FORMULAS Y TABLAS
ESTRUCTURAS DE HORMIGÓN ARMADO
ALUMNO/A: …………………………………………………………
1º Ed. 2013 (EHE 08; EC2)
ESTE TEXTO ES PARA REALIZAR LOS EXAMENES DE LAS ASIGNATURAS ESTRUCTURAS DE HORMIGÓN Y HORMIGÓN PRETENSADO. ESTA TERMINANTEMENTE PROHIBIDO ESCRIBIR NADA EN ÉL. SERÁ REVISADO POR EL PROFESOR Y PUEDE SIGNIFICAR LA DESCALIFICACIÓN DEL EXAMEN.
1
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INDICE 1 2 3 4 5 6 7 8 9 10
MATERIALES Y BASES DE CALCULO ELU. VIGAS Y COLUMNAS. FLEXION SIMPLE Y COMPUESTA ELU. VIGAS Y COLUMNAS. COMPRESIÓN COMPUESTA ELU. PANDEO ELU. CORTANTE. PUNZONAMIENTO. RASANTE. TORSION. ELS. FISURACIONES. DEFORMACIONES. VIBRACIONES VIGAS EN T CIMENTACIONES LOSAS/PLACAS MUROS/PANTALLAS
03 19 26 43 46 55 62 67 86 92
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1.MATERIALES Y BASES DE CALCULO Componentes del hormigón Cementos Tipos de cementos más comunes en construcción
Cementos comunes CEM: CEM I: Cemento portland CEM II: Cementos portland con adiciones Escoria de alto horno Humo de sílice Puzolanas naturales Cenizas volantes Caliza Mixto CEM III: Cemento de alto horno CEM IV: Cemento puzolánico CEM V: Cemento compuesto
(CEM I) (CEM II/A ó B-S) (CEM II/A-D) (CEM II/A ó B-P) (CEM II/A ó B-V) (CEM II/A-L) (CEM II/A ó B-M) (CEM III/A ó B) (CEM IV/A ó B) (CEM V/A)
Cementos blancos BL , tipos I, II y V Cementos para usos especiales ESP VI-1 y 2 Cemento de aluminato de calcio CAC/R (endurecimiento rápido) Cementos con características adicionales: Cementos resistentes a sulfatos o aguas de mar (SR, MR) Cementos de bajo calor de hidratación (BC)
RC 97. Tipos de cemento y composiciones: proporción en masa (1) Tipo de Cem. CEM I
Escoria de Humo de Puzolanas Cenizas Caliza Componentes Denominación Designación Clínker K Horno alto Sílice D Naturales Volantes L Minoritarios S P V Adiciones (2) (2) Cemento CEM I 95-100 0-5 —— —— —— —— —— Portland Cem Portland CEM II/A-S 80-94 6-20 0-5 —— —— —— —— con escoria CEM II/B-S 65-79 21-35 0-5 —— —— —— —— Cem Portland + CEM II/A-D 90-94 6-10 0-5 —— —— —— —— humo de sílice Cem Portland CEM II/A-P 80-94 6-20 0-5 —— —— —— —— CEM II con puzolana CEM II/B-P 65-79 21-35 0-5 —— —— —— —— Cem Portland + CEM II/A-V 80-94 6-20 0-5 —— —— —— —— ceniza volante CEM II/B-V 65-79 21-35 —— 0-5 —— —— —— —— —— —— —— Cem Portland CEM II/A-L 80-94 6-20 0-5 con caliza Cem Portland CEM II/A-M 80-94 6-20 (4) (5) Mixto (3). CEM II/B-M 65-79 21-35 (4) (5) (6) CEM III Cemento de CEM III/A 35-64 36-65 0-5 —— —— —— —— horno Alto CEM III/B 20-34 66-80 0-5 —— —— —— —— CEM IV Cemento CEM IV/A 65-89 11-35 (4) 0-5 —— —— —— —— Puzolánico CEM IV/B 45-64 36-55 (4) 0-5 CEM V Compuesto CEM V/A 40-64 18-30 1830 0-5 —— ——
1. 2.
Los valores de la tabla tabla se refieren al al núcleo de cemento, entendiéndose entendiéndose por tal el "clínker" "clínker" y las las adiciones con exclusión del sulfato de calcio (regulador de fraguado) y de los aditivos. Los componentes componentes minoritarios adicionales pueden ser "filler", o uno o más de los componentes componentes principales, principales, a
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RC 97. Prescripciones mecánicas y físicas de los cementos comunes Resistencia a compresión [N/mm ] Clase resistente
Resistencia inicial 2 días
7 días
32,5
-
16,0
32,5R (1)
13,5
-
42,5
13,5
-
42,5 R (1)
20,0
-
52,5
20,0
-
52,5 R (1)
30,0
-
Tiempo de fraguado
Resistencia normal a 28 días 32,5
Principio [Minutos]
Final [Horas]
Expansión [mm]
52,5 60
42,5
62,5
52,5
-
12
10
45
(1) R = Alta resistencia inicial
En general se acepta utilizar: Tipo de hormigón Hormigón en masa Hormigón armado Hormigón pretensado
Tipo de cemento Cementos comunes Cementos para usos especiales (BC, etc.) Cementos comunes Cementos comunes tipo CEM I y II/ A-D
Áridos
SUSTANCIAS PERJUDICIALES
Terrones de arcilla, determinados con arreglo al método de ensayo indicado en la UNE 7133:58. Partículas blandas, determinadas con arreglo al método de ensayo indicado en la UNE 134:58. Material retenido por el tamiz 0,063 UNE EN 933-2:96 y que flota en líquido de peso específico 2, determinado con arreglo al método de ensayo indicado en la UNE EN 1744-1:98. Compuestos totales de azufre expresados en SO3= y referidos al árido seco, determinados con arreglo al método de ensayo indicado en la UNE EN 17441:98. Sulfatos solubles en ácidos, expresados en SO 3= y referidos al árido seco, determinados con arreglo al método de ensayo indicado en la UNE EN 17441:98. Cloruros expresados en CI- y referidos al Hormigón armado u hormigón en árido seco, determinados con arreglo al masa que contenga armaduras método de ensayo indicado en la UNE EN para reducir la fisuración. 1744-1:98. Hormigón pretensado.
Parábola de Fuller: P E = 100
d
Cantidad máxima en % del peso total de la muestra rido rido fino grueso 1.00 0.25 --
5.00
0.50
1.00
1.00
1.00
0.80
0.80
0.05
0.05
0.03
0.03
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Curva de Bolomey: P E = a + (100-a)
d D
Cortesía de Jiménez Montoya. Consistencia Seca y Plástica Blanda Fluida
Valores de a Rodados Machacados 10 12 11 13 12 14
Cortesía de Jiménez Montoya
Tamices normalizados por las normas UNE, que son los que se indican:
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Límites Superior Inferior
4 mm 0 20
Material retenido acumulado, en % en peso, en los tamices 2 mm 1 mm 0.5 mm 0.25 mm 0.125 mm 0.063 mm 4 16 40 70 82 (1) 38 60 82 94 100 100
El módulo de finura representa finura representa el tamiz medio, o sea, la abertura o tamaño medio de la partícula retenida en él y se describe para las arenas, cuál es el módulo de finura mínimo y máximo que tienen que tener para emplearse en hormigones. Este módulo, las clasifica en finas, medias y gruesas. El coeficiente de forma ( forma ( ) de un árido, es obtenido a partir de un conjunto de n granos representativos de dicho árido, mediante la expresión:
V1 V2 ..... V n
6
d 3 d 3 .....d 3 1
2
n
donde es: = coeficiente de forma; = volumen de cada grano; V i i = = la mayor dimensión de cada grano, es decir, la distancia entre los dos planos paralelos y tangentes a d i i = ese grano que estén más alejados entre sí de entre todos los que sea posible trazar. El tamaño máximo del árido grueso será grueso será menor que las dimensiones siguientes: a) 0,8 veces la distancia horizontal libre entre vainas o armaduras que no formen grupo, o entre un borde de la pieza y una vaina o armadura que forme un ángulo mayor que 45o con la dirección de hormigonado. b) 1,25 veces la distancia entre un borde de la pieza y una vaina o armadura que forme un ángulo no mayor que 45o con la dirección de hormigonado. c) 0,25 veces la dimensión mínima de la pieza, excepto en los casos siguientes: - Losa superior de los forjados, donde el tamaño máximo del árido será menor que 0,4 veces el espesor mínimo. - Piezas de ejecución muy cuidada (caso de p refabricación en taller) y aquellos elementos en los que el efecto pared del encofrado sea reducido (forjados que se encofran por una sola cara), en cuyo caso será menor que 0,33 veces el espesor mínimo. Los áridos se designan como: d /D – IL - N donde. d /D, indica fracción granulométrica comprendida entre un tamaño mínimo d y y un tamaño máximo D, en mm, IL, indica forma de Presentación ( R, rodado; T, triturado (machacado); M, mezcla), N, indica Naturaleza (C, calizo; S, silíceo; G, granito; O, ofita; B, basalto; D, dolomítico; Q, traquita; I, fonolita; V, varios; A, artificial; R, reciclado). Agua Exponente de hidrógeno pH
5
Sustancias disueltas
15 gr/litro
Sulfatos SO4= excepto para cemento SR que llega a 5 gr/litro
1 gr/litro
Ion cloruro Cl
Para hormigón pretensado
1 gr/litro
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Aditivos
Inclusores de aire: aire: para elaborar hormigones menos densos, más ligeros. Debido a la rotura de capilares interiores, aumentan la resistencia a heladas, en general. Fluidificantes: Fluidificantes: aumentan la docilidad y permiten disminuir el agua de amasado. Superfluidificantes: Superfluidificantes: son mejores que los anteriores porque permiten disminuir mayor cantidad de agua. De los más usados en la práctica. Aceleradores de fraguado (o de endurecimiento) endurecimiento):: para prefabricados, climas muy fríos, etc. Retardadores de fraguado: fraguado: grandes macizos con gran desprendimiento desprendimiento de calor de hidratación. Son incorporados en todos los hormigones de Central actualmente, para permitir un tiempo mínimo para transporte y colocación adecuada. Inhibidores de fraguado: fraguado: similares a los anteriores pero para transportar hormigones a muy largas distancias (camiones de transporte o Mixers). Anticongelantes: Anticongelantes: climas muy fríos. fríos. Impermeabilizantes: Impermeabilizantes: estructuras estancas. Inhibidores de corrosión de armaduras: armaduras: ambientes muy agresivos. Inhibidores de reacción álcali-áridos: álcali-áridos: evitan la reacción entre el cemento y los áridos. Para el bombeo: bombeo: reducen el rozamiento en la tubería de bombeo. Hormigones proyectados: disminuyen proyectados: disminuyen el descuelgue o caída. Para inyecciones: inyecciones: aumentan la consistencia. Colorantes: fin estético.
Adiciones Recogidas y aceptadas en EHE: Ceniza volante: volante: son las cenizas que salen por las chimeneas de las centrales termoeléctricas, en estas chimeneas hay unos filtros para que estas cenizas cenizas sean recogidas y no salgan salgan al ambiente. Humo de sílice: sílice: procede de la producción de microprocesadores. Directamente de la fundición de sílice. No recogidas en EHE : Escoria: Escoria: las sustancias vítreas que flotan en el acero en los procesos de fundidos. Son impurezas del acero que se retiran y que se aprovechan para el cemento a precio muy barato. Puzolana natural : es un tipo de piedra natural de naturaleza volcánica procedente de Puzuoli (Italia) o de similares características. Esta piedra ya la conocían los romanos y la empleaban para sus construcciones. Filler calizo: calizo: es el polvo de las canteras de caliza, residuos de triturado en las canteras. De todas las adiciones es la menos contaminante, y junto con el humo de sílice la que más resistencia aporta al cemento.
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Esta armadura se utiliza en lo que se conoce habitualmente como “mallazo” y las mallas estándar pueden
ser de retícula cuadrada de 15x15 [cm] y 20x20 [cm] o rectangular de 15x30 [cm] y 20x30 [cm]. Aunque también se hacen adaptándose a la necesidad de obra. Los diámetros nominales de los alambres lisos o corrugados, más empleados en las armaduras básicas electrosoldadas se ajustan a: 5 - 6 - 7 - 8 - 9 - 10 y 12 [mm] Las características mecánicas de los aceros Tipo de acero
Acero soldable
Acero soldable con características características especiales de ductilidad
Designación Límite elástico (f y) en N/mm Límite rotura (f s) en N/mm
2
2
Alargamiento de rotura, ε u,5 (%) (sobre
B 400 S
B 500 S
B 400 SD
B 500 SD
≥ 400
≥ 500
≥ 400
≥ 500
≥ 440
≥ 550
≥ 480
≥ 575
≥ 14
≥ 12
≥ 20
≥ 16
≥ 5.0
≥ 5.0
≥ 7.5
≥ 7.5
≥ 7.5
≥ 7.5
≥ 10.0
≥ 10.0
≥ 1.05
≥ 1.05
1.20 ≤ f s/f y ≤ 1.35
1.15 ≤ f s/f y ≤ 1.35
---
---
≤ 1.20
≤ 1.25
base de 5 ) Alargamiento
Acero suministrado
total bajo carga
en barra
máxima,
Acero suministrado
εmax (%)
en rollo
Relación f s/f y Relación f y real /f y nominal
Existen dos tipos de aceros:
Acero trefilado (T) Acero de alta soldabilidad (S)
La nomenclatura de las barras de acero utilizadas es B 400 S (ó SD), B 500 S (ó SD) y B 500 T. Donde:
B indica barra de acero 400/500 es es el valor del límite elástico en Mpa S indica que es soldable (certificado por AENOR) SD significa que es soldable y de alta ductilidad (también certificado) T significa significa que está trefilado para mallas electrosoldadas. Tipo de acero para alambres Ensayo de tracción
Ensayo de doblado-
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Propiedades del Hormigón fresco Las consistencias EHE son: Tipo de consistencia Seca Plástica Blanda Fluida Líquida
Asiento en cm 0-2 3-5 6-9 10 - 15 16 - 20
EC 2 indica las siguientes consistencias: Tipo de consistencia S1 (seca) S2 (plástica) S3 (blanda) S4 (fluida)
Asiento en cm 1-4 5-9 10 - 15 >16
De la consistencia del hormigón dependerá, el método de vibrado: Tipo de consistencia Seca Plástica Blanda Fluida Líquida
La relación Agua - Cemento (A/C)
Tipo de Compactación Vibrado enérgico Vibrado Normal Vibrado Normal o picado con barra Picado con barra No necesario
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Tiempos de desencofrado: Elementos Lados de vigas y columnas Losas o placas (no prefab.) Fondo de vigas de hasta 7 m Idem de luces mayores
Tiempos 3 días 8 días 21 días 3 x Luz
Para EHE: Temperatura superficial del hormigón (ºC) Encofrado vertical Losas Fondos de encofrado Puntales Vigas Fondos de encofrado Puntales
> 24º
> 16º
> 8º
> 2º
9 horas
12 horas
18 horas
30 horas
2 días 7 días
3 días 9 días
5 días 13 días
8 días 20 días
7 días 10 días
9 días 13 días
13 días 18 días
20 días 28 días
Características Características mecánicas del hormigón endurecido Determinación de la resistencia del hormigón Tabla de conversión entre distintas probetas que es: Coeficiente de conversión a la probeta cilíndrica de 15 x 30 cm Tipo de Dimensiones (cm) probeta Límites de variación 15 x 30 --10 x 20 0.94 a 1.00 Cilindro 25 x 50 1.00 a 1.10 10 0.70 a 0.90 15 0.70 a 0.90 Cubo 20 0.75 a 0.90 30 0.80 a 1.00 15 x 15 x 45 0.90 a 1.20 Prisma 20 x 20 x 60 0.90 a 1.20
Valores medios 1.00 0.97 1.05 0.80 0.80 0.83 0.90 1.05 1.05
Indicado en una tabla es: Edad del hormigón (días) Hormigón de endurecimiento normal Hormigón con endurecimiento endurecimiento rápido
3 0.40 0.55
7 0.65 0.75
28 1.00 1.00
90 1.20 1.15
360 1.35 1.20
Resistencia Característica y resistencia de Cálculo Resistencia media: f m
f , ci
n
siendo, n el número de probetas y f ci las resistencias últimas de cada probeta.
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EHE define una relación entre la resistencia característica real y la resistencia media: fck fcreal f cm 1 1.64 1.64
Valores de Resistencias a Compresión: 20; 25; 30; 35; 40; 45; 50; 55; 60; 70; 80; 90; 100
(en N/mm2 o Mpa):
La de 20 vale sólo para HM, entre 25 y 50 son de resistencia normal, y valores mayores de 50 corresponden a Alta Resistencia (AR). Resistencia a Tracción: fct ,m 0.30 3 f ck 2 para fck 50 N / mm 2 fct ,m 0.58 3 f ck 2 para fck 50 N / mm 2
La resistencia media a flexotracción, flexotracción, f ct ,m ,fl , viene dada por la siguiente expresión que es función del canto total del elemento (h) en mm: fct ,m,fl max 1.6 h / 10 1000 fct ,m; f ct , m
Luego, con estos valores medios se halla la resistencia característica a tracción como: tracción como: fct ,k 0.70 f ct ,m
Resistencia de cálculo del hormigón en compresión f f cd cc ck c
Donde: 0.85 cc 1 . f f ctd ct ct ,k c
Tipificación del hormigón Para EHE: T – – R / / C / / TM / A / A,, donde:
T tipo tipo de hormigón: en masa (HM), hormigón armado (HA) u hormigón pretensado (HP) R es es la resistencia característica especificada en N/mm2 C es la letra inicial del tipo de consistencia (S, P, B, F y L) TM es es el tamaño máximo del árido en milímetros A es A es la designación del ambiente (se verá más adelante)
Para EC2: densidad normal (C), baja densidad o ligeros (LC) y alta densidad o pesados (HC). Siempre C denota concrete. concrete. Se tiene: C25/30; C30/37; C40/50; C45/55, C50/60, etc., donde el primer valor es la resistencia cilíndrica y el segundo la cúbica. Los hormigones van desde C16/20 hasta C90/105 y superiores.
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Edad del hormigón, en días Hormigón de endurecimiento normal Hormigón con endurecimiento rápido
3 0.40 0.55
7 0.65 0.75
28 1.00 1.00
90 1.20 1.15
360 1.35 1.20
Bases de Cálculo Valores característicos y ponderados de las acciones F d = f F k k d = Donde: F d d , , valor de cálculo. f , coeficiente parcial de seguridad de la acción considerada. F k k, valor representativo (en general característico, extraído de su norma correspondiente). Peso propio del hormigón (Usar ( Usar las siguientes densidades de EC2 y EHE): 2300 kg/m3 2400 kg/m3 2500 kg/m3
HM: HA y HP:
si si
2 f ck ck ≤ 50 N/mm 2 f ck ck > 50 N/mm
Coeficientes de seguridad
Para ELU Tipo de acción
Permanente Pretensado Permanente de valor no constante Variable Accidental
Situación persistente o transitoria Efecto Efecto desfavorable favorable G = 1.00 G = 1.35 P = 1.00 P = 1.00
Situación accidental Efecto Efecto favorable desfavorable G = 1.00 G = 1.00 P = 1.00 P = 1.00
G* = 1.00
G* = 1.50
G* = 1.00
G* = 1.00
= 0.00 --
= 1.50 --
= 0.00 A = 1.00
= 1.00 A = 1.00
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2,iQk ,i : valores representativos cuasi permanentes de las acciones variables con la acción determinante
o con la acción accidental Ak : valor característico de la acción accidental AE ,k : valor característico de la acción sísmica Coeficientes parciales de seguridad de los materiales Situación de proyecto Persistente o transitoria Accidental
Hormigón ( c ) 1.50 1.30
Curvas de tensión - deformación para el acero y el hormigón f f cd cc ck c
-
;
f ctd
f ct ,k c
; 0.85 cc 1.00
para el hormigón
c 0
cu
Ecuación de la parábola n c c f cd 1 1 ; 0
0 c c 0
Acero pasivo y activo( s ) 1.15 1.00
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cu
Cortesía de Jiménez Montoya
Vida útil de una estructura Tipo de estructura Estructuras de carácter temporal Elementos reemplazables que no forman parte de la estructura principal (por ejemplo, barandillas, apoyos de tuberías) Edificios (o instalaciones) instalaciones) agrícolas o industriales y obras marítimas Edificios de viviendas u oficinas, puentes u obras de paso de longitud total inferior a 10 metros y estructuras de ingeniería civil (excepto obras marítimas) de repercusión económica baja o media Edificios de carácter monumental o de importancia especial Puentes de longitud total igual o superior a 10 metros y otras estructuras de ingeniería civil de repercusión económica alta
Hipótesis y bases de cálculo en hormigón armado El método de los Estados Límites: ELD. Estado Límite de Durabilidad
Vida útil nominal Entre 3 y 10 años Entre 10 y 25 años Entre 15 y 50 años 50 años 100 años 100 años
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a n i r a M
a e r é a
IIIa
-elementos de estructuras marinas, por
-edificaciones -edificaciones en las proximidades de la
encima del nivel de pleamar
costa
corrosión por
-elemento exteriores de estructuras
-puentes en las proximidades de la costa
cloruros
situadas en las proximidades de la línea
-zonas aéreas de diques, pantalanes y
costera (a menos de 5 km)
otras obras de defensa litoral -instalaciones -instalaciones portuarias
a d i g r e m u s
e d a n o z n E
corrosión por IIIb
cloruros
-elementos de estructuras marinas
-zonas sumergidas de diques y otras
sumergidas permanentemente, permanentemente, por debajo
obras de defensa litoral
del nivel mínimo de bajamar
-cimentaciones -cimentaciones y zonas sumergidas de pilas de puentes -zonas situadas en el recorrido de marea
s a e r a m
corrosión por
-elementos de estructuras marinas situadas de diques y otras obras de defensa litoral
cloruros
en la zona de carrera de mareas.
IIIc
situadas en el recorrido de marea.
con cloruros de origen
IV
diferente del
-zonas de pilas de puentes sobre el mar,
corrosión por cloruros
medio marino
-instalaciones -instalaciones en contacto con agua con
-piscinas
cloruros, no relacionados con el ambiente
-pilas de pasos superiores o pasarelas
marino
en zonas de nieve
-superficies expuestas a sales de deshielo
-estaciones -estaciones de tratamiento de agua.
no impermeabilizadas.
Clases Específicas de exposición (relativas a otros procesos distintos de corrosión) Clase
Subclase l i b é d
Des
Qa
Tipo de proceso
ataque químico
Descripción
Ejemplos
-elementos situados en ambientes con contenidos
-instalaciones industriales, con sustancias
de sustancias químicas capaces de provocar la
débilmente agresivas según tabla 8.2.3.b.
alteración del hormigón con velocidad lenta lenta (ver
-construcciones en áreas industriales, con
Tabla 8.2.3.b)
agresividad débil.
-elementos en contacto con agua de mar
-dolos, bloques y otros elementos para diques
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Clasificación de la agresividad química TIPO TIPO DE DE EXPOS EXPOSIC ICII N TIPO DE MEDIO
PARÁMETROS
AGRESIVO
AGUA
SUELO
Qa
Qb
Qc
ATAQUE
ATAQUE
ATAQUE
DÉBIL
MEDIO
FUERTE
6,5 - 5,5
5,5 - 4,5
< 4,5
CO2 AGRESIVO (mg CO2/ l)
15 - 40
40 - 100
> 100
IÓN AMONIO (mg NH4+ / l)
15 - 30
30 - 60
> 60
IÓN MAGNESIO (mg Mg2+ / l)
300 - 1000
1000 - 3000
> 3000
IÓN SULFATO (mg SO42- / l)
200 - 600
600 - 3000
> 3000
RESIDUO SECO (mg / l)
75 – 150
50 – 75
<50
GRADO DE ACIDEZ BAUMANN-GULLY
> 20
(*)
(*)
IÓN SULFATO (mg SO42- / kg de suelo seco)
2000 - 3000
3000-12000
> 12000
VALOR DEL pH
Máxima relación agua/cemento y mínimo contenido de cemento CLASE DE EXPOSICIÓN
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Recubrimientos mínimos [mm] mínimos [mm] para las clases generales de exposición I y II Clase de exposición
Tipo de cemento
I
Cualesquiera CEM I
II a
II b
Resistencia característica del hormigón [Mpa]
Vida útil de proyecto (tg) [años]
f ck 25 25 f ck 40 f ck 40
50 15
100 25
15 10
25 20 30 25
Otros tipos o empleo de adiciones al hormigón
25 f ck 40
CEM I
f ck 40
20 15
25 f ck 40
20
30
f ck 40
15
25
25 f ck 40
25
35
f ck 40
20
30
Otros tipos o empleo de adiciones al hormigón
Recubrimientos mínimos [mm] mínimos [mm] para las clases generales de exposición III y IV Hormigón Armado
Tipo de cemento CEM III/A, CEM III B, CEM IV, CEM II/B-S, B-P, B-V, A-D ó con
Vida útil de proyecto (tg) [años]
Clase de exposición III a
III b
IIIc
IV
50
25
30
35
35
100
30
35
40
40
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Recubrimientos nominales (a nominales (a aplicar): rnom rmin r
Donde es: r nom , es el recubrimiento nominal o o de aplicación r min , es el recubrimiento mínimo obtenido mínimo obtenido de la tabla correspondiente r , es el margen de recubrimiento, recubrimiento, que es función del control de ejecución y se adoptará: 0 mm, en elementos prefabricados con control intenso de ejecución, 5 mm, para elementos hormigonados in situ y nivel intenso de control y, 10 mm, en el resto de los casos. Control del valor máximo de abertura de fisura. w máx máx [mm]
Clase de exposición
Hormigón armado
Hormigón pretensado
I
0,4
0,2
IIa, IIa, IIb, IIb, H
0,3
0,2
IIIa, IIIb, IV, F, Qa (2)
0,2
(2)
(2)
Descompresión
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VIGAS (CG mínima en tracción. Se recomienda poner 30 % en compresión) MUROS (cuantía mínima vertical en la cara de tracción. En Arm. Horiz. la cara de compresión se recomienda poner el 30 %) Arm. Vert. *Referida al espesor de la capa de compresión hormigonada
3,3 4.0 1,2
1000 [‰] (para sección rectangular); en vigas se suele usar Se define CG (A / b h) 10 controlar mejor la cuantía máxima (ductilidad seccional). s
2,8 3,2 0,9 Ac b d ,
que permite
Estados Límites de Servicio (ELS) Control de: -
Fisuraciones Deformaciones Vibraciones
2. ELU. VIGAS Y COLUMNAS. COLUMNAS. FLEXIÓN SIMPLE Y COMPUESTA Dominios de deformación. ELU ante tensiones normales ( fck 50Mpa , cc 0.85 )
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1 0.827 0.8278 8 0.8278 0.8278 1 2.416 2.416 (Expresión de la TABLA)
T U A fyd
;
A f yd b d f cd
T U A fyd b d fcd
Si se coloca armadura de tracción y de compresión, se distinguen los valores de Capacidad mecánica con dos subíndices, 1 y 2, respectivamente. respectivamente. Así: T1 U1 A1 fyd ; T2 U2 A2 2
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Tabla de Capacidades Mecánicas [kN] para acero B 400 S Diám. [mm] 6 8 10 12 14 16 20 25 32 40
1 9.83 17.48 27.32 39.34 53.54 69.93 109.22 170.74 279.74 437.09
2 19.67 34.97 54.64 78.67 107.09 139.87 218.43 341.48 341.48 559.32 874.18 874.18
3 29.50 52.45 81.96 118.01 160.63 209.8 327.65 512.22 512.22 839.21 1311.3 1311.3
NUMERO DE BARRAS 4 5 6 7 39.33 49.17 58.99 68.83 69.93 87.42 104.9 122.39 109.27 136.59 163.91 191.23 157.35 196.69 236.03 275.37 214.17 267.72 321.26 374.8 279.74 349.67 419.61 489.54 436.87 546.09 655.3 764.52 682.96 853.69 1024.4 1195.2 1195.2 1119 1398.7 1678.4 1958.2 1748.4 1748.4 2185.5 2622.5 3059.6 3059.6
8 78.66 139.87 218.55 314.71 428.35 559.48 873.74 1365.9 1365.9 2237.9 3496.7 3496.7
9 88.49 157.35 245.86 354.05 481.89 629.41 982.96 1536.6 2517.6 3933.8
10 98.33 174.83 273.18 393.39 535.44 699.35 1152.7 1707.4 2797.4 4370.9
8 98.33 174.83 273.18 393.38 535.43
9 110.62 196.69 307.33 442.56 602.37
10 122.91 218.54 341.48 491.74 669.30
Tabla de Capacidades Mecánicas [kN] para acero B 500 S Diám. [mm] 6 8 10 12 14
1 12.29 21.85 34.15 49.17 66.93
NUMERO DE BARRAS 2 3 4 5 6 7 24.58 36.87 49.17 61.46 73.75 86.04 43.71 65.56 87.42 109.27 131.13 152.98 68.29 102.44 136.59 170.74 204.89 239.03 98.34 147.52 196.69 245.86 295.03 344.21 133.86 200.79 267.72 334.65 401.58 468.50
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3) A) Si 0 0 ; prescindiendo de la colaboración del hormigón, puede tomarse: 1 (1 2 )
donde era: ' 2 d2 d r d B) Para 0 0 0.310 ; este es el caso más frecuente. Se fija el máximo en REC 0.310 . con 0 se obtiene 0 , y luego: 0 2 (1 2 )
Para el caso en que 0.1667 , debe hacerse 2 0 , con lo que directamente se halla 0 .
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0.310 , se entra en tabla con 103 0 / f yd (para f yd en kg/cm2) ó 102 0 / f yd (para f yd en 3. Si 0
N/mm2). Con este valor se encuentra 0 y . 0 2 (1 2 )
Indices de Sobrerresistencia y de Sobreseguridad
Coeficiente o índice de sobrerresistencia
Donde es:
T ' T T
100 [%]
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– c) la tercera parte de la longitud básica de anclaje para barras traccionadas y los dos tercios de dicha longitud para barras comprimidas.
Tipos de anclajes (varían anclajes (varían las longitudes, Ver EHE):
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Tabla universal para flexión simple o compuesta 0.0890 0.1042 0.1181 0.1312 0.1438
CC 0.85
CC 0.90
CC 0.95
CC 1.00
0.0300 0.0400 0.0500 0.0600 0.0700
0.0310 0.0415 0.0522 0.0630 0.0739
0.0318 0.0424 0.0529 0.0635 0.0741
0.0328 0.0439 0.0553 0.0667 0.0782
0.0335 0.0447 0.0559 0.0671 0.0782
0.0346 0.0464 0.0583 0.0704 0.0826
0.0353 0.0471 0.0588 0.0706 0.0824
0.0365 0.0488 0.0614 0.0741 0.0869
D O M
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3. ELU. VIGAS Y COLUMNAS. COMPRESIÓN COMPUESTA
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Estructura Gimnasios o palacios de deportes Salas de fiestas o conciertos sin asientos fijos Salas de fiestas o conciertos con asientos fijos
Frecuencia [Hz] > 8.0 > 7.0 > 3.4
7. VIGAS EN T Definición geométrica de sección en T b be
Donde es: hf b1
b1
Ala
h
Nervio bw
Unas fórmulas muy utilizadas son: a) Para alas comprimidas: be bw
1 l b 5
donde, l es es la distancia entre puntos de momento nulo En el caso particular de vigas de borde, con ala en un solo sentido: be bw
1 l b 10
b) Para alas traccionadas: be bw 8 hf b
Cuando sean vigas de borde: be bw 4 hf b
b: anchura de la cabeza be: anchura eficaz de la cabeza b1: vuelo eficaz del ala bw : anchura del nervio h: altura total hf :: altura de la cabeza
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b 'w bw 2 bo , si bo h0 b 'w bw 2 ho , si bo h0
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Ecuaciones de equilibrio y compatibilidad para secciones T
hf
M 0
y=0.36 y=0.36 d d M n
y=0.36 y=0.36 d d x=y/ 0.80= 0.80=0.45 d 0.45 d
y=hf x=y/ 0.80= 0.80=1.25 h 1.25 hf
Suponiendo un valor de cc 0.85 , la armadura se encontrará como: 1.
Para Md M 0 ,
2.
Para M0 Md M n ,
M d b d 2 f cd
T b d fcd
Se puede hacer: 0
Md M 0 bw (d hf )2 fcd
0 0.85
b
hf
bw d hf
Donde de forma simplificada se puede hacer: (1 ) 0. 0.85
hf b bw d hf
Finalmente: T bw (d hf ) fcd
También se puede usar: T 0.85 fcd b hf
3.
Md M 0 0.90 d hh
Para Md M n , es necesaria la colocación de armadura comprimida y, de forma aproximada: 2
Md M n d d 2
1 0.85 fcd 0.36 bw d (b bw ) hf 2
Comprobación de la sección en T ( cc 0.85 ) Para realizar una comprobación ( Md M u ), se parte de considerar que no existe armadura de compresión y, desde la primera ecuación de equilibrio, se halla el valor de la profundidad del bloque comprimido (y (y ): ): 0 0.85 fcd bw y b bw hf A1 f yd A f
0.85 f
b
b
h
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Mu 0.85 fcd b y d 0.5 y
2.
Para hf y 0.36 d , la fibra cae en el nervio y el momento se deduce como: Mu 0.85 fcd bw y d 0.5 y b bw hf d 0.5 hf
3. Para y 0.36 d . Este es el caso no habitual. Aquí, el valor de y hallado hallado puede no ser correcto. Esta sección está conceptualmente mal diseñada. Método aproximado para Md M n Un método aproximado y sencillo de utilizar es el que se da en las siguientes tablas. El método tiene más precisión para valores de Md M 0 . Aquí se saca directamente el valor de . En las tablas se entra con los valores de hf /d , b/bw y y . Donde es:
M d b d f cd
El uso de los valores obtenidos de estas tablas garantizan estar en el D3 y se dan los valores límites de y en cada combinación posible. Lógicamente será necesario interpolar (doble interpolación) con bastante frecuencia. h /d =0.10 f =0.10 0.037 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190
lim lim
1
2
4
0.038 0.042 0.052 0.063 0.074 0.085 0.096 0.107 0.119 0.119 0.131 0.143 0.155 0.167 0.179 0.192 0.206 0.219 0.327
0.038 0.042 0.052 0.063 0.074 0.085 0.096 0.107 0.119 0.132 0.144 0.158 0.172 0.187 0.203 0.220 0.238 0.204
0.038 0.042 0.052 0.063 0.074 0.085 0.096 0.108 0.121 0.121 0.136 0.152 0.171
0.142 0.142
0.448
0.267
0.176 0.176
b/bw 5 0.038 0.042 0.052 0.063 0.074 0.084 0.096 0.108 0.122 0.138 0.158
6
8
10
0.038 0.042 0.052 0.063 0.074 0.084 0.096 0.109 0.124 0.142
0.038 0.042 0.052 0.063 0.074 0.084 0.096 0.110 0.128
0.038 0.042 0.052 0.063 0.074 0.084 0.096 0.111
0.130
0.122
0.112
0.105
0.158
0.146
0.130
0.121
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hf /d =0.15 =0.15 0.075 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.210 0.220
lim lim hf /d =0.20 =0.20 0.115 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.210 0.220 0.230 0.240
lim lim
1
2
4
0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.155 0.155 0.167 0.179 0.192 0.206 0.219 0.233 0.247 0.261 0.327
0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.155 0.168 0.182 0.196 0.211 0.227 0.244 0.263 0.283 0.222
0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.156 0.156 0.171 0.188 0.207
0.170 0.170
0.448
0.288
0.20 0.2077
1
2
4
0.125 0.131 0.137 0.155 0.167 0.179 0.192 0.192 0.206 0.219 0.233 0.247 0.261 0.276 0.291 0.327
0.125 0.131 0.137 0.155 0.167 0.179 0.193 0.206 0.221 0.236 0.252 0.270 0.289 0.310 0.240
0.125 0.131 0.137 0.155 0.167 0.179 0.193 0.193 0.209 0.226
0.448
0.310
b/bw 5 0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.157 0.173 0.192
6
8
10
0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.157 0.175
0.079 0.085 0.096 0.107 0.119 0.131 0.143 0.159
0.079 0.085 0.096 0.107 0.119 0.131 0.143
0.160
0.153
0.144
0.139
0.192
0.181
0.168
0.160
b/bw 5 0.125 0.131 0.137 0.155 0.167 0.179 0.194 0.210
6
8
10
0.125 0.131 0.137 0.155 0.167 0.179 0.194 0.212
0.125 0.131 0.137 0.155 0.167 0.180 0.195
0.125 0.131 0.137 0.155 0.167 0.180
0.196 0.196
0.188
0.182
0.175
0.170
0.240 0.240
0.226
0.216
0.205
0.198
Cálculo a cortante en secciones T o doble T
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8.CIMENTACIONES Tipos de suelos COMPOSICIÓN ROCAS
Formaciones Formaciones Geolog icas
CARACTERISTICAS
Notable resistencia a compresión 2 Isotropas sin visible estratificación 30 a 60 kp/cm
CLASIFICACIÓN
2 Estratificadas con visible estratificación 10 a 20 kp/cm
Gravas y gravillas con d>2mm COMPOSICIÓN
Arena s gruesas y me dia s 2>d>0 ,2 m m Arena s fina s 0,2>d >0,0 6 m m Limos inorgánicos d<0,06 mm
TERRENOS SIN COHESIÓN
CARA CARACTE CTERI RISTIC STICAS AS
Predomin Predominaa resis resisten tencia cia al corte corte Graveras
CLASIFICACIÓN
2 Predominan gravas y gravillas 4 a 8 kp/cm
Areno sos 2 Predominan arenas gruesas y medias 2,5 a 5 kp/cm gruesos Areno sos 2 Predominan arenas finas 1,6 a 3,2 kp/cm finos
COMPOSICIÓN CARACTERISTICAS
TERRENOS COHERENTES
TERRENOS DEFICIENTES
Arcil la s Predomina l a resist resistencia encia a la cohesión Arcil lo sos duros
CLASIFICACIÓN
2 Se romp en di fici lm ent e a ma no 4 kp/cm
2 Arcil lo sos sem sem idu ros Se a ma san di fici lm ent e a ma no 2 a 4 kp/cm
Arcil lo sos blan dos
2 Se am asan f acil me nte 1a 2 kp/cm
Arcil lo sos fluid os
Fluy en < 1 kp/cm2
COMPOSICIÓN
Fangos, terrenos organicos, rellenos
CARACTERISTICAS
No aptos para cimentación Fangos inorganicos
CLASIFICACIÓN
Terrenos organicos Terrenos de relleno
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ELS. Cálculo de Vuelco, deslizamiento y tensiones
1) Vuelco: M ≥M e
v
(N + P) (a/2) ≥ (M+V h)γ
1
2) Deslizamiento: (N + P) tg φ ≥ γ V → Arenas
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Clasificación de las cimentaciones de hormigón estructural Los encepado encepados y zapatas de cimentación pueden clasificarse en rígidos y flexibles. Cim entac ion es rígid as
.- Los encepados cuyo vuelo v en la dirección principal de mayor vuelo es menor que 2h 2h. .- Las zapatas cuyo vuelo v en la dirección principal de mayor vuelo es menor que 2h 2h. .- Los pozos de cimentación. .- Los elementos masivos de cimentación: contrapesos, muros masivos de gravedad, etc. Según EHE: hmin = 25 cm para zapatas; y hmin = 40 cm para encepados
Para zapata rígida El método general de análisis más adecuado es el de bielas y tirantes. No obstante puede utilizarse el modelo general de flexión simple ya visto. Para zapatas rectangulares sometidas a flexocompresión recta, siempre que se pueda despreciar el efecto del peso de la zapata y de las tierras situadas sobre ésta. La armadura principal se obtendrá para resistir la tracción T d indicada en el modelo, que resulta: d indicada Td Td =
N d
(a a0 ) (N d d centrada) 6.8 d R 1d x1 = Asf yd T 1 0,85 d
(N d d excéntrica)
Con: 2 f yd y σ 2d : tensiones obtenidas: R 1d 400 N/mm ; σ 1d y yd 1d : Fuerza resultante del trapecio hasta a/4 en el ancho de
la zapata; x1, la distancia del centro de gravedad del trapecio a la línea de carga de N 1d. Se
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2 con f yd corresponde al axil de cálculo del pilote más cargado. yd d corresponde 400 N/mm y donde N d
Cimentaciones flexibles flexibles
.- Los encepados cuyo vuelo v en en la dirección principal de mayor vuelo es mayor que 2h 2h. . .- Las zapatas cuyo vuelo v en en la dirección principal de mayor vuelo es mayor que 2h 2h. .- Las losas de cimentación. Cálculo a flexión. Sección de referencia S 1 La sección de referencia que se considerará para el cálculo a flexión, se define como a continuación se indica: es plana, perpendicular a la base de la zapata o encepado y tiene en cuenta la sección total de la zapata o encepado. Es paralela a la cara del soporte o del muro y está situada detrás de dicha cara a una distancia igual a 0,15a, siendo a la dimensión del soporte o del muro medida ortogonalmente a la sección que se considera. El canto útil de esta sección de referencia se tomará igual al canto útil de la sección paralela a la sección S1 situada en la cara del soporte o del muro. En todo lo anterior se supone que el soporte o el muro son elementos de hormigón. Si no fuera así, la magnitud 0,15a 0,15a se sustituirá por: - 0,25a 0,25a, cuando se trate de muros de ladrillo o mampostería. - La mitad de la distancia entre la cara del soporte y el borde de la placa de acero, cuando se trate de soportes metálicos sobre placas de reparto de acero.
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Cálculo a Flexión por Método General de Flexión (Usar para pa ra todos los casos):
1) Tensión uniforme
Vuelos:
v = MAX (v (v a;v b) Md
b
2
(v a 0.15 a0 )2
Calculo a flexión Simple con Tabla
2) Tensión no uniforme
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V d ≤ V u Donde: V d : cortante actuante y V u: cortante resistido V d = σ 1 b (v -d ) (Aplicar rectángulo + triángulo si la distribución de tensiones es trapecial hasta S 2. También es aceptable utilizar un valor medio) (100 ρ1 f ck )1/3 * b d V u = V cu= 0.12 ξ (100 1/2
/d ) ξ = 1 + (200 /d
(engranaje de los áridos; d en mm)
(b d ) ≤ 0.02 ρ = As / (b 1
Cálculo a Punzonamiento
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- La armadura va en Emparrillado - En rectangular el emparrillado en
una dirección no puede ser < 20% del total de la otra dirección - En el caso de zapatas cuadradas la disposición de armaduras será idéntica en ambas direcciones. En el caso de zapatas rectangulares la armadura principal (paralela al lado a) se distribuye uniformemente. La armadura paralela al lado menor se reparte de forma que la armadura transversal necesaria As necesaria As se distribuya en una proporción 2b ab
[%]
En un ancho b alrededor del soporte, y el resto en las bandas. - La EHE establece CG CG mínimas (igual que losas), y se deben respetar siempre (se calculan separadamente para cada dirección de armado, son 2 y 1.8 por mil para B400S y B500S respec)
Zapata con 2 Momentos
e x
M x N
ey
M y N
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En zona D (en u punto 5 interior): 5
P t 5 a b
adm
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Cimentaciones Profundas- Pilotes Tope estructural de un pilote (simplificación de CTE-SE-C)
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También se verifica esta cuantía mínima (
de EC2
2
Para pilotes con
Ac 0,5m (D 80 cm)
0,5%
Para pilotes con
0,5 Ac 1m (80
As=2500 mm
Para pilotes con
Ac
2
1m (D>115 cm)
Cálculos de pilotes con N y M
0,25%
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