Lecture 5 Flexure

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Lecture 5 - Flexure

• January 24, 2003
• CVEN 444

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Lecture Goals

• Structures
• Basic Concepts
• Rectangular Beams

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Class of Structures
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Class of Structures
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Class of Structures
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Class of Structures
Abutment
Retaining Wall
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Class of Structures
Deformed Frame
Reinforced Frame
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Flexural Stress
The beam is a structural member used to support
the internal moments and shears. It would be
called a beam-column if a compressive force
existed. C T M C(jd)
T(jd)
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Flexural Stress
The stress in the block is defined as s (My)
/ I Sxx I / (ymax) The equation for Sxx modulus
for calculating maximum compressive stress.
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Flexural Stress
There are 5 stages the concrete through which the
beam goes.
Stage 1 No external loads self weight.
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Flexural Stress
There are 5 stages the concrete through which the
beam goes.
Stage 2 the external load P cause the bottom
fibers to equal to modulus of rupture of the
concrete. Entire concrete section was effective,
steel bar at tension side has same strain as
surrounding concrete.
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Flexural Stress
There are 5 stages the concrete through which the
beam goes.
Stage 3 The tensile strength of the concrete
exceeds the rupture fr and cracks develop. The
neutral axis shifts upward and cracks extend to
neutral axis. Concrete loses tensile strength
and steel starts working effectively and resists
the entire tensile load.
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Flexural Stress
There are 5 stages the concrete through which the
beam goes.
Stage 4 The reinforcement yields. Stage 5
Failure of the beam.
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Flexural Stress
The three stages of the beam. Stage 1 No
external loads acting on the beam. Stage 3
Service loading on the beam. Stage 5 Beam
failure.
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Flexural Stress
The moment-curvature diagram show the five stages
of the beam. The plot is of the curvature angle,
f , verse the moment. f (e / y) s / E /
y (My / I) / E / y f M / ( E I )
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Flexural Stress
The beam fails first in shear and the second beam
fails in bending moment.
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Flexural Stress
There are three types of flexural failure of a
structural member. Steel may reach its yield
strength before the concrete reaches its maximum.
(Under-reinforced section). Steel reaches yield
at same time as concrete reaches ultimate
strength. (Balanced section). Concrete may fail
before the the yield of steel due to the presence
of a high percentage of steel in the section.
(Over-reinforced section).
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Flexural Stress
Steel may reach its yield strength before the
concrete reaches its maximum. (Under-reinforced
section).
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Flexural Stress
Steel reaches yield at same time as concrete
reaches ultimate strength. (Balanced section).
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Flexural Stress
Concrete may fail before the the yield of steel
due to the presence of a high percentage of steel
in the section. (Over-reinforced section).
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Flexural Stress
The flexural strain and stress distribution of
beam from a test beam.
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Flexural Stress
Strain measured in test of eccentrically loaded
columns for a tied and spiral columns.
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Flexural Stress-Example
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-Definitions
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 Mechanics of Materials Properties
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