Elastic Flexural Analysis for Serviceability

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Elastic Flexural Analysis for Serviceability
August 11, 2003 CVEN444
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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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Calculate the Deflections
(1) Instantaneous (immediate) deflections (2)
Sustained load deflection
Instantaneous Deflections due to dead loads(
unfactored) , live, etc.
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Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases
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Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases
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Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases
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Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases
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Sustained Load Deflections
Creep causes an increase in concrete strain
Curvature increases
Increase in compressive strains cause increase in
stress in compression reinforcement (reduces
creep strain in concrete)
Compression steel present
Helps limit this effect.
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Sustained Load Deflections
Sustain load deflection l Di
Instantaneous deflection
ACI 9.5.2.5
at midspan for simple and continuous beams at
support for cantilever beams
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Sustained Load Deflections
x time dependent factor for sustained load
Also see Figure 9.5.2.5 from ACI code
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Sustained Load Deflections
For dead and live loads
DL and LL may have different x factors for LT (
long term ) D calculations
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Sustained Load Deflections
The appropriate value of Ic must be used to
calculate D at each load stage.
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Serviceability Load Deflections - Example
Show in the attached figure is a typical interior
span of a floor beam spanning between the girders
at locations A and C. Partition walls, which may
be damaged by large deflections, are to be
erected at this level. The interior beam shown
in the attached figure will support one of these
partition walls. The weight of the wall is
included in the uniform dead load provided in the
figure. Assume that 15 of the distributed
dead load is due to a superimposed dead load,
which is applied to the beam after the partition
wall is in place. Also assume that 40 of the
live load will be sustained for at least 6 months.
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Serviceability Load Deflections - Example
fc 5 ksi fy 60 ksi
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Serviceability Load Deflections - Example
Part I Determine whether the floor beam meets the
ACI Code maximum permissible deflection criteria.
(Note it will be assumed that it is acceptable
to consider the effective moments of inertia at
location A and B when computing the average
effective moment of inertia for the span in this
example.) Part II Check the ACI Code crack width
provisions at midspan of the beam.
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Serviceability Load Deflections - Example
Deflection before glass partition is installed
(85 of DL)
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Serviceability Load Deflections - Example
Compute the centroid and gross moment of inertia,
Ig.
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Serviceability Load Deflections - Example
The moment of inertia
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Serviceability Load Deflections - Example
The moment capacity
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Serviceability Load Deflections - Example
Determine bending moments due to initial load
(0.85 DL) The ACI moment coefficients will be
used to calculate the bending moments Since the
loading is not patterned in this case, This is
slightly conservative
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Serviceability Load Deflections - Example
The moments at the two locations
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Serviceability Load Deflections - Example
Moment at C will be set equal to Ma for
simplicity, as given in the problem statement.
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Serviceability Load Deflections - Example
Assume Rectangular Section Behavior and calculate
the areas of steel and ratio of Modulus of
Elasticity
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Serviceability Load Deflections - Example
Calculate the center of the T-beam
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Serviceability Load Deflections - Example
The centroid is located at the As lt 4.5 in. tf
Use rectangular section behavior
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Serviceability Load Deflections - Example
The moment of inertia at midspan
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Serviceability Load Deflections - Example
Calculate average effective moment of inertia,
Ie(avg) for interior span (for 0.85 DL) For beam
with two ends continuous and use Ig for the two
ends.
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Serviceability Load Deflections - Example
Calculate instantaneous deflection due to 0.85
DL Use the deflection equation for a fixed-fixed
beam but use the span length from the centerline
support to centerline support to reasonably
approximate the actual deflection.
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Serviceability Load Deflections - Example
Calculate additional short-term Deflections (full
DL LL)
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Serviceability Load Deflections - Example
Calculate additional short-term Deflections (full
DL LL) Let Mc Ma - 2000 k-in for
simplicity see problem statement
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Serviceability Load Deflections - Example
Assume beam is fully cracked under full DL LL,
therefore I Icr (do not calculate Ie for now).
Icr for supports
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Serviceability Load Deflections - Example
Class formula using doubly reinforced rectangular
section behavior.
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Serviceability Load Deflections - Example
Class formula using doubly reinforced rectangular
section behavior.
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Serviceability Load Deflections - Example
Calculate moment of inertia.
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Serviceability Load Deflections - Example
Weighted Icr
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Serviceability Load Deflections - Example
Instantaneous Dead and Live Load Deflection.
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Serviceability Load Deflections - Example
Long term Deflection at the midspan
Dead Load (Duration gt 5 years)
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Serviceability Load Deflections - Example
Long term Deflection use the midspan information
Live Load (40 sustained 6 months)
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Serviceability Load Deflections - Example
Total Deflection after Installation of Glass
Partition Wall.
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Serviceability Load Deflections - Example
Check whether modifying Icr to Ie will give an
acceptable deflection
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Serviceability Load Deflections - Example
Check whether modifying Icr to Ie will give an
acceptable deflection
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Serviceability Load Deflections - Example
Floor Beam meets the ACI Code Maximum permissible
Deflection Criteria. Adjust deflections
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Serviceability Load Deflections - Example
Adjust deflections
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Serviceability Load Deflections - Example
Part II Check crack width _at_ midspan
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Serviceability Load Deflections - Example
Assume
For interior exposure, the crack width _at_ midspan
is acceptable.