment is obtained by multiplying MA, HA, and VA by the proper abutment constants, a, 72, ß, and y on sheet 1, transferred to arch points by using functions of the angle A Φ on sheet 1.

А

126. Tangential and Angular Movements.-The computation form for tangential and angular movements, sheet 4, is shown in figure 43. The indicated calculations for tangential deflections are based on equations 121 and 124, and calculations for angular movements are based on equations 119 and 122, chapter V. The procedure is as follows:

1. Tabular D-term constants at the upper right corner of the sheet are copied from tables 17 to 21, inclusive, chapter IX.

2. Total tangential deflections and total angular movements are computed by performing the indicated operations, which are similar to those outlined for computing radial deflections.

127. Use of Forms.-Since deflection summations are sometimes multiplied by large factors and since the number of significant figures in the summations is often reduced by algebraic additions, the use of seven significant figures is necessary to provide sufficient accuracy. Suggestions regarding general computation forms are given in the following paragraphs.

1. Computations on sheet 1 should be independently checked, because a slight discrepancy may produce errors of considerable magnitude in subsequent calculations.

3

2. Total values of A1, B1, C1, B2, C2, and B, on sheet 2 should be checked. Since these values can be used for any unit load, they are computed only once for each arch element. Total values of load constants D1, D2, and D. should also be verified.

3

3. On the upper part of sheet 3, only the total values of M, H, and V at the crown and abutment points should be calculated at first. With these and other values already calculated, radial deflections on sheet 3 and tangential deflections and angular movements on sheet 4 may be determined at the crown of the arch and checked as follows: Tangential deflections and angular movements for symmetrical loads are zero.

Radial deflections, tangential deflections, and angular movements for the two sides of the arch are equal for nonsymmetrical loads.

If crown movements satisfy these conditions, calculations of quantities for other arch points are resumed and carried to completion.

4. Inspection of plotted deflection curves may indicate appreciable errors at the arch quarter-points, as discussed in section 137.