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F2009abn
twenty two
concrete construction:
materials & beamsConcrete Beams 1
Lecture 22
Architectural Structures
ARCH 331
lecture
http:// nisee.berkeley.edu/godden
ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN ARCH 331
DR. ANNE NICHOLS
SPRING 2018
F2008abnConcrete Beams 2
Lecture 22
Foundations Structures
ARCH 331
Concrete Beam Design
• composite of concrete and steel
• American Concrete Institute (ACI)
– design for maximum stresses
– limit state design
• service loads x load factors
• concrete holds no tension
• failure criteria is yield of reinforcement
• failure capacity x reduction factor
• factored loads < reduced capacity
– concrete strength = f’c
F2008abnConcrete Beams 3
Lecture 22
Foundations Structures
ARCH 331
Concrete Construction
• cast-in-place
• tilt-up
• prestressing
• post-tensioning
http:// nisee.berkeley.edu/goddenarch.mcgill.ca
F2008abnConcrete Beams 4
Lecture 22
Foundations Structures
ARCH 331
Concrete Beams
• types
– reinforced
– precast
– prestressed
• shapes
– rectangular, I
– T, double T’s, bulb T’s
– box
– spandrel
2
F2008abnConcrete Beams 5
Lecture 22
Foundations Structures
ARCH 331
Concrete Beams
• shear
– vertical
– horizontal
– combination:
• tensile stresses
at 45
• bearing
– crushing
http://urban.arch.virginia.edu
Concrete Beams 6
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Concrete
• low strength to weight ratio
• relatively inexpensive
– Portland cement
• types I - V
– aggregate
• course & fine
– water
– admixtures
• air entraining
• superplasticizers
F2008abnConcrete Beams 7
Lecture 22
Foundations Structures
ARCH 331
Concrete
• hydration
– chemical reaction
– workability
– water to cement ratio
– mix design
• fire resistant
• cover for steel
• creep &
shrinkage jci-web.jp
Concrete Beams 8
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Concrete
• placement (not pouring!)
• vibrating
• screeding
• floating
• troweling
• curing
• finishing
3
F2008abnConcrete Beams 9
Lecture 22
Foundations Structures
ARCH 331
Reinforcement
• deformed steel bars (rebar)
– Grade 40, Fy = 40 ksi
– Grade 60, Fy = 60 ksi - most common
– Grade 75, Fy = 75 ksi
– US customary in # of 1/8”
• longitudinally placed
– bottom
– top for compression
reinforcement
(nominal)
F2008abnConcrete Beams 10
Lecture 22
Foundations Structures
ARCH 331
Reinforcement
• prestressing strand
• post-tensioning
• stirrups
• detailing
– development length
– anchorage
– splices
http:// nisee.berkeley.edu/godden
F2008abnConcrete Beams 11
Lecture 22
Foundations Structures
ARCH 331
Composite Beams
• concrete
– in compression
• steel
– in tension
• shear studs
F2008abn
• plane sections remain plane
• stress distribution changes
11 1
E yf E
R 2
2 2
E yf E
R
Concrete Beams 12
Lecture 22
Foundations Structures
ARCH 331
Behavior of Composite Members
4
Concrete Beams 13
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Transformation of Material
• n is the ratio of E’s
• effectively widens a material to get
same stress distribution
1
2
E
En
F2008abnConcrete Beams 14
Lecture 22
Foundations Structures
ARCH 331
Stresses in Composite Section
• with a section
transformed to one
material, new I
– stresses in that
material are
determined as usual
– stresses in the other
material need to be
adjusted by n
concrete
steel
E
E
E
En
1
2
c
transformed
Myf
I
s
transformed
Mynf
I
Concrete Beams 15
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Reinforced Concrete - stress/strain
Concrete Beams 16
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Reinforced Concrete Analysis
• for stress calculations
– steel is transformed to concrete
– concrete is in compression above n.a. and
represented by an equivalent stress block
– concrete takes no tension
– steel takes tension
– force ductile failure
5
Concrete Beams 17
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Location of n.a.
• ignore concrete below n.a.
• transform steel
• same area moments, solve for x
0)(2
xdnAx
bx s
Concrete Beams 18
Lecture 22
Foundations Structures
ARCH 331
F2008abn
T sections
• n.a. equation is different if n.a. below
flangef f
bw
bw
hf hf
0)(22
xdnAhx
bhxh
xhb sf
wff
ff
F2008abnConcrete Beams 19
Lecture 22
Foundations Structures
ARCH 331
ACI Load Combinations*
*can also use old
ACI factors
• 1.4D
• 1.2D + 1.6L + 0.5(Lr or S or R)
• 1.2D + 1.6(Lr or S or R) + (1.0L or 0.5W)
• 1.2D + 1.0W + 1.0L + 0.5(Lr or S or R)
• 1.2D + 1.0E + 1.0L + 0.2S
• 0.9D + 1.0W
• 0.9D + 1.0E
F2008abnConcrete Beams 20
Lecture 22
Foundations Structures
ARCH 331
Reinforced Concrete Design
• stress distribution in bending
Wang & Salmon, Chapter 3
b
As
a/2
TTNA
CCc a=
1c
0.85f’c
actual stress Whitney stressblock
dh
6
F2008abnConcrete Beams 21
Lecture 22
Foundations Structures
ARCH 331
Force Equations
• C = 0.85 f cba
• T = Asfy
• where
– f c = concrete compressive
strength
– a = height of stress block
– 1 = factor based on f c
– c = location to the n.a.
– b = width of stress block
– fy = steel yield strength
– As = area of steel reinforcement
a/2
T
a=1c
0.85f’c
C
1
40000.85 (0.05) 0.65
1000
cf
F2008abnConcrete Beams 22
Lecture 22
Foundations Structures
ARCH 331
• T = C
• Mn = T(d-a/2)
– d = depth to the steel n.a.
• with As
– a =
– Mu Mn = 0.9 for flexure*
– Mn = T(d-a/2) = Asfy (d-a/2)
Equilibrium
a/2
T
Ca=1c
0.85f’c
d
bf
fA
c
ys
85.00.25
* 0.65 ( ) 0.65(0.005 )
t y
y
F2008abnConcrete Beams 23
Lecture 22
Foundations Structures
ARCH 331
• over-reinforced
– steel won’t yield
• under-reinforced
– steel will yield
• reinforcement ratio
–
– use as a design estimate to find As,b,d
– max is found with steel 0.004 (not bal)
– *with steel 0.005, = 0.9
Over and Under-reinforcement
bd
Aρ s
http://people.bath.ac.uk/abstji/concrete_video/virtual_lab.htm
Concrete Beams 24
Lecture 22
Foundations Structures
ARCH 331
F2008abn
As for a Given Section
• several methods
– guess a and iterate
1. guess a (less than n.a.)
2.
3. solve for a from Mu = Asfy (d-a/2)
4. repeat from 2. until a from 3. matches a in 2.
y
cs
f
baf.A
850
ys
u
fA
Mda
2
7
Concrete Beams 25
Lecture 22
Foundations Structures
ARCH 331
F2008abn
As for a Given Section (cont)
• chart method
– Wang & Salmon Fig. 3.8.1 Rn vs.
1. calculate
2. find curve for f’c and fy to get
3. calculate As and a
• simplify by setting h = 1.1d
2bd
MR nn
Concrete Beams 26
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Reinforcement
• min for crack control
• required
• not less than
• As-max :
• typical cover
– 1.5 in, 3 in with soil
• bar spacing
)(3
bdf
fA
y
c
s
)bd(f
Ay
s
200
cover
spacing
)d.(a 37501
F2008abnConcrete Beams 27
Lecture 22
Foundations Structures
ARCH 331
Shells
http:// nisee.berkeley.edu/godden
F2008abnConcrete Beams 28
Lecture 22
Foundations Structures
ARCH 331
Annunciation Greek Orthodox Church
• Wright, 1956
http://www.bluffton.edu/~sullivanm/
8
Concrete Beams 29
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Annunciation Greek Orthodox Church
• Wright, 1956
Concrete Beams 30
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Cylindrical Shells
• can resist tension
• shape adds “depth”
• not vaults
• barrel shells
F2008abnConcrete Beams 31
Lecture 22
Foundations Structures
ARCH 331
Kimball Museum, Kahn 1972
aasarchitecture.com
Concrete Beams 32
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Kimball Museum, Kahn 1972
• outer shell edges
9
Concrete Beams 33
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Kimball Museum, Kahn 1972
• skylights at peak
Concrete Beams 35
Lecture 22
Foundations Structures
ARCH 331
F2008abn
Approximate Depths