Lecture 31 Beam Deflection

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Lecture 31 - Beam Deflection

• April 5, 2001
• CVEN 444

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Lecture Goals

• Serviceability
• Moments and centroids

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Deflection Control
Reasons to Limit Deflection
Visual Appearance ( 25 ft. span 1.2 in.
) Damage to Non-structural Elements - cracking
of partitions - malfunction of doors /windows
(1.)
(2.)
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Deflection Control
Disruption of function - sensitive machinery,
equipment - ponding of rain water on
roofs Damage to Structural Elements - large
ds than serviceability problem - (contact w/
other members modify load paths)
(3.)
(4.)
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Allowable Deflections
ACI Table 9.5(a) min. thickness unless ds are
computed ACI Table 9.5(b) max. permissible
computed deflection
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Allowable Deflections
Flat Roofs ( no damageable nonstructural elements
supported)
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Allowable Deflections
Floors ( no damageable nonstructural elements
supported )
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Allowable Deflections
Roof or Floor elements (supported nonstructural
elements likely damaged by large ds)
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Allowable Deflections
Roof or Floor elements ( supported nonstructural
elements not likely to be damaged by large
ds )
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Allowable Deflections
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Moment of Inertia for Deflection Calculation
For (intermediate
values of EI)
Brandon derived
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Moment of Inertia for Deflection Calculation
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Moment of Inertia for Deflection Calculation
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Moment Vs curvature plot
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Moment Vs Slope Plot
The cracked beam starts to lose strength as the
amount of cracking increases
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Moment of Inertia
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Deflection Response of RC Beams (Flexure)
A- Ends of Beam Crack B - Cracking at midspan C -
Instantaneous deflection under service load C -
long time deflection under service load D and E -
yielding of reinforcement _at_ ends midspan
Note Stiffness (slope) decreases as cracking
progresses
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Deflection Response of RC Beams (Flexure)
The maximum moments for distributed load acting
on an indeterminate beam are given.
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Deflection Response of RC Beams (Flexure)
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Uncracked Transformed Section
(n-1) is to remove area of concrete
Note
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Cracked Transformed Section
Finding the centroid of singly Reinforced
Rectangular Section
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Cracked Transformed Section
Singly Reinforced Rectangular Section
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Cracked Transformed Section
Doubly Reinforced Rectangular Section
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Uncracked Transformed Section
Moment of inertia (uncracked doubly reinforced
beam)
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Cracked Transformed Section
Finding the centroid of doubly reinforced
T-Section
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Cracked Transformed Section
Finding the moment of inertia for a doubly
reinforced T-Section
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Stiffness of Reinforced Concrete Sections -
Example
Given a doubly reinforced beam with h 24 in, b
12 in., d 2.5 in. and d 21.5 in. with 2 7
bars in compression steel and 4 7 bars in
tension steel. The material properties are fc
4 ksi and fy 60 ksi. Determine Igt, Icr ,
Mcr(), Mcr(-), and compare to the NA of the
beam.