Post on 15-Jan-2016
Lecture GoalsLecture Goals
Doubly Reinforced beamsT Beams and L Beams
Analysis of Doubly Analysis of Doubly Reinforced SectionsReinforced Sections
Effect of Compression Reinforcement on the Strength and Behavior
Less concrete is needed to resist the T and thereby moving the neutral axis (NA) up.
TC
fAT
ys
Analysis of Doubly Analysis of Doubly Reinforced SectionsReinforced Sections
Effect of Compression Reinforcement on the Strength and Behavior
12
2sc
1c
and
2; C
ReinforcedDoubly
2;
ReinforcedSingly
aa
adfAMCC
adfAMCC
ysn
ysn
Reasons for Providing Reasons for Providing Compression Compression ReinforcementReinforcement
Reduced sustained load deflections. Creep of concrete in compression
zone transfer load to compression steel reduced stress in concrete less creep less sustained load deflection
Reasons for Providing Reasons for Providing Compression ReinforcementCompression Reinforcement
Effective of compression reinforcement on sustained load deflections.
Reasons for Providing Reasons for Providing Compression ReinforcementCompression Reinforcement
Increased Ductility
reduced stress block depth
increase in steel strain larger curvature are obtained.
Reasons for Providing Reasons for Providing Compression ReinforcementCompression Reinforcement
Effect of compression reinforcement on strength and ductility of under reinforced beams.
b
Reasons for Providing Reasons for Providing Compression ReinforcementCompression Reinforcement
Change failure mode from compression to tension. When bal addition of As strengthens.
Effective reinforcement ratio = (’)
Compression zone
allows tension steel to yield before crushing of concrete.
Reasons for Providing Reasons for Providing Compression ReinforcementCompression Reinforcement
Eases in Fabrication - Use corner bars to hold & anchor
stirrups.
Effect of Compression Effect of Compression ReinforcementReinforcement
Compare the strain distribution in two beams with the same As
Effect of Compression Effect of Compression ReinforcementReinforcement
Section 1: Section 2:
Addition of A’s strengthens compression zone so that less concrete is needed to resist a given value of T. NA goes up (c2 <c1) and s increases (s2 >s1).
1c
ss1
11cc1c
ss
85.0
85.0 85.0
bf
fAc
cbfbafCT
fAT
1c
ssss2
21css
2css
1cs
ss
85.0
85.0
85.0
bf
fAfAc
cbffA
baffA
CCT
fAT
Doubly Reinforced Doubly Reinforced BeamsBeams
Under reinforced Failure ( Case 1 ) Compression and tension steel
yields ( Case 2 ) Only tension steel yields
Over reinforced Failure ( Case 3 ) Only compression steel yields ( Case 4 ) No yielding Concrete crushes
Four Possible Modes of Failure
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Strain Compatibility Check Assume s’ using similar triangles
s
s
0.003
'
'*0.003
c d c
c d
c
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Strain Compatibility
Using equilibrium and find a
s s yc s
c
s s y y
1 1 c 1 c
0.85
'
0.85 0.85
A A fT C C a
f b
A A f d fac
f b f
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Strain Compatibility The strain in the compression steel is
s cu
1 c
y
1
0.851 0.003
'
d
c
f d
d f
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Strain Compatibility Confirm
;E ys
s
yys
f
y y1 cs 3
y s
0.851 0.003
' E 29 x 10 ksi
f ff d
d f
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSectionsStrain Compatibility Confirm
y1 c
y
1 c
y y
870.85
' 87
0.85 87'
87
ff d
d f
f d
d f f
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSectionsFind c
confirm that the tension steel has yielded
s y c s y
ss s y1
c 1
0.85
0.85
A f f ba A f
A A fc a c
f b
ys cu y
sE
fd c
c
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
If the statement is true than
else the strain in the compression steel
n s s y s y2
aM A A f d A f d d
s sf E
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Return to the original equilibrium equation
s y s s c
s s s c 1
s s cu c 1
0.85
0.85
1 0.85
A f A f f ba
A E f b c
dA E f b c
c
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Rearrange the equation and find a quadratic equation
Solve the quadratic and find c.
s y s s cu c 1
2c 1 s s cu s y s s cu
1 0.85
0.85 0
dA f A E f b c
c
f b c A E A f c A E d
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Find the fs’
Check the tension steel.
s s cu1 1 87 ksid d
f Ec c
ys cu y
sE
fd c
c
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Another option is to compute the stress in the compression steel using an iterative method.
1 c3s
y
0.8529 x 10 1 0.003
'
f df
d f
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSectionsGo back and calculate the equilibrium with fs’
s y s s
c sc
1
s
0.85
1 87 ksi
A f A fT C C a
f b
ac
df
c
Iterate until the c value is adjusted for the fs’ until the stress converges.
Analysis of Doubly Analysis of Doubly Reinforced Rectangular Reinforced Rectangular SectionsSections
Compute the moment capacity of the beam
n s y s s s s2
aM A f A f d A f d d
Limitations on Reinforcement Limitations on Reinforcement Ratio for Doubly Reinforced Ratio for Doubly Reinforced beamsbeams
Lower limit on
same as for single reinforce beams.
yy
cmin
200
3
ff
f
(ACI 10.5)
Example: Doubly Reinforced Example: Doubly Reinforced SectionSection
Given:
f’c= 4000 psi fy = 60 ksi
A’s = 2 #5 As = 4 #7
d’= 2.5 in. d = 15.5 in
h=18 in. b =12 in.
Calculate Mn for the section for the given compression steel.
Example: Doubly Reinforced Example: Doubly Reinforced SectionSection
Compute the reinforcement coefficients, the area of the bars #7 (0.6 in2) and #5 (0.31 in2)
2 2s
2 2s
2s
2s
4 0.6 in 2.4 in
2 0.31 in 0.62 in
2.4 in0.0129
12 in. 15.5 in.
0.62 in0.0033
12 in. 15.5 in.
A
A
A
bd
A
bd
Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionCompute the effective reinforcement ratio and minimum
y
c
y
min
0.0129 0.0033 0.00957
200 2000.00333
60000
3 3 4000or 0.00316
60000
0.0129 0.00333 OK!
eff
f
f
f
Example: Doubly Reinforced Example: Doubly Reinforced SectionSection
Compute the effective reinforcement ratio and minimum
1 c
y y
0.85 87'
87
0.85 0.85 4 ksi 2.5 in. 870.0398
60 ksi 15.5 in. 87 60
f d
d f f
0.00957 0.0398 Compression steel has not yielded.
Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionInstead of iterating the equation use the quadratic method
2c 1 s s cu s y s s cu
2
2 2
2
2
2
0.85 0
0.85 4 ksi 12 in. 0.85
0.62 in 29000 ksi 0.003 2.4 in 60 ksi
0.62 in 29000 ksi 0.003 2.5 in. 0
34.68 90.06 134.85 0
2.5969 3.8884 0
f b c A E A f c A E d
c
c
c c
c c
Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionSolve using the quadratic formula
2
2
2.5969 3.8884 0
2.5969 2.5969 4 3.8884
23.6595 in.
c c
c
c
Example: Doubly Reinforced Example: Doubly Reinforced SectionSection
Find the fs’
Check the tension steel.
s s cu
2.5 in.1 1 87 ksi
3.659 in.
27.565 ksi
df E
c
s
15.5 in. 3.659 in.0.003 0.00971 0.00207
3.659 in.
Example: Doubly Reinforced Example: Doubly Reinforced SectionSection
Check to see if c works
2 2s y s s
c 1
2.4 in 60 ksi 0.62 in 27.565 ksi
0.85 0.85 4 ksi 0.85 12 in.
3.659 in.
A f A fc
f b
c
The problem worked
Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionCompute the moment capacity of the beam
s y s s s s
2
2
2
2
2.4 in 60 ksi 0.85 3.659 in.15.5 in.
2 0.62 in 27.565 ksi
0.62 in 27.565 ksi 15.5 in. 2.5 in.
1991.9 k - in. 166 k - ft
n
aM A f A f d A f d d
Example: Doubly Reinforced Example: Doubly Reinforced SectionSectionIf you want to find the Mu for the problem
u u
3.66 in.0.236
15.5 in.
0.375 0.9
0.9 166 k - ft
149.4 k - ft
c
d
c
d
M M
From ACI (figure R9.3.2)or figure (pg 100 in your text)
The resulting ultimate moment is
Analysis of Flanged Analysis of Flanged SectionSection
Floor systems with slabs and beams are placed in monolithic pour.Slab acts as a top flange to the beam; T-beams, and Inverted L(Spandrel) Beams.
Analysis of Flanged Analysis of Flanged SectionsSectionsPositive and Negative Moment Regions in a T-beam
Analysis of Flanged Analysis of Flanged SectionsSections
If the neutral axis falls within the slab depth analyze the beam as a rectangular beam, otherwise as a T-beam.
Analysis of Flanged Analysis of Flanged SectionsSectionsEffective Flange Width
Portions near the webs are more highly stressed than areas away from the web.
Analysis of Flanged Analysis of Flanged SectionsSections
Effective width (beff)
beff is width that is stressed uniformly to give the same compression force actually developed in compression zone of width b(actual)
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10.2
T Beam Flange:
eff
f w
actual
4
16
Lb
h b
b
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10.3
Inverted L Shape Flange
eff w
f w
actual w
12
6
0.5* clear distance to next web
Lb b
h b
b b
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10
Isolated T-Beams
weff
wf
42bb
bh
Various Possible Geometries Various Possible Geometries of T-Beamsof T-Beams
Single Tee
Twin Tee
Box