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chain 100 feet long being the unit generally adopted by American engineers for field ineasurements, any circular arc having that radius, of 5,730 feet, is called a one-degree curve, for the reason that one chain is equivalent to au arc of one degree at the circumference.

2. The circumferences of circles vary directly as their radii: hence, in any circular arc struck with half that radius, or 2,565 feet, one hundred feet at the circumference would sublend an angle of two degrees at the centre. Such an are is called a two-degree curve. If one-third of the primary radius of 5,730 feet, or 1,910 feet, be used, the arc is called a threedegree curve; and so on.

3. It should be borne in mind, however, that these measurements are supposed to be made around the arc itself, and no! on lines of chords. Since field measurements with the chain are always made on the lines of the chords, which are shorter between given points at the circumference than the lines of the arcs, as a bowstring is shorter than the bow, it is plain that, in advancing towards the centre of the one-degree curve by a series of concentric circles having radii equal to one-half, one-third, &c., of the primary radius, the chord 100 feet long differs more and more in length from the arc subtended by it, the bow being more and more arched in relation to the string. Thus, in the circle having a radius equal to one-twentieth of the primary radius, the chord 100 feet long subtends an angle of 20° 06', at the centre, instead of 20°, and the arc is 100.5 feet in length, instead of 100 feet. In order, therefore, that the chord of 100 feet may subtend arcs of 1o, 2o, 3o, &c., in regular succession, the radii of these successive arcs must be somewhat greater than the above method by subdivision of the primary radius would make them; though, as might be inferred from the extreme case given by way of illustration, the difference is not appreciable in ordinary field practice, and radii, together with all the functions dependent on them, may usually be held to vary as the degree of curvature, or central angle per 100 feet chord, varies.

XVIII.

TO FIND THE RADIUS, THE APEX DISTANCE, THE

LENGTH, THE DEGREE, ETC., OF A CURVE.

D

K

М.

F

H

N

1. Let DB, A L be two straight lines intersecting at D. Lay off equal distances, D A, D B; erect perpendiculars at A and

B, meeting at G, and connect A B, DG. From the centre G, with radius GA, draw the curve A H B.

The point D will be the P. I.; A and B, tangent points; D A, D B, the tangents, or apex distances, which denote hyAD; DH, the external secant, or S; HN, the middle ord, or O. Let the long chord A B, tonnecting the tan

gent points, be called C, and G A or G B, the radius, R. Call the deflection angle to a chord of 100 feet D, as before.

2. By XVI. 3 and 4, angle D A B = DB A = A GD=DGB = 11.

3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO

FIND THE APEX DIST. AD. AD= R X tan. # I.

Example. R= 1,910.1, I = 35° 24'.

Then AD= R tan. I = 1,910.1 X 0.3191 609.5. 4. Measure from the P. I. equal distances, D M, D F, along the tangents. Measure, also, M F and D K, the distance from D to the middle point of MF. Then, by reason of similarity in the triangles MDK, D AG,

MK:DK:: A G: AD::R:T

=

Erample.
Let MK 190.5, DK= 60.8, R: = 1910.1.
Then R

Ꭱ 1910.1
DK= 60.8
MK = 190.5 (a.c.)

3.281056
1.783904
7.720105

AD= 609.6

2.785065

5. If 100-feet chords be used, find the ap. dist. in Table XVI. corresponding to the given angle I. Divide that tabular ap. dist. by the degree of curvature corresponding to the given radius: the quotient will be the required ap. dist. Thus, Tab. A D corresponding to 35° 24' = 1,828.7, which, divided by 3, the degree of curvature, gives 609.6, the ap. dist. sought.

6. GIVEN THE INTERSECTION ANGLE I AND AP. DIST. A D,

TO FIND RADIUS R.

Transposing the equation in (3),

R= AD = tan. + I= A D X cot. f I.

Example AD=609.6, I=35° 24' R=AD cot: ! T=609.6 X 3.1334=1910.1.

By a like transposition of the equation in (4),

R= A DXMK:) K.

7. If 100-feet chords be used, find in Table XVI. the ap. dist. corresponding to the given angle 1. Divide that tabular datum by the given ap. dist.; the quotient will be the degree of curvature in degrees and decimals. The radius corresponding to this degree of curvature may be found by (12), by Table X., or, with sufficient accuracy for ordinary practice, by dividing 5,730, the radius of a 1° curve, by it.

Thus, in the foregoing example, the tabular ap. dist. cor responding to 35° 24' is 1,828.7. Dividing by 609.6, we ve3 for the degree of curvature; and 5,730 divided by 3 gives

3. GIVEN THE INTERSECTION ANGLE I AND CHORD AB=C,

CONNECTING THE TANGENT POINTS, TO FIND RADIUS R.

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Example I=359 24', C=1161.4.

Then R=# C = sin. f 1,=580.7 = 0.304 = 1910.2.

9. If 100-feet chords be used, find in Table XVI. the chord corresponding to the given angle 1. Divide that chord by the given chord, for the degree of curvature in degrees and decimals. Determine the corresponding radius by (17), by Table X., or, for ordinary practice, by dividing 5,730 by it.

Thus, in the foregoing example, the · tabular chord corresponding to angle 35° 24' would be 3,484.2, which, divided by the given chord, 1,161.4, gives 3 for the degree of curvature, and 5,730 divided by 3 makes the radius R : 1,910 feet.

10. GIVEN THE INTERSECTION ANGLE I AND THE DEGREE OF CURVATURE OR DEFLECTION ANGLE D, WITH 100-FEET CHORDS, TO DETERMINE THE LENGTH OF THE LONG CHORD C, THE MIDDLE ORD. O, THE EXTERNAL SECANT S, OR THE APEX DIST. A D.

Take from the proper column in Table XVI., the number corresponding to the intersection angle, and divide it by the degree of curvature: the quotient will be the length required.

Example. A 4o curve, I= 50° 16'; to find the several functions above named.

Table XVI. gives the designated functions of a 1o curve as follows: C 4,867.3, O 542.4, S 599.3, A D 2,688.2. Dividing by 4 the degree of curvature, we have for the corresponding functions of a 4o curve as follows: C 1,216.8, O 135.6, S 149.8,

11. GIVEN C, O, S, OR A D, OF ANY CURVE, AND D, THE DEGREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I.

Multiply the given function C, O, S, or AD, by the degree of curvature, D: the product will be found in the proper column of Table XVI., corresponding to the required angle.

E.cample 1. Given AD= 515, D = 5°; to find I.

Then A D XD = 2,575, which corresponds in Table XVI. to 48° 24' = I.

Example 2. Given C = 1,656, D - 3°; to find 1.

Then C XD = 4,968, which corresponds in Table XVI. to 51° 23' =1.

12. GIVEN C, O, S, OR A D, OF ANY CURVE, AND THE INTERSECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D.

Take from the proper column of Table XVI. the number corresponding to the given angle I, and divide that tabular number by the length of the given part; the quotient will be D, in degrees and decimals.

Example 1.
Given A D= 587, I = 22° 26'; to find D.
The A D corresponding to I in Table XVI. is 1,136.3.

Then 1,136.3 = 587 = 1.935 = 1° 56' = D.

Example 2.
Given S= 64, I = 30° 25', to find D.
The Ex. Sec. corresponding to I in Table XVI. is 208

Then 208 • 64 = 3.25 = 3° 15' = D.

13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION

ANGLE D, TO FIND THE LENGTH OF THE CURVE. Divide I by D: the quotient will be the number of chord lengths in the curve.

If the degree of curvature is a whole munber, the more cone

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