Lecture 34 Elastic Flexural Analysis for Serviceability

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Lecture 34 - Elastic Flexural Analysis for
Serviceability

• November 20, 2001
• CVEN 444

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Lecture Goals

• Serviceability
• Crack width
• Moments of inertia

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Introduction
Recall
Ultimate Limit States Lead to
collapse Serviceability Limit States Disrupt
use of Structures but do not cause
collapse
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Introduction
Types of Serviceability Limit States - Excessive
crack width - Excessive deflection -
Undesirable vibrations - Fatigue (ULS)
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Crack Width Control
Cracks are caused by tensile stresses due to
loads moments, shears, etc..
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Crack Width Control
Cracks are caused by tensile stresses due to
loads moments, shears, etc..
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Crack Width Control
Bar crack development.
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Crack Width Control
Temperature crack development
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Crack Width Control
Reasons for crack width control?

• Appearance (smooth surface gt 0.01 to 0.013
  public concern)
• Leakage (Liquid-retaining structures)
• Corrosion (cracks can speed up occurrence
  of corrosion)

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Crack Width Control
Corrosion more apt to occur if (steel oxidizes
rust )

• Chlorides ( other corrosive substances) present
• Relative Humidity gt 60
• High Ambient Temperatures (accelerates chemical
  reactions)
• Wetting and drying cycles
• Stray electrical currents occur in the bars.

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Limits on Crack Width
ACI Codes Basis
0.016 in. for interior exposure 0.013 in.
for exterior exposure
max.. crack width
Cracking controlled in ACI code by regulating the
distribution of reinforcement in beams/slabs.
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Limits on Crack Width
Gergely-Lutz Equation
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Limits on Crack Width
Gergely-Lutz Equation
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Limits on Crack Width
ACI Code Eqn 10-5 ( limits magnitude of z term )
Note w 0.076b z (b 1.2 for beams)
Interior exposure critical crack width 0.016
in. ( w 16 ) z 175k/in Exterior
exposure critical crack width 0.013 in. (
w 13 ) z 145k/in
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Limits on Crack Width
Tolerable Crack Widths
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Limits on Crack Width
Thin one-way slabs Use b 1.35
z 155 k/in (Interior Exposure) z 130 k/in
(Exterior Exposure)
fs service load stress may be taken as
1.55 average load factor
f - strength reduction
. factor for flexure
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Example-Crack
Given A beam with bw 14 in. Gr 60 steel 4 8
with 2 6 in the second layer with a 4 stirrup.
Determine the crack width limit, z for exterior
and interior limits (145 k/in and 175 k/in.).
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Example-Crack
Compute the center of the steel for the given
bars.
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Example-Crack
The locations of the center of the bars are
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Example-Crack
Compute the center of the steel for the given
bars.
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Example-Crack
Compute number of equivalent bars, n. Use the
largest bar. Compute the effective tension area
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Example-Crack
The effective service load stress is Compute
the effective tension area
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Example-Crack
The limits magnitude of z term. 122.9 k/in. lt
145 k/in. - Interior exposure 122.9 k/in. lt
175 k/in. - Exterior exposure Crack width
is or w 0.0112 in.
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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
For normal weight concrete
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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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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.
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Reinforced Concrete Sections - Example
The components of the beam
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Reinforced Concrete Sections - Example
The compute the n value and the centroid, I
uncracked
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Reinforced Concrete Sections - Example
The compute the centroid and I uncracked
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Reinforced Concrete Sections - Example
The compute the centroid and I for a cracked
doubly reinforced beam.
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Reinforced Concrete Sections - Example
The compute the centroid for a cracked doubly
reinforced beam.
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Reinforced Concrete Sections - Example
The compute the moment of inertia for a cracked
doubly reinforced beam.
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Reinforced Concrete Sections - Example
The critical ratio of moment of inertia
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Reinforced Concrete Sections - Example
Find the components of the beam
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Reinforced Concrete Sections - Example
Find the components of the beam
The neutral axis
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Reinforced Concrete Sections - Example
The strain of the steel
Note At service loads, beams are assumed to act
elastically.
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Reinforced Concrete Sections - Example
Using a linearly varying e and s Ee along the
NA is the centroid of the area for an elastic
center
The maximum tension stress in tension is
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Reinforced Concrete Sections - Example
The uncracked moments for the beam
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Homework-12/2/02
Problem 8.7