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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
Visual Appearance
( 25 ft. span 1.2 in. )
Damage to Non-structural Elements
- cracking of partitions
- malfunction of doors /windows
(1.)
(2.)
Reasons to Limit Deflection
visiblegenerallyare*
250
1l
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Deflection Control
Disruption of function
- sensitive machinery, equipment
- ponding of rain water on roofs
Damage to Structural Elements
- large s 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 s are
computed
ACI Table 9.5(b) = max. permissible computeddeflection
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AllowableDeflectionsFlat Roofs ( no damageable nonstructural elements
supported)
180
instLL
l
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AllowableDeflectionsFloors ( no damageable nonstructural elements
supported )
180
instLLl
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Allowable Deflections
Roof or Floor elements (supported nonstructural elementslikely damaged by large s)
480
l
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Allowable Deflections
Roof or Floor elements ( supported nonstructural elementsnot likely to be damaged by large
s )
240
l
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Allowable Deflections
Deflection occurring after attachment of
nonstructural elements
Need to consider the specific structures
function and characteristics.
allow
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Moment of Inertia for Deflection Calculation
For (intermediate values of EI)gtecr III
Brandon
derived cr
a
a
cr
gt
a
a
cr
e *1* IM
M
IM
M
I
Cracking Moment =Moment of inertia of transformed cross-section
Modulus of rupture =
t
gr
y
If
c5.7 f
Mcr =Igt =
fr =
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Moment of Inertia for Deflection Calculation
cr
a
a
crgt
a
a
cre *1* I
M
MI
M
MI
Distance from centroid to extreme tension fiber
maximum moment in member at loading stage for
which Ie (
) is being computed or at any previousloading stage
Moment of inertia of concrete section neglect
reinforcement
yt =
Ma =
Ig =
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Moment of Inertia for Deflection Calculation
3
a
crcrgcre
cr
3
a
crg
3
a
cre
or
*1*
M
MIIII
I
M
MI
M
MI
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Moment Vs curvature plot
EIM
EI
M
slope
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Moment Vs Slope Plot
The cracked beam starts to lose strength as the amountof cracking increases
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Moment of Inertia
psi33 c1.5
cc fE
For wc = 90 to 155 lb/ft3
psi57000 cc fE
For normal weight concrete
(ACI 8.5.1)
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Deflection Response of RC Beams (Flexure)
A- Ends of Beam CrackB - Cracking at midspan
C - Instantaneous deflection
under service load
C - long time deflection under
service load
D and E - yielding of
reinforcement @ 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 actingon an indeterminate beam are given.
12
2wlM
12
2
wlM
24
2wlM
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Deflection Response of RC Beams (Flexure)
For Continuous beams
ACI 9.5.2.4
ACI Com. 435
Weight Average
e21emideavge 25.050.0 IIII
e21emideavge 15.070.0
:continousends2
IIII
1emideavge 15.085.0
:continousend1
III
2end@
1end@
midspan@
ee2
ee1
emide
II
II
II
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Uncracked Transformed Section
Part (n) =Ej /Ei Area n*Area yi yi*(n)A
Concrete 1 bw*h bw*h 0.5*h 0.5*bw*h2
As n As (n-1)As d (n-1)*As*d
As n As (n-1)As d (n-1)*As*d
An*ii Any **
*
ii
*
iii *
An
Anyy
Note:(n-1) is to remove area
of concrete
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Cracked Transformed Section
s
s
i
ii 2nAyb
dnAy
yb
AAyy
Finding the centroid of singly Reinforced RectangularSection
022
0
2
2
ss2
ss
2
ss
2
b
dnAy
b
nAy
dnAynAyb
dnAy
ybynAyb
Solve for the quadratic for y
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Cracked Transformed Section
022 ss2
b
dnAy
b
nAy
Note:
c
s
E
En
Singly Reinforced Rectangular Section
2s3
cr
3
1ydnAybI
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Cracked Transformed Section
0
212212 ssss2
b
dnAAny
b
nAAny
Note:
c
s
E
En
Doubly Reinforced Rectangular Section
2s2
s
3
cr 1
3
1ydnAdyAnybI
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Cracked Transformed Section
Finding the centroid of doubly reinforced T-Section
0
212
2122
w
ss
2
we
w
sswe2
b
dnAAntbb
y
b
nAAnbbty
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Cracked Transformed Section
Finding the moment of inertia fora doubly reinforced T-Section
steel
2
s
2
s
beam
3
w
flange
2
e
3
ecr
1
3
1
212
1
ydnAdyAn
tybt
ytbybI
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Stiffness of Reinforced Concrete Sections
- ExampleGiven 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 materialproperties are fc = 4 ksi and fy= 60 ksi.
Determine Igt, Icr , Mcr(+), Mcr(-), and compare to the NA of
the beam.