Lecture 6 Flexure

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Lecture 6 - Flexure

• September 13, 2002
• CVEN 444

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Lecture Goals

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

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Flexural Stress
Example Consider a simple rectangular beam( b x h
) reinforced with steel reinforcement of As.
Determine the centroid ( neutral axis, NA ) and
moment of inertia Izz of the beam for an ideal
beam (no cracks). Determine the NA and moment of
inertia, Izz, of beam if the beam is cracked and
tensile forces are in the steel only.
(1) (2)
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Example
Ec Modulus of Elasticity - concrete Es
Modulus of Elasticity - steel As Area of
steel d distance to steel b width h height
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Example
Centroid (NA)
Moment of Inertia
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Example (uncracked)
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Example - (cracked)
For a cracked section the concrete is in
compression and steel is in tension. The strain
in the beam is linear.
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Example - (cracked)
Using Equilibrium
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Example - (cracked)
Using Hookes law
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Example - (cracked)
Using a compatibility condition.
Substitute into the first equation.
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Example - (cracked)
Substitute in for the strain relationship.
Rearrange the equation into a quadratic equation.
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Example - (cracked)
Use a ratio of areas of concrete and steel.
Modify the equation to create a non-dimensional
ratio.
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Example - (cracked)
Use the quadratic formula
Solve for the centroid by multiplying the result
by d.
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Example - (cracked)
The moment of inertia using the parallel axis
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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.
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Example
A 7 bar has an As 0.6 in2
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Example
The uncracked centroid is
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Example
The uncracked moment of inertia
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Example
The cracked centroid is defined by
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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 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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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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Example
From equilibrium
The neutral axis is
37
Example
Check to see whether or not the steel has yielded.
Check the strain in the steel
Steel yielded!
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Example
Compute moment capacity of the beam.