text: You are on page 1of 4
  
  Problem 19.
  
  21
  24
  
  0.9 in2
  
  14
  
  The beam shown has 30-ft simple span; fc = 5000 psi, fpu = 250,000 psi, and the initial prestress is 160,000 psi and
  
  fpy = 0.85 fpu
  
  Given:
  
  b = 14 in L = 30 ft or
  h = 24 in L = 360 in
  d = 21 in fc = 5000 psi
  As = 0.9 in2 fpi = 160,000 psi
  
  a) Calculate the concrete stress in the top and bottom of the beam at midspan immediately after the tendons are cut.
  (Ans. ftop= +0.183 ksi, fbottom= -1.041 ksi)
  
  Solution:
  
  (@ midspan)
  
  3
  Pi = fpi As I=
  12
  (14")(24")3
  Pi = (160,000 psi)( 0.9 in2) I=
  12
  Pi = 144,000 lb or 144 kips I = 16,128 in4
  
  c = 12 in Ac = (14)(24)
  e = 9 in Ac = 336 in2
  
  1
  W b = (14)(24)(150 3 ) ( 12 )3
  
  W b = 29.167
  
  2
  Md = 8
  
  (29.167 )(360 )2
  
  Md = 8
  
  Md = 472,505.4 lb-in
  
  ftop = - +
  -
  
  144,000 lb 144,000 lb 9 (12 ) 472,505.4 lb in (12 )
  
  ftop = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  fbottom = - -
  +
  
  fbottom = -1,041.29 or -1.04129 --------------------------- answer
  
  b) Recalculate the stresses at midspan after assumed prestress losses in the tendons of 20%. (Ans. ftop = +0.076 ksi,
  fbottom = - 0.762 ksi)
  
  Peff = Pi - 0.20(Pi)
  
  Peff = 144,000 lb - 0.20(144,000 lb)
  
  Peff = 115,200 lb
  
  ftop = - +
  -
  
  115,200 lb 115,200 lb 9 (12 ) 472,505.4 lb in (12 )
  
  ftop = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  fbottom = - -
  +
  
  fbottom = -780.139 or -0.780139 ------------------------------ answer
  
  c) What maximum service liveload can this beam support in addition to its own weight if allowable stresses of 0.45fc in
  compression and 6 fc in tension are permitted? (ans. 1.182k/ft)
  
  Peff = 115,200 lb
  
  Allowable:
  
  ftop = 0.45 fc --------- compression MTotal = MD + ML
  
  2 ( ) 2
  fbottom = 6 fc --------- tension Mt = 8
  + 8
  
  ftop = - +
  
  115,200 lb 115,200 lb 9 (12 ) (12 )
  
  -0.45 fc = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  (12 )
  - 0.45 (5000 psi ) = - 342.857 + 771.429 16,128 in 4
  
  (12 )
  - 2,250 psi = - 342.857 + 771.429 16,128 in 4
  
  (12 )
  - 2,678.572 psi = 16,128 in 4
  
  Mt = + 3,600,000.768 lb-in
  
  2 ( ) 2
  Mt= 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  3,600,000.768 lb-in = 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  3,600,000.768 lb-in = 8
  + 8
  
  (360 )2
  3,600,000.768 lb-in = 472,505.4 lb-in +
  8
  
  (360 )2
  3,127,495.369 lb-in = 8
  
  = 193.055 or
  
  1 12
  = 193.055 (1000 )( 1
  ) = 2.317
  
  fbottom = -
  +
  
  115,200 lb 115,200 lb 9 (12 ) (12 )
  
  6 fc = - - +
  336 in 2 16,128 in 4 16,128 in 4
  
  (12 )
  6 5000 psi = - 342.857 - 771.429 +
  16,128 in 4
  
  (12 )
  424.264 psi = - 342.857 - 771.429 + 16,128 in 4
  
  (12 )
  1538.55 psi = 16,128 in 4
  
  Mt = 2,067,811.2 lb-in
  
  2 ( ) 2
  Mt= 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  2,067,811.2 lb-in = 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  2,067,811.2 lb-in = 8
  + 8
  
  (360 )2
  2,067,811.2 lb-in = 472,505.4 lb-in +
  8
  
  (360 )2
  1,595,305.8 lb-in = 8
  
  = 98.476 or
  
  1 12
  = 98.476 (1000 )( 1
  ) = 1. 182
  
  ( the least governs)!!
  
  Therefore, = = 1. 182 ----------------- answer
explanation: The document is a detailed solution to a structural engineering problem involving the calculation of concrete stresses in a prestressed beam. It contains multiple questions and their corresponding answers, specifically:
  
  1. Calculating the concrete stress in the top and bottom of the beam at midspan immediately after the tendons are cut.
  2. Recalculating the stresses at midspan after assumed prestress losses in the tendons of 20%.
  3. Determining the maximum service liveload the beam can support in addition to its own weight, given specific allowable stresses.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: You are on page 1of 4
  
  Problem 19.
  
  21
  24
  
  0.9 in2
  
  14
  
  The beam shown has 30-ft simple span; fc = 5000 psi, fpu = 250,000 psi, and the initial prestress is 160,000 psi and
  
  fpy = 0.85 fpu
  
  Given:
  
  b = 14 in L = 30 ft or
  h = 24 in L = 360 in
  d = 21 in fc = 5000 psi
  As = 0.9 in2 fpi = 160,000 psi
  
  a) Calculate the concrete stress in the top and bottom of the beam at midspan immediately after the tendons are cut.
  (Ans. ftop= +0.183 ksi, fbottom= -1.041 ksi)
  
  Solution:
  
  (@ midspan)
  
  3
  Pi = fpi As I=
  12
  (14")(24")3
  Pi = (160,000 psi)( 0.9 in2) I=
  12
  Pi = 144,000 lb or 144 kips I = 16,128 in4
  
  c = 12 in Ac = (14)(24)
  e = 9 in Ac = 336 in2
  
  1
  W b = (14)(24)(150 3 ) ( 12 )3
  
  W b = 29.167
  
  2
  Md = 8
  
  (29.167 )(360 )2
  
  Md = 8
  
  Md = 472,505.4 lb-in
  
  ftop = - +
  -
  
  144,000 lb 144,000 lb 9 (12 ) 472,505.4 lb in (12 )
  
  ftop = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  fbottom = - -
  +
  
  fbottom = -1,041.29 or -1.04129 --------------------------- answer
  
  b) Recalculate the stresses at midspan after assumed prestress losses in the tendons of 20%. (Ans. ftop = +0.076 ksi,
  fbottom = - 0.762 ksi)
  
  Peff = Pi - 0.20(Pi)
  
  Peff = 144,000 lb - 0.20(144,000 lb)
  
  Peff = 115,200 lb
  
  ftop = - +
  -
  
  115,200 lb 115,200 lb 9 (12 ) 472,505.4 lb in (12 )
  
  ftop = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  fbottom = - -
  +
  
  fbottom = -780.139 or -0.780139 ------------------------------ answer
  
  c) What maximum service liveload can this beam support in addition to its own weight if allowable stresses of 0.45fc in
  compression and 6 fc in tension are permitted? (ans. 1.182k/ft)
  
  Peff = 115,200 lb
  
  Allowable:
  
  ftop = 0.45 fc --------- compression MTotal = MD + ML
  
  2 ( ) 2
  fbottom = 6 fc --------- tension Mt = 8
  + 8
  
  ftop = - +
  
  115,200 lb 115,200 lb 9 (12 ) (12 )
  
  -0.45 fc = - +
  336 in 2 16,128 in 4 16,128 in 4
  
  (12 )
  - 0.45 (5000 psi ) = - 342.857 + 771.429 16,128 in 4
  
  (12 )
  - 2,250 psi = - 342.857 + 771.429 16,128 in 4
  
  (12 )
  - 2,678.572 psi = 16,128 in 4
  
  Mt = + 3,600,000.768 lb-in
  
  2 ( ) 2
  Mt= 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  3,600,000.768 lb-in = 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  3,600,000.768 lb-in = 8
  + 8
  
  (360 )2
  3,600,000.768 lb-in = 472,505.4 lb-in +
  8
  
  (360 )2
  3,127,495.369 lb-in = 8
  
  = 193.055 or
  
  1 12
  = 193.055 (1000 )( 1
  ) = 2.317
  
  fbottom = -
  +
  
  115,200 lb 115,200 lb 9 (12 ) (12 )
  
  6 fc = - - +
  336 in 2 16,128 in 4 16,128 in 4
  
  (12 )
  6 5000 psi = - 342.857 - 771.429 +
  16,128 in 4
  
  (12 )
  424.264 psi = - 342.857 - 771.429 + 16,128 in 4
  
  (12 )
  1538.55 psi = 16,128 in 4
  
  Mt = 2,067,811.2 lb-in
  
  2 ( ) 2
  Mt= 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  2,067,811.2 lb-in = 8
  + 8
  
  29.167 (360 )2 (360 )2
  
  2,067,811.2 lb-in = 8
  + 8
  
  (360 )2
  2,067,811.2 lb-in = 472,505.4 lb-in +
  8
  
  (360 )2
  1,595,305.8 lb-in = 8
  
  = 98.476 or
  
  1 12
  = 98.476 (1000 )( 1
  ) = 1. 182
  
  ( the least governs)!!
  
  Therefore, = = 1. 182 ----------------- answer
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed solution to a structural engineering problem involving the calculation of concrete stresses in a prestressed beam. It contains multiple questions and their corresponding answers, specifically:
  
  1. Calculating the concrete stress in the top and bottom of the beam at midspan immediately after the tendons are cut.
  2. Recalculating the stresses at midspan after assumed prestress losses in the tendons of 20%.
  3. Determining the maximum service liveload the beam can support in addition to its own weight, given specific allowable stresses.
  
  Label: multiple-question-and-answer