Lecture 9 Flexure

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Lecture 9 - Flexure

• February 4, 2002
• CVEN 444

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Lecture Goals

• Basic Concepts
• Rectangular Beams
• Balanced Beams

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Example - Class
For the given beam find the maximum allowable
moment and the neutral axis fy 40 ksi, fc 3
ksi. (1)If the reinforcement consists of 3
6 (2)If the reinforcement consists of 3 9
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Analysis of Doubly Reinforced Sections
Effect of Compression Reinforcement on the
Strength and Behavior
Less concrete is needed to resist the T and
thereby moving the neutral axis (NA) up.
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Analysis of Doubly Reinforced Sections
Effect of Compression Reinforcement on the
Strength and Behavior
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Reasons for Providing Compression Reinforcement

• Reduced sustained load deflections.
• Creep of concrete in compression zone
• transfer load to compression steel
• reduced stress in concrete
• less creep
• less sustained load deflection

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Reasons for Providing Compression Reinforcement
Effective of compression reinforcement on
sustained load deflections. Fig 5-14 MacGregor
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Reasons for Providing Compression Reinforcement

• Increased Ductility reduced stress block
  depth increase in steel strain larger
  curvature are obtained.

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Reasons for Providing Compression Reinforcement
Effect of compression reinforcement on strength
and ductility of under reinforced beams.
r lt rb Fig 5-15 MacGregor
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Reasons for Providing Compression Reinforcement

• Change failure mode from compression to tension.
  When r gt rbal, addition of As strengthens.
  Compression zone allows tension
  steel to yield before crushing of
  concrete. Effective reinforcement ratio (r -
  r)

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Reasons for Providing Compression Reinforcement

• Eases in Fabrication use corner bars to hold
  anchor stirrups.

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Effect of Compression Reinforcement
Compare the strain distribution in two beams with
the same As
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Effect of Compression Reinforcement
Section 1
Section 2
Addition of As strengthens compression zone so
that less concrete is needed to resist a given
value of T. NA goes up (c2 ltc1) and es
increases (es2 gtes1).
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Doubly Reinforced Beams
Four Possible Modes of Failure

• Under reinforced Failure
• ( Case 1 ) Compression and tension steel yields
• ( Case 2 ) Only tension steel yields
• Over reinforced Failure
• ( Case 3 ) Only compression steel yields
• ( Case 4 ) No yielding Concrete crushes

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Analysis of Doubly Reinforced Rectangular Sections
Case 1 Assume As yields fsfy
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Analysis of Doubly Reinforced Rectangular Sections
Case 1(cont.) Using equilibrium and find a
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Analysis of Doubly Reinforced Rectangular Sections
Case 1 (cont.) Check assumption
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Analysis of Doubly Reinforced Rectangular Sections
Case 1 (cont.) Confirm Calculate
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Analysis of Doubly Reinforced Rectangular Sections
Case 2 Assume As has not yielded
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Analysis of Doubly Reinforced Rectangular Sections
Case 2(cont.) Using similar triangles
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Analysis of Doubly Reinforced Rectangular Sections
Case 2 (cont.) Equilibrium Solve for NA,
depth c (quadratic equation in terms of NA depth
c)
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Analysis of Doubly Reinforced Rectangular Sections
Case 2 (cont.) Check Assumptions Calculate