FACULTAD DE MINAS U.N..P
1
ING.CARLOS CALLE G.
CONTENIDO GENERALIDADES PROLOGO
CAPITULO I 1.
TRIANGULACION REDES DE TRIANGULACION. RED DE TRIANGULOS. RED DE CUADRILATEROS. RED DE POLÍGONOS. CONDICION DE TRIANGULO. MEDICION DE ANGULOS Y BASE. CLASES DE TRIANGULOS.
2.
PLANEAMIENTO DE UNA TRIANGULACION INFORMACIÓN BASICA. RECONOCIMIENTO DEL TERRENO. MONUMENTACION DE HITOS. MEDICON DE LA BASE. CORRECCION POR LONGITUD VERDADERA. CORRECCION POR TEMPERATURA. CORRECCION POR HORIZONTALIDAD. CORRECCION POR CATENARIA. CORRECCION POR TENSIÓN. MEDICON DE ANGULOS. COMPENSACION DE BASE. COMPENSACIÓN DE ANGULOS. RED TRIANGULOS ASIMPLES. RED DE CUADRILATEROS. COMPENSACIÓN CON PUNTO CENTRAL. RESISTENCIA DE FIGURA. CALCULO DE LADOS. CALCULO DE AZIMUTS.
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FACULTAD DE MINAS U.N..P
2
ING.CARLOS CALLE G.
CALCULO DE COORDENADAS. CALCULO DE AREAS. CALCULO DE COTAS. DIBUJO DE LA RED. CONFIGURACIÓN. LIBRETA DE CAMPO.
CAPITULO II CAMINOS GENERALIDADES 1. ETAPAS DEL TRAZO. 2. CURVAS CIRCULARES HORIZONTALES. ELEMENTOS DE UNA CURVA. DETERMINACIÓN DE LOS ELEMENTOS. REPLANTEO DE CURVAS CIRCULARES. 3. REPLANTEO POR DEFLEXIONES CON VISIBILIDAD DESDE EL PC. 4. REPLANTEO POR DEFLEXIONES CON PUNTOS DE CAMBIO. 5. SECCIONES LONGITUDINALES. 6. SECCIONES TRANSVERSALES. 7. RASANTES. 8. AREAS Y VOLÚMENES.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
3
GENERALIDADES Triangulación es un sistema de redes de apoyo que sirven para dar mejor coherencia a los levantamientos.I Las triangulaciones son usadas para terrenos relativamente extensos, siendo estos los que tienen menor error con respecto a las poligonales, Para iniciar una red, para ambos casos es necesario hacer un reconocimiento del terreno y diseñar el sistema adecuado teniendo en consideración la naturaleza del levantamiento, después de la inspección se procede a la monumentación de hitos en cada vértice los cuales deben cumplir las características adecuadas; la medida de los hitos son relativos, dependiendo del grado de precisión.
25 Cm 25 cm
60 cm
25 cm
60cm
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FACULTAD DE MINAS U.N..P
4
ING.CARLOS CALLE G.
PROLOGO Es indudable que actualmente estamos entrando cada vez más a la era de la informática, para el cual debemos estar preparados de acuerdo al avance de la tecnología para desarrollar nuevos modelos matemáticos, esto nos permitirá realizar algoritmos, para el caso especifico del curso desarrollaremos paso a paso como llegar al resultado final del problema. En el presente texto nos ocuparemos exclusivamente al desarrollo práctico de los contenidos, como, TRIANGULACION Y CAMINOS, sabiendo que para hacer un levantamiento topográfico es de vital importancia conocer las principales redes de apoyo para tener el éxito esperado, como es de esperar el estudiante debe estar en la capacidad de desarrollar algoritmos para una Triangulación el cual será un gran aporte dando consistencia al levantamiento topográfico. Dentro de una poligonación veremos desde el reconocimiento del terreno, monumentación de hitos en los vértices, cálculos de ángulos, distancias y llegar al objetivo final de obtener las coordenadas rectangulares y cotas para poder graficar, el mismo que será mediante un programa CAD y realizar la impresión respectiva, de la misma manera estaremos procediendo con la triangulación desarrollando secuencialmente todos los pasos hasta llegar al resultado final, de esta manera contribuyendo con todo los que lleven el curso y los que están relacionados directa o indirectamente a la especialidad. El Autor
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FACULTAD DE MINAS U.N..P
5
ING.CARLOS CALLE G.
CAPITULO I 1.-TRIANGULACION La red de triángulos es un sistema de apoyo para levantamientos topográficos de terrenos relativamente extensos, la triangulación comprende una serie de procesos, entre ello tenemos el reconocimiento del terreno, monumentación de hitos, medición de base, ángulos, compensación, cálculo de coordenadas y cotas; la disposición de los triángulos son generalmente figuras geométricas que se determinan por principio geométrico con la suma de sus ángulos internos. Así en un triángulo la suma de sus ángulos internos debe ser 180° y los ángulos alrededor de un punto 360°, al realizar una triangulación la longitud de sus lados esta en función al seno de su ángulo opuesto, para calcular los lados de una red de triangulación solamente se mide la base, o sea un solo lado y los siguientes se calcula mediante fórmulas trigonométricas, con el avance tecnológico y los equipos electrónicos (Distanciómetro y Estación total) se miden directamente sus lados y a este método se denomina trilateración.
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FACULTAD DE MINAS U.N..P
1.1-
ING.CARLOS CALLE G.
6
REDES DE TRIANGULACION. - El tipo de red a emplearse está en función al levantamiento topográfico y la extensión o zonas donde se monumentarán puntos de 1er, 2do. orden u otras de menor precisión, entre ellos tenemos:
1.1.1.- Red de triángulos.- Se determina ese tipo de red cuando no se requiere mucha precisión y es diseñado
generalmente
para
trazos
de
carreteras, canales y ferrocarriles.
6 A
2
4
Carreteras
B
1
3
`
5
7
1.1.2.- Red de Cuadriláteros, sistema que se decide para alcanzar una precisión mayor, y es utilizado para comunicación de túneles, dirección de labores subterráneas.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
7
A C B
E D F
1.1.3.- Red de polígonos con punto central.- Cuando no es preciso hacer un cuadrilátero se puede realizar polígonos con punto central, con la misma precisión que la red de cuadriláteros.
B
G
A C
H
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FACULTAD DE MINAS U.N..P
8
ING.CARLOS CALLE G.
1.2- Condición de triángulos.- Para que un programa de triangulación resulte satisfactorio debe tenerse en cuenta que los ángulos deben estar dentro del rango o sea no < de 30° ni > de 150° porque los lados están en función al seno, los ángulos cerca a 0° y 180° tienden a error, y la suma de ángulos internos de un polígono debe cumplir la condición geométrica, 180*(n-2) y sus lados deben estar en función de 1 a 3, en redes de cuadriláteros o polígonos con punto central debe cumplir la condición geométrica y trigonométrica. Dentro de la condición trigonométrica tenemos que: (Lg Senimpares) = (Lg Sen pares)
1.3- Medición de ángulos y base.-La medición de ángulos puede realizarse por los métodos ya conocidos, por reiteración o repetición dependiendo de la precisión que se quiere alcanzar, la diferencia vertical se puede medir
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
9
1.4.- Clases de triangulaciones.- Las triangulaciones pueden clasificarse por el orden de su precisión de acuerdo a: a).- El error de cierre angular en los triángulos. b).- La discrepancia que resulta de medir la base de cierre y calculada. c).- Precisión de la medición de la base. d).- Longitud máxima de sus lados. De acuerdo a lo mencionado podemos clasificar en triangulaciones de 1er , 2do y 3er . Orden.
DESCRIPCIÓN Error de cierre de base Error de cierre angular en triangulacion.
1er ORDEN 1/25000
2do. ORDEN 1/10000
3er. ORDEN 1/5000
8”
15”
30”
Longitud máx. de lados 50-200Km. 15-40 Km. (Km)
1.5-10 Km.
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FACULTAD DE MINAS U.N..P
10
ING.CARLOS CALLE G.
2.- PLANEAMIENTO DE UNA TRIANGULACION 1. Información básica....(gabinete) 2. Reconocimiento del terreno (campo) 3. Monumentación de hitos (campo) 4. Medición de base (campo) 5. Medición de ángulos (campo) 6. Compensación de base.(gabinete) 7. Compensación de ángulos.(gabinete) 8. Cálculo resistencia de figura.(gabinete) 9. Cálculos de lados.(gabinete) 10. Cálculo de azimut (magnético, verdadero, U.T.M.) 11. Cálculo de coordenadas (magnéticos, verdadero y U.T.M.) 12. Cálculo de cotas. 13. Dibujo de red. 14. Configuración a partir de la red. 15. Puntos auxiliares. 16. Informe.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
11
2.2.- RECONOCIMIENTO DEL TERRENO. Consiste en hacer una evaluación insitú de la zona donde se proyectará la Red ubicando ubican do adecuadamente los puntos pu ntos o vértices para monumentar los hitos, de tal manera que los puntos deben ser visibles de un vértice a otro.
2.3.- MONUMENTACION DE HITOS . La señalización es una etapa de importancia dependiendo de ella el resultado
final
de
la
Red
de
triángulos,
la
monumentación de hitos se hará con buen criterio, pudiendo ser desde hitos de concreto con placas de metal grabados o con un hierro de acero al centro.
2.4.- MEDICION DE BASE. Dentro del reconocimiento insitú se ubicará la zona adecuada para medir la base, esta distancia puede medirse con métodos convencionales o electrónicos, la medición electrónica se realiza con un
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
12
horizontalidad, para hacer las correcciones respectivas en gabinete.
2.4.1- Corrección por Longitud Verdadera.- La cinta por el constante uso, temperatura, tensión sufre una cierta dilatación aumentando en milímetros su longitud verdadera, al realizar una medición por tramos se está cometiendo un error acumulativo en todo el circuito, la corrección se realiza aplicando la fórmula
Lc
Lm * Lr
Ln
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
13
SOLUCIÓN:
Ln= 30 m. Lm= 189.80 Lc= ¿ Lr= 29.996
Lc
Lm * Lr
Ln
189 18 9.80 * 29.996 99 6
30
189 18 9.775 77 5 mts.
2.4.2- Corrección por Temperatura.- La temperatura de ambiente puede afectar mucho a la cinta, la medición de base debe hacerse a una temperatura aproximada de calibración, generalmente las winchas vienen calibradas a 20° C.
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FACULTAD DE MINAS U.N..P
14
ING.CARLOS CALLE G.
Ejemplo No 2. Con una cinta de 50m graduada a 20ºC se mide dos tramos, AB 50 mts a 23ºC y BC = 38.25 a 18ºc, ¿cual es la corrección por temperatura?
SOLUCIÓN: Si.
Ct = ? L = 50 y 38.25 m. = 88.25 m. K = 0.000012 T = 23º C y 18º C to = 20o C Ct = LK (t-to)
Remplazando valores. Ct (AB) = 50 (0.000012) (23-20)
= 0.00180
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FACULTAD DE MINAS U.N..P
15
ING.CARLOS CALLE G.
Donde: Ch = Corrección por horizontalidad. h= Diferencia vertical del tramo
L = longitud del tramo
Ejemplo No 3. Encontrar la corrección de una base de 85.48 m. medido con wincha de 30 mts. Teniendo el desnivel entre AB, 0.08m, BC, 0.25m y CD, 0.15m.
SOLUCIÓN: Ch = ? h = 0.18, 0.25, 0.15m respectivamente.
L = 30, 30, 25.48 respectivamente.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
16
2.4.4- Corrección por catenaria.- La cinta al ser suspendida de sus extremos forma una catenaria, la corrección será la diferencia que existe entre la cuerda y el arco formado por los extremos, para corregir aplicamos la fórmula: WL Cc 24 P L
2
Donde: Cc = Corrección por catenaria. L = Longitud de catenaria. W = Peso de la cinta en kg/m.l. P = Tensión aplicada en kg.
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FACULTAD DE MINAS U.N..P
TRAMO LONGITUD
17
ING.CARLOS CALLE G.
W= Kg/m.l.
p
Cc
AB
30
0.025
10
-0.00703
BC
30
0.025
5
-0.02812
CD
20.45
0.025
10
-0.00223
Corrección total
-0.03738
Distancia corregida. 80.45 – 0.03738 0.03738 = 80.413m.
2.4.5- Corrección por Tensión.- Cuando en la cinta se
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FACULTAD DE MINAS U.N..P
18
ING.CARLOS CALLE G.
Ejemplo No 5. Del ejemplo anterior encontrar la corrección por tensión si para el tramo AB 8Kg, BC 10Kg, CD 15Kg.
SOLUCIÓN: C p
= Corrección por tensión.
L
= 30, 30, 20.45m
P
= 8Kg, 10Kg y 15kg.
Po
= 10Kg
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FACULTAD DE MINAS U.N..P
19
ING.CARLOS CALLE G.
Ida + vuelta Base Corregida = 2
Base = LC + CT - CH - CC + CP
2.5-Medición de ángulos.- En el desarrollo de una triangulación es importante determinar el grado de precisión que se requiere y el objetivo de la red, en función a estos parámetros se puede fijar el método de medición de ángulos, pudiendo ser por repetición para poca precisión y por
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FACULTAD DE MINAS U.N..P
20
ING.CARLOS CALLE G.
c) Compensación para redes de polígonos con punto central.
2.7.1-.Red de Triángulos simples.- Para compensar una red de triángulos podemos realizar de dos formas: a) Compensación de estación, cuando la suma de los ángulos alrededor del punto sea 360º. b) Compensación del triángulo, comparar que la suma de los ángulos internos del sea 180º.
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FACULTAD DE MINAS U.N..P
21
ING.CARLOS CALLE G.
Ejemplo 06. Compensar las siguientes redes de triángulos, los ángulos son promedios de una lectura por repetición.
1) 38o 20’
6) 58o 07’
11) 255o 29’
2) 72o 40’
7) 46o 25’
12) 238o 43’
3) 69o 02’
8) 93o 14’
13) 321o 39’
4) 52o 14’
9) 40o 23’
14) 124o 29’
5) 69o 38’
10) 319o 36’
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
22
359o 59’
= 360°
Er.C = 359o 59’-360 = -1’ f c C = +1’/2 =30” sumando +30” a los ángulos 1 y 13
38o 20’30’’ + 321o 39’30” =360o
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FACULTAD DE MINAS U.N..P
23
ING.CARLOS CALLE G.
b) Compensando por i=180°, Se suma los ángulos internos, la diferencia que existe al restar 180° se divide entre 3, el resultado se suma o resta a cada ángulo.
Comp. de Vert.
Vert. Compensado
1
38°20’30”
38°19’38.333”
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ING.CARLOS CALLE G.
los mismos que se encuentran superpuestos entre sí. En la figura se tiene los siguientes triángulos:
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ING.CARLOS CALLE G.
ABC, ACD, ABD, BCD, en cada uno de ellos la suma de los ángulos debe ser 180°. ABC = 3+4+5+6 = 180°
ACD = 2+7+8+1 = 180° ABD = 1+2+3+4 = 180°
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FACULTAD DE MINAS U.N..P
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ING.CARLOS CALLE G.
algebraicamente con signo cambiado a cada ángulo. 3) Se compara los ángulos opuestos por el vértice en la intersección de las diagonales, estas deben ser iguales, la discrepancia se divide entre 4, el
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FACULTAD DE MINAS U.N..P
62
O
= Punto medio.
PC.
= Principio de curva.
PT.
= Principio de tangente.
T
= Tangente.
R
= Radio.
E
= External (M-V)
I
= Angulo de intersección.
V
= Punto de intersección.
G
= Grado de curva.
LC
= Longitud de curva (PC-M-PT)
C
= Cuerda (PC-N-PT)
ING.CARLOS CALLE G.
Por principio Geométrico G = I
2.2.-DETERMINACIÓN DE LOS ELEMENTOS. - TANGENTE.- Dentro del alineamiento AA’ entre el tramo PC y V es la tangente, el mismo que se calcula con T R * Tg
G 2
- CUERDA.- Tramo comprendido entre PC y PT. C 2 R * Sen
-
G
2
LONGITUD DE CURVA. - Tramo comprendido entre
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
63
- EXTERNA.- Distancia del punto máximo de la curva al vértice (M-V) E T * Tg
G 4
Las fórmulas expuestas de los cuatro elementos de curva circular horizontal es fundamentalmente para hacer cálculos y ubicar los puntos sobre la curva para un posible replanteo.
EJEMPLO 1: Calcular los elementos de curva de un radio de 95 m, conociendo los alineamientos AA’=343°20’ Y BB’=295°35’, El PC. se encuentra en el alineamiento AA’
SOLUCION. 1) Croquis Por principio geométrico Se tiene que G=I. calculamos I en función de los Azimuts
A
343°20’
PC R=95m
de AA’ Y BB’ I=180°-(343°20’ -295°35’) I=132°15’ G=I=132°15’
O
I
PT 295°35’
2) cálculo de elementos
B
V
B’ A’
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FACULTAD DE MINAS U.N..P
64
ING.CARLOS CALLE G.
G 132 13 2°15' T = R * Tg = 95 * Tg = 214 21 4.632 63 2 m. 2 2 G 132 13 2°15' C = 2R * Sen = 2 * 95 * Sen = 173 17 3.742 74 2 m. 2 2 13 2°15'*95 ΠGR Π *132 LC = = = 219 21 9.279 27 9 m. 180 18 0° 180 18 0° G 132 13 2°15' E = T * Tg = 214 21 4.632 63 2* Tg = 139 13 9.717 71 7 m. 4 4
EJEMPLO 2. En el problema anterior ubicar las estacas sobre la curva cada 30 mts. replanteando desde el PC.
SOLUCION. 1) La longitud de curva en el problema anterior es 219.279 mts, se pide replantear cada 30 mts. N° de estacas = 219.279 / 30 = 7.3093.
se tiene 7 tramos cada 30 mts y un tramo de 9.279 mts. 2) Calculamos el grado de curva (G) para 30, 60, 90, 120, 150, 180, 210 y 219,279mts, Si para 219.279mts es 132°15’, entonces para 30mts será 18°05’36.2”; (se o btiene por regla de
tres simple), con igual procedimiento se calcula para las demás distancias.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
65
G / 2 G 7 / G 2 6
G 5 / 2 / G 4 2 / 2 G 3
/ 2 G 2 / 2 G
4
3
1/
2
5
2
6
1 7
PC
PT G
7 G 6 G
5 G
. m 5 9 = R
4 G
3 G
2 G
1 G
O
3) Cálculo de cuerdas para cada punto. Por la fórmula C = 2R * Sen G/2
PUNTO PC-1 PC-2 PC-3 PC-4 PC-5
LONGITUD CUERDA DE CURVA (m). 30 29.875 60 59.008 90 86.672 120 112.180 150 134.897
GRADO DE CURVA 18°05’36.2” 36°11’12.4” 54°16’40.6” 72°22’24.8” 90°28’01”
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
66
2.3.-REPLANTEO
DE
CURVAS
CIRCULARES
HORIZONTALES. Conociendo los elementos de curva circular horizontal podemos calcular el estacado del tramo sobre la longitud de la curva, existen diferentes métodos para replantear las curvas circulares, por condición del terreno enunciaremos los dos métodos más usuales por ángulo de deflexión; el primero es cuando la visibilidad es total de la curva desde el PC. y el segundo método es cuando no es visible la curva desde el PC.(con puntos de cambio). 3.-REPLANTEO POR ANGULOS DE DEFLEXIÓN CON
VISIBILIDAD DESDE EL PRINCIPIO DE CURVA (PC.)
I
G / 2
T A N G E N T E
G 3 / 2 G 2 / 2 G 1/ 2
2
1
3
PC PT
G
I O D R A
3
G
2
G
1
G
O
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FACULTAD DE MINAS U.N..P
67
ING.CARLOS CALLE G.
longitud de arco y cuerda, por geometría tenemos que G = I para ubicar el punto 1 se debe calcular la cuerda PC-1 en función al grado de curva G1, de igual manera para ubicar el punto 2 calcular la cuerda PC-2 en función del grado de curva G2, así sucesivamente hasta la cuerda mayor PC-PT. Para replantear se estaciona el teodolito en PC con el limbo horizontal en el alineamiento o Tangente con 0°0’0”, desde el cual giramos al
punto 1 con un ángulo de G1/2 (mitad del grado de curva para la longitud del arco.) y con una distancia de PC-1 (cuerda). Para el punto 2 medimos un ángulo de G2/2 y una cuerda de PC-2, de ésta manera procedemos para los demás puntos.
EJEMPLO 3. Se tiene una curva circular de 90 m. de radio y un ángulo de intersección de 130°, se quiere replantear cada 60 m.
SOLUCION
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
68
G 130 13 0° T = R * Tg = 90 * Tg = 193 19 3.006 00 6 m. 2 2 G 130 13 0° C = 2 * R * Sen = 2 * 90 * Sen = 163 16 3.135 13 5m. 2 2 13 0° * 90 ΠGR Π *130 LC = = = 204 20 4.204 20 4m. 180 18 0° 180 18 0° G 130 13 0° E = T * Tg = 193 19 3.006 00 6* Tg = 122 12 2.958 95 8m. 4 4
T = 1 9 3 . 0 0 6 m .
6 5 7 5 ° ° 1 0 0 7 ' ' 0 3 8 4 4 0 " ° 1 .3 1 ' 1 9 " 4 9 ° 0 .5 5 ' " 5 4 .8 "
204.20 204.204
163.135
° 0 3 " 1 5 = G 8. " 2 9 ' 3 5 " ' 3 5 3 ° . 2 4 9 °
m. 5 9 9 8 8. m. 5 6 0 3 1. 1 m. 1 5 6 4 1. 5 1
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
69
3) Calcular el grado de curva (G) y cuerda para una longitud de arco de 60, 120, 180 y 204.204m.de acuerdo al cálculo de estacas. Si para una longitud de arco de 204.204m. corresponde un ángulo de 130° y para 60m de arco corresponderá 38°11’49.5”(regla de tres s imple), con el mismo procedimiento
se calcula para 120, 180m. Para calcular la cuerda aplicamos su fórmula: C=2RSenG/2. Del punto PC-1= 2*90*Sen38°11’49.5”/2 = 58.895m. PC-2= 2*90*Sen76°23’39”/2 = 111.306m. de esta manera calculamos las cuerdas.
RESUMEN. PTOS
LONG. DE CURVA.
PC-1
60
GRADO DE CURVA. 38°11’49.5”
PC-2
120
76°23’39”
111.306
38°11’49.5”
PC-3
180
114°35’28.5” 151.465
57°17’44.3”
PC-PT 204.204 130°00’00”
CUERDA (m)
58.895
ÁNG. DEFLEX. G/2 19°05’54.8”
163.135
65°00’00”
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
70
y una distancia (cuerda) de 151.45m. y al PT tenemos la mitad del grado de curva (G) 65° y una cuerda principal de 163.135m. de esta manera queda demostrado.
4.- REPLANTEO POR ANGULOS DE DEFLEXIÓN CON PUNTOS DE CAMBIO. Por principio geométrico tenemos que el ángulo de PC al punto 1 es igual a G/2, o sea la mitad del grado de curva G. En el gráfico para la longitud de arco PC-1 el ángulo de deflexión será G1/2, mitad del grado de curva G1, El ángulo de deflexión en el punto 1 será (G1+G2)/2, La deflexión para el punto 2 será (G2+G3)/2, así sucesivamente hasta el último punto.
2 /
1
G
2 2)/ ( G 1+ G
( G 2 + G 3 )/ 2
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FACULTAD DE MINAS U.N..P
71
ING.CARLOS CALLE G.
EJEMPLO.4. En el problema anterior, replantear con puntos de cambio suponiendo no existe visibilidad al extremo opuesto desde PC.
SOLUCION. 1) En el problema anterior tenemos ubicado tres puntos cada 60 mts. y un tramo de 24.204 mts. 2) El grado de curva calculado para 60 mts. es 38°11’49.5” 3) El grado de curva para 24.204 mts. es 15°24’31.4” 4) Las cuerdas calculadas para 60 mts. de arco es 58.895 mts. y para 24.204 mts. es 24.131 mts. 5) Calculamos la deflexión para cada punto de acuerdo al principio geométrico. Angulo de deflexión en PC = G1/2 Angulo de deflexión en 1 = (G1+G2)/2 Angulo de deflexión en 2 = (G2+G3)/2 Angulo de deflexión en 3 = (G3+G4)/2
RESUMEN. LONG.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
72
V
I
/ 2
.
/ 2 G 2 ) G 1 + ( G 2 .
1
.
T
1
T
(G2+G3)/2
58.895
5 8 .8 9 5
5 8 9 . 5 8
3 2 4 . 1 3
PC G
G2 R
1
G
G 3
G
4
R=90m.
O
PT .
( G 3 + G 4 )/ 2
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FACULTAD DE MINAS U.N..P
73
ING.CARLOS CALLE G.
de 58.895 mts. y finalmente ubicamos el equipo en el último
punto 3, con el mismo procedimiento medimos un ángulo (G3+G4)/2=26°48’10.45” y una distancia de 24.131 mts, de esta manera queda replanteado los tres puntos sobre la curva.
EJEMPLO 5. En el levantamiento del eje de una carretera se tiene el rumbo del PC al punto de intersección V N68°32’E y del punto de intersección al PT S16°44’W, de acuerdo a las características del
terreno pide diseñar una carretera de 120 mts de radio y replantear cada 35 mts. desde el PC. SOLUCION.
'E N68°32 'E I
128°12'
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FACULTAD DE MINAS U.N..P
74
ING.CARLOS CALLE G.
G 128 12 8°12' T = R * Tg = 120 12 0* Tg = 247 24 7.130 13 0 mts. 2 2 G 128 12 8°12' C = 2 * R * Sen = 2 *120 12 0* Sen = 215 21 5.894 mts. 2 2 12 0*128 12 8°12' ΠRG Π *120 LC = = = 268 26 8.501 50 1mts. 180 18 0° 180 18 0° G 128 12 8°12' E = T * Tg = 247 24 7.13 * Tg = 154 15 4.724 72 4 mts. 4 4
3) Cálculo del número de estacas. Conociendo la longitud de curva calculamos el número de estacas No de estac.= 268.501/35 = 7.671, entonces tenemos 7 tramos de 35 mts y uno de 23.501 m. 4) Cálculo del grado de curva para 35 m. y 23.501 m. Si para 268.501 (LC) corresponde un grado de 128°12’ y para 35 m. será 16°42’40.67”, de igual manera el grado para 23.501 m será
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FACULTAD DE MINAS U.N..P
75
ING.CARLOS CALLE G.
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FACULTAD DE MINAS U.N..P
76
ING.CARLOS CALLE G.
6) Cálculo del ángulo de deflexión. Este ángulo viene a ser la mitad G/2 del grado de curva G. como muestra en la última columna del cuadro. Si, del PC-1, G es 16°42’40.7” y G/2 es 8°21’20.4”, PC-2, G es 33°25’21.2” y G/2 es 16°42’40.7”, así sucesivamente hasta el último punto.
CONCLUSIÓN. Para replantear ubicamos el teodolito en PC. Visamos el alineamiento ó el vértice V con 0°00’00” en el limbo horizontal luego giramos al punto 1 con un ángulo G1/2 (8°21’20.4”) y una distancia de 34.876 m. (cuerda), para el punto
2 medimos con un ángulo de G2/2 (16°42’40.7”) y una cuerda de 69.012 m. así sucesivamente hasta visar el PT con un ángulo G/2 (64°06’) y una cuerda de 215.894 m.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
77
4) Para calcular el ángulo de deflexión para cada punto se aplica de acuerdo al principio Geométrico de la siguiente manera:
PC
G 1/2
1
(G1+G2)/2
2 3 1 2 0 M .
V
N68°32'E
G1
G 2 G
3
G
w ' (G2+G3)/2 4 4 ° 4 6 1 5 S
' 2 1 ° 8 2 1
(G3+G4)/2
4
G 5
G
6
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
78
punto 1 visando al PC con el limbo Horizontal en 180°00’00”, 180°00’00”,
en ésta basculamos el anteojo 180° quedando en su proyección ar al punto 2 midiendo un ángulo (G1+G2)/2 = en 0°0’00”, gir ar 16°42’40.7” y una cuerda de 34.876 mts. así sucesivamente
hasta llegar hasta el penúltimo punto con los mismos valores por tener distancias y grados de curvas iguales, en el último tramo,
punto
7
varía
el
ángulo
y
la
cuerda
en
(G7+G8)/2=13°57’58” y una distancia de 23.464 mts. de esta manera queda establecido todo los puntos de la curva.
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FACULTAD DE MINAS U.N..P
79
ING.CARLOS CALLE G.
EJEMPLO 7. La ubicación de estacas en un alineamiento que tiene un rumbo de S62°20’E, Llegando al punto de intersección con una longitud del
proyecto de 3460 m. o correspondiente a la progresiva Km 3+460m. a partir de ésta, cambia de dirección a S42°51’W, se quiere replantear cada 25 mts. en cantidades enteras con un radio de 80 mts, calcular las progresivas, ángulo de deflexión y cuerdas para cada punto.
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FACULTAD DE MINAS U.N..P
80
ING.CARLOS CALLE G.
1) La distancia del proyecto hasta el punto de intersección “V” es 3460 mts correspondiente a la progresiva Km 3+460 2) El grado de curva “G” es igual a I=62°20’+42°51’=105°11’, entonces G=I= 105°11’
3) cálculo de los elementos de curva G 105 10 5°11' T = RTg RTg = 80Tg = 104 10 4.60 m. 2 2 10 5°11' ΠRG Π * 80 *105 LC = = = 146 14 6.864 86 4 m. 180 18 0° 180 18 0° G 105 10 5°11' C 2RSen 2 * 80 * Sen 127 12 7 092 09 2
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FACULTAD DE MINAS U.N..P
81
ING.CARLOS CALLE G.
6) Los siguientes puntos sobre la curva será: (en el cuadro se muestra desde el punto 1).
PUNTO
DISTANCIA
PROGRESIVA.
PC
3355.4
3+355.4
1
3375
3+375
2
3400
3+400
3
3425
3+425
4
3450
3+450
5
3475
3+475
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FACULTAD DE MINAS U.N..P
82
ING.CARLOS CALLE G.
7) Para llegar al PT se suma la Longitud de curva al PC, entonces, 3355.4+146.864=3502.264 (Km 3+502.3), Hasta el momento se ha calculado las distancias sobre la curva y sus progresivas de los seis puntos, del PC al PT.
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FACULTAD DE MINAS U.N..P
ING.CARLOS CALLE G.
83
RESUMEN. PTO
LONG. DE CURVA
GRADO DE CURVA (G)
CUERDA (m)
ÁNG. DE DEFLEX.(G/2)
PC-1
19.60
14°02’14.75” 19.551
7°01’07.38”
PC-2
44.60
31°56’32.35” 44.025
15°58’16.18”
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FACULTAD DE MINAS U.N..P
84
ING.CARLOS CALLE G.
EJEMPLO 8. En el problema anterior con los elementos de curva calculados replantear cada 30 mts. en cantidades enteras con puntos de cambio. SOLUCION: 1) Graficando el croquis, se tiene calculado los elementos de
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FACULTAD DE MINAS U.N..P
85
ING.CARLOS CALLE G.
2) La progresiva de PT es Km 3+502.3, ésta se obtiene sumando la Longitud de Curva al PC. 3) El primer punto sobre la curva es Km 3+360 por ser un cantidad inmediata entera que se obtiene sumando 4.60 mts (3355.4 + 4.6 = 3360 = Km 3+360, en la siguiente tabla
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FACULTAD DE MINAS U.N..P
86
ING.CARLOS CALLE G.
5) CONCLUSION. Calculado los ángulos de deflexión para cada punto y sus respectivas cuerdas iniciamos el replanteo estacionar el teodolito en PC, cuya progresiva es Km 3+355.4 desde el cual hacemos la vista atrás al punto de intersección con el limbo horizontal en 0°00’00” luego giramos al punto 1
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FACULTAD DE MINAS U.N..P
SOLUCION.
87
ING.CARLOS CALLE G.
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FACULTAD DE MINAS U.N..P
88
2) Coordenadas parciales de PC,
ING.CARLOS CALLE G.
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FACULTAD DE MINAS U.N..P
PUNTOS ÁNG. D.= G/2
89
ING.CARLOS CALLE G.
AZIMUT
DH=C
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FACULTAD DE MINAS U.N..P
PROBLEMA PROPUESTO.
90
ING.CARLOS CALLE G.
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FACULTAD DE MINAS U.N..P
91
ING.CARLOS CALLE G.
Los perfiles se dibujan en el plano vertical.
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