Marcus

CALCULO Y DIMENSIONAMIENTO DE LOSAS ESTABILIDADES III G 02

Doc. Arq. Ronald Edwin Almaraz Pinto Aux. Univ. Rodrigo Terrazas Sejas

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Tabla 1 Placas libremente apoyadas por su contorno

Tipo 1 de apoyo

(carga uniforme)

1. Momentos con vrtices sin distorsin:

2. Fuerzas de vrtice:

3. Momentos con vrtices sin armadura

de torsin (normas DIN 1045, 23.1):

1. Valores numricos segn Marcus

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

27.4 25.1 25.1 24.1 23.3 22.7 22.1 21.5 21.3 21.0 20.8 20.6 20.6 20.7 21.0

27.4 28.7 30.2 31.8 33.6 35.8 37.3 39.5 41.7 44.3 46.8 49.7 52.8 67.6 84.2

24.0 24.2 24.4 24.7 25.0 25.3 25.6 26.0 26.4 26.8 27.2 27.7 28.2 30.3 33.2

1.35 1.35 1.34 1.33 1.32 1.30 1.28 1.27 1.25 1.23 1.22 1.20 1.18 1.13 1.12

2. Valores numricos segn Czerny

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

27.2 25.8 24.6 23.7 22.9 22.3 21.8 21.4 21.0 20.7 20.5 20.4 20.3 20.3 20.8

27.2 28.9 30.7 32.7 34.9 37.5 40.2 43.0 45.9 48.9 52.0 54.9 57.9 69.3 80.6

21.6 21.7 21.8 21.9 22.1 22.4 22.7 23.1 23.5 23.9 24.4 24.9 25.4 27.3 30.2

3. Reacciones de apoyo y esfuerzos cortantes

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.250 0.238 0.227 0.217 0.208 0.200 0.192 0.185 0.179 0.173 0.167 0.161 0.156 0.139 0.125

0.250 0.262 0.273 0.283 0.292 0.300 0.308 0.315 0.321 0.327 0.333 0.339 0.344 0.361 0.375

0.68 0.69 0.70 0.70 0.71 0.71 0.72 0.72 0.73 0.73 0.73 0.73 0.73 0.73 0.74

0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.83 0.84 0.85 0.86 0.87 0.91 0.93

Los momentos se representan mediante sus vectores. Los trazos son las secciones donde actan los

momentos. La armadura correspondiente est dispuesta perpendicularmente a estas superficies. Lo

dicho es extensivo a todos los tipos de apoyo.



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Tabla 2 a Borde mayor empotrado

Tipo 2 de apoyo

(carga uniforme)





4. Placas continuas: Momentos de tramo


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

Con

armadura de

torsin

29.9 29.2 28.6 28.2 28.0 27.9 27.8 27.9 28.0 28.2 28.5 28.8 29.2

36.8 39.8 43.2 47.0 51.1 55.7 60.6 65.9 71.6 77.9 84.4 91.5 98.8

11.2 11.2 11.2 11.3 11.4 11.6 11.8 12.0 12.3 12.6 12.9 13.2 13.5

0.09 0.11 0.14 0.17 0.20 0.23 0.26 0.29 0.31 0.34 0.37 0.40 0.42

0.34 0.38 0.43 0.47 0.52 0.57 0.62 0.67 0.71 0.77 0.80 0.84 0.88

Sin

armadura de

torsin

23.9 23.7 23.5 23.5 23.6 23.8 24.0 24.5 24.6 25.0 25.4 25.9 26.4

31.7 34.8 38.1 41.7 45.6 49.9 54.6 59.8 65.4 71.4 77.8 84.6 91.8

0.05 0.06 0.07 0.09 0.11 0.14 0.18 0.21 0.24 0.27 0.30 0.33 0.35

0.25 0.30 0.35 0.39 0.44 0.49 0.54 0.59 0.64 0.70 0.75 0.79 0.83


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

31.4 30.7 30.0 29.7 29.4 29.2 29.1 29.2 29.4 29.6 29.8 30.1 30.4 32.0 34.2

41.2 45.4 49.6 54.0 58.5 62.9 67.3 71.7 76.0 79.7 83.4 86.9 90.4 106 118

11.9 11.9 12.0 12.1 12.2 12.4 12.6 12.8 13.0 13.2 13.5 13.8 14.1 15.1 16.6

26.2 26.8 27.4 28.1 28.8 29.6 30.5 31.3 32.2 33.1 34.1 35.1 36.1 40.3 44.8


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.183 0.175 0.167 0.160 0.153 0.147 0.141 0.136 0.131 0.126 0.122 0.118 0.115 0.102 0.092

0.402 0.412 0.422 0.431 0.440 0.447 0.455 0.461 0.468 0.474 0.479 0.484 0.488 0.504 0.517

0.232 0.238 0.244 0.249 0.254 0.259 0.263 0.267 0.270 0.274 0.277 0.280 0.282 0.292 0.299

0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.56

1.16 1.18 1.20 1.22 1.23 1.24 1.24 1.25 1.26 1.26 1.27 1.27 1.27 1.27 1.26

0.63 0.64 0.65 0.66 0.68 0.69 0.70 0.70 0.71 0.71 0.72 0.72 0.73 0.74 0.75



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Tabla 2 b Borde menor empotrado

Tipo 2 de apoyo

(carga uniforme)







1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

Con

armadura de

torsin

36.8 34.2 32.0 30.2 28.6 27.3 26.3 25.5 24.8 24.2 23.8 23.4 23.1

29.9 30.8 31.9 33.1 34.4 35.8 37.3 38.8 40.4 42.2 44.0 46.0 48.2

11.2 11.3 11.5 11.8 12.2 12.7 13.2 13.8 14.4 15.2 16.1 17.1 17.9

0.34 0.30 0.28 0.25 0.23 0.21 0.20 0.18 0.17 0.16 0.16 0.15 0.14

0.09 0.07 0.05 0.04 0.02 0.01

Sin

armadura de

torsin

31.7 29.4 27.4 25.8 24.5 23.4 22.6 21.9 21.3 20.8 20.5 20.3 20.2

23.9 24.3 24.9 25.7 26.6 27.6 28.7 30.0 31.3 32.8 34.5 36.4 38.6

0.25 0.21 0.19 0.16 0.14 0.12 0.11 0.09 0.08 0.08 0.08 0.07 0.07

0.05 0.04 0.03 0.02 0.01


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

41.2 37.9 35.1 32.9 31.1 29.6 28.3 27.2 26.3 25.6 24.9 24.4 24.0 23.0 22.8

29.4 30.3 31.7 33.1 34.7 36.5 38.6 40.8 43.1 45.7 48.4 51.1 53.8 65.2 77.6

11.9 11.9 12.0 12.0 12.1 12.3 12.5 12.7 12.9 13.1 13.3 13.6 13.9 15.1 16.4

26.2 25.8 25.5 25.3 25.2 25.1 25.2 25.4 25.6 25.8 26.1 26.5 26.9 28.6 30.8


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.402 0.388 0.378 0.366 0.355 0.342 0.331 0.320 0.310 0.300 0.289 0.280 0.272 0.241 0.217

0.232 0.226 0.218 0.212 0.205 0.198 0.191 0.184 0.179 0.173 0.167 0.161 0.156 0.139 0.125

0.183 0.193 0.202 0.211 0.220 0.230 0.239 0.248 0.256 0.264 0.272 0.280 0.286 0.310 0.329

1.16 1.20 1.23 1.26 1.29 1.31 1.33 1.35 1.37 1.38 1.39 1.40 1.41 1.44 1.46

0.63 0.65 0.66 0.67 0.68 0.68 0.69 0.70 0.70 0.71 0.71 0.71 0.72 0.73 0.74

0.57 0.60 0.62 0.64 0.66 0.69 0.70 0.72 0.74 0.76 0.78 0.80 0.81 0.88 0.90



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Tabla 3 a Los dos bordes mayores empotrados

Tipo 3 de apoyo

(carga uniforme)

1. Momentos:



1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

37.5 37.5 37.6 37.9 38.3 38.8 39.4 40.0 40.8 41.7 42.6 43.5 44.3

55.7 62.0 69.0 76.6 84.9 94.0 104 114 126 138 151 165 180

14.4 14.7 15.0 15.4 15.8 16.2 16.7 17.2 17.7 18.2 18.7 19.2 19.8

0.37 0.43 0.50 0.57 0.64 0.71 0.78 0.85 0.92 0.99 1.05 1.11 1.15

1.04 1.15 1.28 1.40 1.51 1.66 1.78 1.90 2.02 2.14 2.24 2.34 2.42


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

35.1 35.0 34.9 35.1 35.3 35.7 36.1 36.6 37.3 37.9 38.7 39.4 40.3 43.9 48.2

61.7 68.0 73.9 79.9 85.8 90.8 95.5 100.3 105 107.5 114 118.5 123 139 154

14.3 14.6 14.9 15.2 15.6 16.0 16.4 16.8 17.3 17.7 18.2 18.7 19.2 21.5 24.0


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.144 0.137 0.131 0.125 0.120 0.115 0.111 0.107 0.103 0.099 0.096 0.093 0.090 0.080 0.072

0.356 0.63 0.369 0.375 0.380 0.385 0.389 0.393 0.397 0.401 0.404 0.407 0.410 0.420 0.428

0.49 0.49 0.49 0.49 0.49 0.49 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48

1.03 1.04 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.04 1.03 1.02



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Tabla 3 b Los dos bordes mayores empotrados

Tipo 3 de apoyo

(carga uniforme)

1. Momentos:



1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

55.7 50.3 46.0 42.4 39.4 36.9 34.9 33.1 31.6 30.3 29.2 28.4 27.5

37.5 37.7 38.1 38.7 39.5 40.4 41.4 42.5 43.8 45.2 46.7 48.3 50.0

14.4 14.2 14.1 14.0 14.1 14.3 14.6 14.9 15.2 15.6 16.1 16.7 17.3

1.04 0.94 0.85 0.77 0.70 0.64 0.59 0.54 0.49 0.45 0.41 0.38 0.35

0.37 0.32 0.27 0.22 0.18 0.14 0.10 0.07 0.05 0.03 0.02 0.01 0.01


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

63.3 56.4 50.7 46.1 42.5 39.5 37.0 34.9 33.2 31.8 30.6 29.5 28.6 26.3 25.0

35.1 35.6 36.2 37.0 38.0 39.2 40.2 42.2 44.0 46.0 48.2 50.6 55.1 66.2 84.8

14.3 14.1 14.0 13.9 13.8 13.8 13.9 13.9 14.0 14.1 14.2 14.4 14.7 15.7 16.8


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.356 0.349 0.341 0.334 0.327 0.320 0.312 0.304 0.297 0.290 0.293 0.275 0.267 0.241 0.217

0.144 0.151 0.159 0.166 0.173 0.180 0.188 0.196 0.203 0.213 0.217 0.225 0.233 0.259 0.283

1.03 1.07 1.11 1.15 1.18 1.21 1.24 1.27 1.29 1.31 1.33 1.35 1.37 1.42 1.44

0.49 0.51 0.53 0.56 0.59 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.82 0.86



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Tabla 4 Dos bordes adyacentes empotrados

Tipo 4 de apoyo

(carga uniforme)


Para se tiene

Fuerzas de vrtice:





1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

Con armadura

de torsin

37.0 35.4 34.1 33.2 32.3 31.8 31.5 31.3 31.2 31.2 31.3 31.4 31.4

37.0 39.2 41.3 43.9 46.5 49.6 53.2 57.2 61.4 65.8 70.4 75.0 80.3

16.0 15.3 14.8 14.5 14.2 14.1 14.0 14.1 14.1 14.2 14.4 14.6 14.8

16.0 16.6 17.6 18.4 19.2 20.0 20.8 21.6 22.4 23.2 24.0 24.8 25.6

0.35 0.36 0.37 0.38 0.39 0.40 0.42 0.44 0.45 0.47 0.49 0.51 0.52

Sin armadura de torsin

1.15 1.15 1.15 1.14 1.14 1.14 1.13 1.13 1.12 1.12 1.11 1.11 1.10

0.26 0.27 0.28 0.29 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

40.2 38.3 36.8 34.6 34.8 34.2 33.8 33.6 33.5 33.4 33.3 33.3 33.4 34.8 35.8

40.2 43.1 46.2 49.4 52.8 57.0 61.9 66.7 71.3 75.5 79.6 83.8 88.0 114 120

14.3 14.1 14.0 13.9 13.8 13.8 13.9 13.9 14.0 14.0 14.4 14.7 15.1 16.0 16.8

14.3 14.6 15.0 15.3 15.7 16.1 16.6 17.1 17.6 18.1 18.6 19.1 19.6 21.8 24.4


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.317 0.302 0.288 0.276 0.264 0.254 0.244 0.235 0.227 0.219 0.211 0.204 0.198 0.176 0.159

0.183 0.175 0.167 0.160 0.153 0.147 0.141 0.136 0.131 0.126 0.122 0.118 0.115 0.102 0.091

0.317 0.332 0.347 0.359 0.371 0.381 0.391 0.400 0.408 0.416 0.424 0.431 0.437 0.459 0.476

0.183 0.191 0.198 0.205 0.212 0.218 0.224 0.229 0.234 0.239 0.243 0.247 0.250 0.263 0.274

1.02 1.04 1.05 1.06 1.07 1.07 1.08 1.08 1.08 1.08 1.09 1.09 1.09 1.09 1.10

0.55 0.55 0.56 0.56 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57

1.02 1.06 1.09 1.12 1.14 1.16 1.18 1.20 1.21 1.21 1.22 1.22 1.22 1.23 1.27

0.55 0.57 0.59 0.61 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.69 0.70 0.72 0.74



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Tabla 5 a Los bordes mayores y uno menor empotrados

Tipo 5 de apoyo

(carga uniforme)

1. Momentos:



1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

44.2 433 42.7 42.4 42.3 42.4 42.6 43.0 43.5 44.0 44.6 45.2 45.9

50.5 54.6 59.3 64.5 70.3 76.5 83.3 90.8 98.6 107 117 126 137

18.0 17.8 17.7 17.8 17.9 18.1 18.3 18.6 19.0 19.4 19.8 20.2 20.6 24.0 25.2 26.4 27.6 28.8 30.0 31.2 32.4 33.6 34.8 36.0 37.2 38.4 0.61 0.65 0.70 0.75 0.81 0.87 0.93 0.99 1.04 1.09 1.14 1.19 1.24

0.84 0.90 0.96 1.02 1.09 1.16 1.22 1.29 1.36 1.43 1.49 1.55 1.60


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

44.1 42.6 41.6 41.0 40.6 40.4 40.3 40.1 40.6 40.9 41.3 41.8 42.5 45.4 49.0

55.9 61.2 66.5 72.0 77.5 83.0 89.5 95.0 101 107 114 121 129 160 194

16.2 16.2 16.3 16.5 16.7 16.9 17.2 17.5 17.8 18.2 18.6 19.0 19.5 21.4 23.6

18.3 18.9 19.5 20.2 209 21.7 22.6 23.5 24.4 25.3 26.2 27.1 28.0 31.7 35.4


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.250 0.237 0.227 0.217 0.208 0.200 0.192 0.185 0.175 0.173 0.166 0.161 0.156 0.138 0.125

0.144 0.137 0.131 0.125 0.120 0.114 0.110 0.107 0.103 0.099 0.096 0.093 0.090 0.080 0.071

0.303 0.313 0.321 0.329 0.336 0.343 0.349 0.354 0.359 0.364 0.369 0.373 0.377 0.391 0.402

0.91 0.93 0.95 0.96 0.96 0.96 0.95 0.95 0.94 0.92 0.91 0.90 0.90 0.90 0.90

0.49 0.49 0.49 0.49 0.49 0.49 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48

0.95 0.97 0.99 1.01 1.03 1.03 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.03 1.03



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Tabla 5 b Los bordes menores y uno mayor empotrados

Tipo 5 de apoyo

(carga uniforme)

1. Momentos:



1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

50.5 47.5 44.3 42.0 40.7 38.7 37.5 36.7 36.0 35.5 35.1 34.8 34.6

44.2 45.4 46.9 48.6 50.7 53.0 55.7 58.5 61.6 64.9 68.5 72.5 76.6

24.0 22.2 20.8 19.7 18.8 18.1 17.6 17.3 17.1 16.9 16.8 16.7 16.7 18.0 18.4 18.9 19.6 20.4 21.3 22.4 23.7 25.1 26.1 27.0 27.9 28.8 0.84 0.80 0.76 0.74 0.72 0.71 070 0.70 0.69 0.69 0.69 0.68 0.68

0.61 0.58 0.55 0.53 0.51 0.50 0.49 0.48 0.47 0.47 0.45 0.46 0.46


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

59.5 54.7 50.7 47.5 44.9 42.9 41.3 40.1 39.2 38.4 37.8 37.7 37.7 37.6 37.5

44.1 43.9 48.1 50.7 53.7 57.1 61.0 65.4 70.4 76.1 82.5 89.5 97 143 202

18.3 17.5 16.9 16.5 16.2 16.0 15.8 15.7 15.7 15.7 15.8 15.9 16.1 16.7 17.6

16.2 16.2 16.3 16.4 16.6 16.9 17.3 17.7 18.1 18.6 19.0 19.5 20.0 22.1 24.5


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.304 0.294 0.284 0.274 0.264 0.254 0.244 0.235 0.227 0.219 0.211 0.204 0.198 0.176 0.159

0.250 0.263 0.275 0.288 0.301 0.314 0.327 0.339 0.350 0.360 0.370 0.378 0.387 0.416 0.437

0.142 0.149 0.157 0.164 0.171 0.178 0.185 0.191 0.196 0.202 0.208 0.214 0.217 0.232 0.245

0.95 0.98 1.00 1.02 1.04 1.05 1.06 1.07 1.08 1.08 1.08 1.08 1.08 1.09 1.10

0.91 0.95 0.99 1.03 1.06 1.09 1.11 1.13 1.15 1.16 1.17 1.18 1.19 1.22 1.24

0.49 0.51 0.53 0.55 0.57 0.59 0.61 0.63 0.64 0.65 0.66 0.67 0.68 0.71 0.73



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Tabla 6 Los cuatro bordes empotrados

Tipo 6 de apoyo

(carga uniforme)

1. Momentos:



1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60

55.7 53.3 51.3 50.1 49.1 48.4 48.0 47.7 47.7 47.8 48.0 48.4 48.8

55.7 58.8 62.3 66.2 70.7 75.6 81.0 87.2 93.5 100.4 108.2 116.4 125

24.0 23.0 22.2 21.7 21.4 21.2 21.1 21.0 21.2 21.4 21.6 21.8 22.1 24.0 25.2 26.4 27.5 28.8 30.0 31.2 32.4 33.6 34.8 36.0 37.2 38.4 1.03 1.04 1.05 1.07 1.10 1.13 1.17 1.21 1.24 1.28 1.31 1.35 1.39


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

56.8 53.5 50.7 48.8 47.2 46.1 45.2 44.9 44.6 44.5 44.4 45.2 46.1 48.8 50.0

56.8 61.2 66.3 72.2 78.9 86.7 93.6 105.6 116.6 128.5 140.5 152 163 190 210

19.4 19.0 18.8 18.7 18.6 18.7 18.8 19.0 19.2 19.5 19.8 20.1 20.5 22.0 24.0

19.4 19.8 20.3 20.9 21.5 22.2 22.9 23.7 24.5 25.3 26.2 27.0 27.9 31.4 35.0


1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.80 2.00

0.250 0.238 0.227 0.217 0.208 0.200 0.192 0.185 0.179 0.173 0.167 0.161 0.156 0.139 0.125

0.250 0.262 0.273 0.283 0.292 0.300 0.308 0.315 0.321 0.327 0.333 0.339 0.344 0.361 0.375

0.89 0.91 0.93 0.93 0.94 0.94 0.95 0.95 0.96 0.96 0.95 0.94 0.94 0.94 0.94

0.90 0.93 0.95 0.97 0.99 1.01 1.02 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.05

Marcus

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