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Sapienza stochino-25-06-2013
Sapienza stochino-25-06-2013
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Sapienza stochino-25-06-2013

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Flexural Models of Reinforced Concrete Beams …

Flexural Models of Reinforced Concrete Beams
Under Blast Load.

Nowadays, the issue of structural safety under blast loading has become a dramatic problem. The tragic news of the terrorist attacks of recent years raise important, urgent questions regarding the real safety and reliability of our buildings. Extreme loads such as impacts, explosions, etc., can occur in everyday life with unexpectedly high frequency. This seminar deals with the flexural failure of Reinforced Concrete beams under blast loads. The main aim is to develop dynamic and energy models capable of evaluating the response of R.C. under explosive load. The seminar is composed of three sections. Various dynamic models are developed in Section 1, taking into account the strain-rate sensitivity of both steel and concrete, as well as other nonlinearities in structural behaviour. In Section 2, the same problem is solved in a different way: starting from the principle of energy conservation. This procedure, less accurate than the previous one, produces very good results regarding midspan displacement, especially as it requires less computation time. In order to better understand the phenomenon, with the aim of identifying the key parameters in structural response, a sensitivity analysis is developed in Section 3. In this way it is possible to attain simple predictive polynomial formulations and, finally, simple table for early structural assessment of beams under blast load during the design procedure.

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  • 1. !"
  • 2. "# " $ % " " $ & %" " % ' !" !"
  • 3. "# " $ % " " $ & %" " % ' !" !"
  • 4. "# " $ " $ ' !" (' %! ) !"
  • 5. "# " $ " $ ' !" (' %! ) !"
  • 6. "# " $ " * + ' $ " !"
  • 7. " '" ! , !" - " " , .$ " " , & %$ " , " "#" $ ' $ "
  • 8. " '" ! , !" - " " , .$ " " , & %$ " , " "#" $ ' $ "
  • 9. " " " , .$ "   "   .-   & ( ) ( ) ( ) (/) &( ) !&( ) &( ) "&( ) ( ) !"
  • 10. & " /   ✁ ✁ ✂ ✄ ✁ ✁ ! " " 0 0 0 " ! / " " ☎ ✆ ✆ ✝ ✞ ✆ ✆ ! " !"
  • 11. " " " !"0 ( -.& -.& ) ✄ ☎ ✄   ✁ ✂ ✁ ✂ " 1 1 1 ✆ ✆ ✆ ✆ ✝ ✆ ✆ ✆ ✆ ✞ ✟ ✠ ✡ ☛ ✟ ✞ " !"
  • 12.   ✁ ✁ ✂ ✄ ✁ ✁ ! " % # 2 " " ! ! ☎ ✆ ✞ ✟ ✞ ✟ ✝ ✝ ✠ ✡ ☛ ✡ ☛ ✆ ☞ ✌ ☞ ✌ " " ! ! ! ! ! " !"
  • 13.   ✁ ✁ ✂ ✄ ✁ ✁ ! ! ! ☎ ✆ ✞ ✟ ✞ ✟ ✝ ✝ ✠ ✡ ☛ ✡ ☛ ✆ ☞ ✌ ☞ ✌ " " ! ! ! ! ! " % # 2 " " " !"
  • 14. &3 " " , ! !   ✁ ✞ ✟ ✞ ✟ ✄ ✄ ✆ ✠ ✡ ✠ ✡ ✁ ☛ ☞ ☛ ☞ " " ! ! ! ! ! ✂ ☎ ☎ ✝ ✌ ☎ ☎ ! ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ ✍ ✎ 4 4 ! ! " " " " ! ! ! ✏ ✑ ✒ ✓ ✔ ✓ ✔ ✕ ✕ ✕ ✕ ✕ ✓ ✔ ✖ ✗ ✗ ✘ ✙ ✚ ✛ ✜ ✛ ✜ ✛ ✜ ✕ ✕ ✕ ✕ ✕ ✢ ✣ ✢ ✣ ✢ ✣ ✙ ✚ ✤ ✥ " !"
  • 15. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✄ ✄ ✄ ✄ ✄ ✞ ✟ ✆ ☎ ☎ ✝ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✄ ✄ ✄ ✄ ✄ ☛ ☞ ☛ ☞ ☛ ☞ ✎ ✏ ✑ ✒ ! ! ✓ ✔ ✕ ✖ ✕ ✖ ✗ ✗ ✘ ✙ ✚ ✙ ✚ ✔ ✛ ✜ ✛ ✜ " " ! ! ! ! ! ✢ ✣ ✣ ✤ ✥ ✣ ✣ ! " !" &3 " " ,
  • 16. " ! " 2! ! " 0 "# " 00 " " % $ ! ! ! " " 0 " ! / ! " ! / $ / ! " 0 5 " " 6 7 " "# 2 " !"
  • 17. + " &3 " "/ " , " 3 " "/ " , 8" " % " " ! " " 9 ✁ ✂ $ $ $ $ $ $ $ $ $ $5 $ $ $ $ : # #               ✌ ✍ ✆ ✆ ✞ ✟ ✆ ✎ ✏ ✠ ✡ ✆ ✆ ✆ ☛ ☞ ✎ ✏ ✄ ☎ ✆ ✆ ✎ ✏ ✄ ✆ ✎ ✏ ✆ ✑ ✒ ✓ $ $5# # ✝ ✝ ✔ ✕ ✖ ✗ ✘ ✗ ✘ ✗ ✘ ✗ ✘ ✗ ✘ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ : : :! # # ✙ ✙ ✙ ✙ ✚ ✚ ✙ ✙ ✙ ✛ ✜ ✢ ✣ ✤ ✤ ✤ ✥ ✦ ✧ ★ ✤ ✤ ✤ ✥ ✦ ✩ ✪ ✫ ✬ ✫ ✬ ✥ ✦ ✤ ✤ ✬ ✤ ✥ ✦ ✤ ✥ ✦ ✭ ✮ ✯ ✯ !" "
  • 18. " 3 " "/ " ✁ ✂ " " " $5 $5 " : # #           ✄     ✌ ✍ ✆ ✆ ✞ ✟ ✆ ✎ ✏ ✠ ✡ ✆ ☛ ☞ ✎ ✏ ☎ ✝ ✑ ✆ ✎ ✏ ☎ ✆ ✎ ✏ ✒ ✓ ✔ ✕ ✖ " " $5 $5 $5 " # # ✗ ✗ ✗ ✗ ✘ ✗ ✗ ✙ ✚ ✛ ✛ ✜ ✢ ✛ ✣ ✤ ✥ ✦ ✧ ★ ✣ ✤ ✩ ✪ ✫ ✛ ✣ ✤ ✩ ✛ ✣ ✤ ✬ ✭ ✮ ✯ ✰ ✯ ✰ ✯ ✰ " " " $5 " : :! #           ✄     ✌ ✍ ✆ ✆ ✞ ✟ ✆ ✎ ✏ ✠ ✡ ✆ ☛ ☞ ✎ ✏ ✝ ☎ ✑ ✆ ✎ ✏ ☎ ✆ ✎ ✏ ✒ ✓ ✱ ✲ ✳ ✲ ✳ ✲ ✳ " " $5 $5 " :! # ✴ ✴ ✴ ✴ ✵ ✴ ✴ ✶ ✷ ✸ ✸ ✹ ✺ ✸ ✻ ✼ ✽ ✾ ✿ ❀ ✻ ✼ ❁ ❂ ❃ ✸ ✻ ✼ ❂ ✸ ✻ ✼ ❄ ❅ ❆ !" + " &3 " "/ " " " " ! " " 9 "
  • 19. !" "
  • 20. *"# $ $ " " $ !" $ $ $   ✁ ✄ "   ✁ ✄ $ $ ! " ✂ ☎ ✆ ✝ $ $ ! ! !! " " ! ✞ ✞ ✞ ✞ ✟ ✠ ✌ ✍ ✡ ☛ ☞ ✎ ✏ ✑ ✒ ✓ ✔ ✕ ✖ !" "
  • 21. " & & " ;< = 4 = " > $ >   ✁ ✁ ☎ ✆ ✆ ✞ ✟ ✄ ✂ ✕ ✠ ✡ ✂ ☛ ☞ ✝ ✌ 1 $ ✍ ✁ ✁ ✆ ✄ ✂ ✂ ✖ $ > ✎ ✎ ✎ ✏ ✑ ✒ ✓ ✔ ✗ ✘ ✔ ✙ ✚ " $ " > ✁ ✁ ✁ ✆ ✞ ✟ ✄ ✂ ✠ ✡ ✂ ☛ ☞ $5 $ $5 $5 > " ✛ ✛ ✜ ✜ ✢ ✣ ✤ ✥ ✦ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✧ ✮ ✯ ✰ ✱ !"
  • 22. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 23. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ " " ." @ " " "A " , .$ " , " / !" ✓ ✔ ( ) ( ) ( ) ✕ ✕ ✖ ✆ ☎ ✝ ✗ ✗ ✓ ✔ ( ) ( ) ( ) ( ) ✕ ✕ ✖ ✖ ✆ ☎ ✆ ☎ ✝ ✗ ✗ ✓ ✔ ( ) ( ) ( ) ( ) ( ) 4 4 4 4 > 4 ✕ ✕ ✖ ✖ ✘ ✘ ✙ ✘ ✙ ✚ ✛ ✛ ✓ ✔ ( ) ( ) ( ) ✕ ✖ ✘ ✙ ✚ ✛ ✛
  • 24. & 0 " " @ " % " ! , .$ " , " / !" ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ 4 4 ! ! " " " " ! ! ! ✖ ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✘ ✘ ✘ ✘ ✘ ✙ ✚ ✚ ✛ ✘ ✘ ✘ ✘ ✘ & 0 " " @ " % " ! ,
  • 25. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ & 0 " " @ " % " ! , .$ " , " / !" 0 (") +" (B) "
  • 26. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" " " " " 0 (") +" (B) " & 0 " " @ " % " ! ,
  • 27. .$ " , " / !" 0 (") +" (B) " " " " C" " $ " " C" " $ " " & 0 " " @ " % " ! ,
  • 28. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" 0 (") +" (B) " " " " C" " $ " " C" " $ " " & 0 " " @ " % " ! ,
  • 29. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" 0 (") +" (B) " " " " C" " $ " " C" " $ " " & 0 " " @ " % " ! ,
  • 30. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" 0 (") +" (B) " " " " C" " $ " " C" " $ " " & 0 " " @ " % " ! ,
  • 31. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" 0 (") +" (B) " " " " C" " $ " " C" " $ " " & 0 " " @ " % " ! ,
  • 32. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 33. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 34. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 35. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 36. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 37. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 38. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " & 0 " " @ " % " ! ,
  • 39. " $ " " .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " " $ " " & 0 " " @ " % " ! ,
  • 40. " $ " " ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ! ! 4 > 4 " " ! ! " " ! ✆ ✂ ✆ ✂ ✆ ✆ ✂ ✂ ✆ ✆ ✂ ✂ ✞ ✟   ✁   ✗ ✠ ✡ ☛ ☞ ✞ ✟   ✁   ✁   ✁ ✞ ✟ ✗ ✁ ✠ ✡ ✠ ✡ ☛ ☞ ☛ ☞   ✁   ✁ ✁ ✄ ☎ ) 4 ( ) ( ) ( ) ✝ ✆ ✂ ✌ ✍ ✎ ✏ ✎ ✏ ✎ ✏ ✁ ✎ ✏ ✎ ✏ ✎ ✏ ✎ ✏ ✑ ✒   ✁ ✁ ✓ .$ " , " / !" C" " $ " " C" " $ " " 0 (") +" (B) " " " " " $ " " & 0 " " @ " % " ! ,
  • 41. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 42. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? ! / " % 0 " % ! # 9? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 43. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? ! / " % 0 " % ! # 9? 4 ! " 0 ! " 3 " "/ " ! " " ! 00 " / " % ? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 44. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? ! / " % 0 " % ! # 9? 4 ! " 0 ! " 3 " "/ " ! " " ! 00 " / " % ? ! " " /$ " % ! " " " % ! # # ? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 45. ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? ! / " % 0 " % ! # 9? 4 ! " 0 ! " 3 " "/ " ! " " ! 00 " / " % ? ! " " /$ " % ! " " " % ! # # ? > ! " " ! 0 $ " 0 0 " " ? +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 46. +! 0 " 2! ! 0 " " 2!" ! ! / " ! " ! " " ( " ) " " " ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ 4 4 ! ! " " " " ! ! !   ✌ ✍ ✞ ✟ ✞ ✟ ✞ ✟ ✎ ✏ ✠ ✡ ✠ ✡ ✠ ✡ ✎ ✏ ☛ ☞ ☛ ☞ ☛ ☞ ✑ ✒ ✄ ✄ ✄ ✄ ✄ ✆ ☎ ☎ ✝ ✄ ✄ ✄ ✄ ✄ ! # " " 0 # 2!" ! " / " /$ #" % ! 3 " " 2! # " ! " 0 " ? ! # 9 D E 1E ! # E 9 1E ? ! / " % 0 " % ! # 9? 4 ! " 0 ! " 3 " "/ " ! " " ! 00 " / " % ? ! " " /$ " % ! " " " % ! # # ? > ! " " ! 0 $ " 0 0 " " ? < ! 0 # ! ! " ! " " ( $ $ ) /$ 2!" ! ! # " "" ! +! 3 " " " " .& 2" ! # " / " " " / / " /$ " 00 ! ' " "# 0 " 0 2!" ! " " # " % ! " 0 ! 2" % 3 " " , !" .$ " , " / !"
  • 47. .- " % . % . 0" % " " % " "# $ " % " +! " 0 5 " 0 " ! # " " !"
  • 48. $ $ $ ; ! ☎ ✄ ☎ ✄ .- !"
  • 49. $ $ $ ; ! ☎ ✄ ☎ ✄ 4 $ $ &$ ;4 ;4 ☎ ☎ ☎ ☎ ✄ ✄ ☎ ☎ " " .- !"
  • 50. $ & &$ ;4 ☎ ✄ ✄ ☎ " " 4 $ $ &$ ;4 ;4 ☎ ☎ ☎ ☎ ✄ ✄ ☎ ☎ " " $ $ $ ; ! ☎ ✄ ☎ ✄ .- !"
  • 51. ; ! ☎ ✄ ☎ ✄ 4 $ $ &$ ;4 ;4 ☎ ☎ ☎ ☎ ✄ ✄ ☎ ☎ " " $ $ $ ; ! ☎ ✄ ☎ ✄ $ & &$ ;4 ☎ ✄ ✄ ☎ " " .- !"
  • 52. 0 &0   ✄ ☎ ; ! ✁ ✂ ✁ ✂ 4 $ $ &$ ;4 ;4 ✁ ✁ ✁ ✁ ✂ ✂ ✁ ✁ " " $ $ $ ; ! ✁ ✂ ✁ ✂ $ & &$ ;4 ✁ ✂ ✂ ✁ " " .- !"
  • 53. ✁ ✂ & &$ $ 0 4     ✄ ☎ ✆ ✝ ✆ ✆ 0 &0 ✞ ✟ ✠ ; ! ✆ ✄ ✆ ✄ 4 $ $ &$ ;4 ;4 ✆ ✆ ✆ ✆ ✄ ✄ ✆ ✆ " " $ $ $ ; ! ✆ ✄ ✆ ✄ $ & &$ ;4 ✆ ✄ ✄ ✆ " " .- !"
  • 54. $ & 0 & &$ ✆ ✄ ✆ " ✁ ✂ & &$ $ 0 4     ✄ ☎ ✝ ✆ ✝ ✝ 0 &0 ✞ ✟ ✠ ; ! ✝ ✄ ✝ ✄ 4 $ $ &$ ;4 ;4 ✝ ✝ ✝ ✝ ✄ ✄ ✝ ✝ " " $ $ $ ; ! ✝ ✄ ✝ ✄ $ & &$ ;4 ✝ ✄ ✄ ✝ " " .- !"
  • 55. " & & & 4;   ☎ ✄ ☎ & & 4; ✁ ✂ ✆ ✂ & " " 0 & &$ & $ $ 0 01 ✝ ✁ ✁ ✁ ✞ ✆ ✟ ✆ ✟ ✂ ✂ & & 01 ✠ ✡ ☛ ☛ ☞ ✌ ✍ ! ✎ ✏ .- !"
  • 56. + 5" % " " ! 3 " " / " " " 3 " 2" ! # " / " " , ✁ ✂ ✁ ✂ ✁ ✂ ✁ ✂ & & & & & & &$   ✄ ✕ ✕ ! " ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✝ ✞ ☎ ✆ & & 0 & 0 & & & 0 &$ & &$ & & ✟ ✟ ✠ ✡ ☛ ☞ ! " " " ☎ ✆ & & 0 & >> <; ! ! ! ! ✡ ✌ ✡ ✌ ✡ ✌ .- !"
  • 57. & !" " " /$ /$ ! " 2 " * (# ) 0 " ! "/ " 2!" ! " " "#" " % ! " ! / " "/ /
  • 58. $ , & 0 " 0 !" # $ F"%! 0 / /B ! 5 2 # " / 0 -' ;> & . ! & / " ! ( -') + / 2 ? !"
  • 59. $ , & 0 " 0 !"
  • 60. % &'( ) *''+&'( , 0 % ! 6" ! " G . 0 ! " > > # + " " > > 0 "# " 0 "# % ! 4 < 1 4 / " " %" >GH H $" % ! > 4 " * 4 * 0 " % ! 5%1 " % ! 0 "# % ! $ " / +! / 2 2 $ , ! " # ! " ! 0 "# A 2!" ! " "# ! " ! " A +!" # ! / 0 #" /$ ! ! " ! / / 3 5%1 !" $ , ! " "
  • 61. $ , & 0 " / 4 I. & 0 " / 14 I. !"
  • 62. $ , ' + # 4 I. + # 14 I. !"
  • 63. $ 4 I. !"
  • 64. 14 I. !" $
  • 65. ." 0 % %" " " / 4 I. # % %" " " / 4 I. !" $
  • 66. " '" ! , !" - " " , .$ " " , & %$ " , " "#" $ ' $ "
  • 67. & %$ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ! ( ) $ ! " ! !   ✁ ✂ ✂ ✄ ✄ ☎ ☎ ☎ ☎ ✌ ✍ ✆ ✝ ✝ ✞ ✎ ✏ ☎ ☎ ☎ ☎ ✑ ✒ ✟ ✠ ☎ ✟ ✠ ✞ ✆ ✝ ✡ ☛ ✡ ☛ ☎ ☞ ✓ ☞ ✓ ✘ ✘ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ( ) ( )" ( )% !" ( ) ( )" % ✕ ( )
  • 68. 4 I. !" $ ,
  • 69. 4 I. !" $ ,
  • 70. 4 I. &'( ) - . " J 4 44 < " + " " J > > " 0 " J !" $ ,
  • 71. " '" ! , !" - " " , .$ " " , & %$ " , " "#" $ ' $ "
  • 72. " "#" $ ' $ " F"%! 2 && >' ; 4 < & &( ) !&( ) &( ) "&( ) 0 % ! > !1 1G 1 6" ! !1 % ! 4 4 % % #   ✄ ☎ % # % # # ✁ ✂ !"
  • 73. " "#" $ ' $ " !" 4 " " % , K " " F"%! % ! K " " 2 2 !
  • 74. F"%! " "#" $ ' $ " / 0 0 , * 1 2 /3 / 1 2 4 " G ; ;G ;;4 < L " G <> ;G; ;; < /" G <4 G ;<< ; 4 ! % G >G G ;< G !"
  • 75. / 4 " "#" $ ' $ " ✁ ✂ %% ✄   ✆ ✘ 3 . & %()# %% %%* ☎ ✝ 3 , " / 2 ! 3 % " % ! % " ! 3 % " % ! ! 0 # ✞ ✟ ✞ ✟ %()# %% %%* ✠ ☎ ✡ ✝ ✡ ' B 3 , " " 0 " " " " # " " $ 2! " " $ 0 " 2" ! " / " " %% !% ✠ ✝ 3 & !"
  • 76. 2 " "#" $ ' $ " / 0 0 , * 1 2 /3 / 1 2 4 " ; ; ; < L " ; >4 > 4G> /" ;4 >G > 4G> 4 ! % <G ; >; 4G !"
  • 77. 2 , " "#" $ ' $ " ( ) ✄ ✂ ✂ 0 / 0 0 , & 3 ' 3 , & < > < ; < ; < - / 0 0 , & 1 3 ' 3 , & 1 G G > ; !"
  • 78. 2 " " "#" $ ' $ " 4 4 4 4 4 4 4 4 ( ) ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ / 0 0 , & ;4< 3 <G< ' B 3 , <G4; & <<< !"
  • 79. 2 " " "#" $ ' $ " 4 4 4 4 4 4 4 4 ( ) ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ / 0 0 , & ;4< 3 <G< ' B 3 , <G4; & <<< !"
  • 80. " +! / 2 / " % # $" " 3 " " 3 " $ " % " / " 0 / 2" " % ! " 0 # ! 2 /$ 0 " 2" ! 0 " + 5" % " " 3 " % 0 " / " " 0 # ! ! " / ! #" / " # " ! ! " / 0 " 0 " #" 2 / " " 0 " % " 0 " / " 0 # " % ! " 0 " % ! ! 0 +! " "#" $ $ " ! # ! 2 ! ! "% " " 0 " ! 0 ! ! : " " % ! 2 " 0 % 0 $ " ! # / " 2 3 ( & D ) " " % ! "% " " ! 0 " " ! $ " / /" " " . # 0 !"
  • 81. " / 56 7 / &89') )89') :89') ;89') <89') G 44 > <G G< 4 <G G> 4 4G 4 4G >> ;4 4G >> ;4 G ;4 G > 4 4 ; > 4 < ;G < ;; > 4 / 56 7 / 989') 9$)89') *89') ,89') &89') G @ M < 4 >> > > ; 4 4 <G 4 4; > < G; >< ; G < 4 4 ;> G; > < G 4 <G F"%! " 2 " !"
  • 82. 7 - 0 - 7 / ==7 - 0 - 7 / == !" - 0 > -?- 0 > -? - 0 > -?- 0 > -? 0 @ $0 @ $ "" 0 @ $0 @ $ ';':;))&9'+,&')&';*;*';':;))&9'+,&')&';*;* 7- 40 (7- 40 ( 4++ $ / $ / +4++ $ / $ / +
  • 83. Str o N GER www.stronger2012.com

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