Lecture 6 Flexure

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Lecture 6 - Flexure

• January 27, 2003
• CVEN 444

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Lecture Goals

• Basic Concepts
• Rectangular Beams
• Non-uniform beams
• Balanced Beams

3
Example
For the following example find centroid and
moment of inertia for an uncracked and cracked
section and compare the results.
Es 29000 ksi Ec 3625 ksi d 15.5 in b 12
in. h 18 in. Use 4 7 bars for the steel.
4
Example
A 7 bar has an As 0.6 in2
5
Example
The uncracked centroid is
6
Example
The uncracked moment of inertia
7
Example
The cracked centroid is defined by
8
Example
The cracked moment of inertia is
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Example
Notice that the centroid changes from 9.47 in. to
5.62 in. and the moment of inertia decreases from
6491 in4 to 2584 in4 . The cracked section loses
more than half of its strength.
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Flexural Stress
Basic Assumptions in Flexure Theory

• Plane sections remain plane ( not true for deep
  beams h gt 4b)
• The strain in the reinforcement is equal to the
  strain in the concrete at the same level, i.e. es
  ec at same level.
• Stress in concrete reinforcement may be
  calculated from the strains using s-e curves for
  concrete steel.

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Flexural Stress
Additional Assumptions for design (for
simplification)

• Tensile strength of concrete is neglected for
  calculation of flexural strength.
• Concrete is assumed to fail in compression, when
  ec (concrete strain) ecu (limit state) 0.003
• Compressive s-e relationship for concrete may be
  assumed to be any shape that results in an
  acceptable prediction of strength.

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Flexural Stress
The concrete may exceed the ec at the outside
edge of the compressive zone.
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Flexural Stress
The compressive force is modeled as Cc k1k3fc
bc at the location x k2c
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Flexural Stress
The compressive coefficients of the stress block
at given for the following shapes. k3 is ratio of
maximum stress at fc in the compressive zone of a
beam to the cylinder strength, fc (0.85 is a
typical value for common concrete)
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Flexural Stress
The compressive zone is modeled with a equivalent
stress block.
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Flexural Stress
The equivalent rectangular concrete stress
distribution has what is known as a b1
coefficient is proportion of average stress
distribution covers.
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Flexural Stress
Requirements for analysis of reinforced concrete
beams
1 Stress-Strain Compatibility Stress at a
point in member must correspond to strain at a
point.
2 Equilibrium Internal forces balances with
external forces
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Flexural Stress
Example of rectangular reinforced concrete beam.
(1) Setup equilibrium.
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Flexural Stress
Example of rectangular reinforced concrete beam.
(2) Find flexural capacity.
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Flexural Stress
Example of rectangular reinforced concrete beam.
(2) Find flexural capacity.
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Flexural Stress
Example of rectangular reinforced concrete beam.
(3) Need to confirm es gt ey
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Flexural Stress Rectangular Example
Example of rectangular reinforced concrete beam.
Given a rectangular beam fc 4000 psi fy 60
ksi (4 7 bars) b 12 in. d 15.5 in. h 18
in. Find the neutral axis. Find the moment
capacity of the beam.
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Flexural Stress Rectangular Example
Determine the area of steel, 7 bar has 0.6
in2. The b value is b1 0.85 because the
concrete has a fc 4000 psi.
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Flexural Stress Rectangular Example
From equilibrium (assume the steel has yielded)
The neutral axis is
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Flexural Stress Rectangular Example
Check to see whether or not the steel has yielded.
Check the strain in the steel
Steel yielded!
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Flexural Stress Rectangular Example
Compute moment capacity of the beam.
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Flexural Stress Non-Rectangular Example
For a non-rectangular beam
For the given beam with concrete rated at fc
6000 psi and the steel is rated at fs 60,000
psi. d 12.5 in.
(a) Determine the area of the steel for a
balanced system for shown area of concrete. (b)
Determine the moment capacity of the beam.
Mn (c) Determine the NA.
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Flexural Stress Non-Rectangular Example
For a non-rectangular beam
The area of the concrete section is
The force due to concrete forces.
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Flexural Stress Non-Rectangular Example
Using equilibrium, the area of the steel can be
found
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Flexural Stress Non-Rectangular Example
Find the center of the area of concrete area
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Flexural Stress Non-Rectangular Example
The moment capacity of the beam is
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Flexural Stress Non-Rectangular Example
Compute the b1 value
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Flexural Stress Non-Rectangular Example
Find the neutral axis