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7/31/2019 Design Analysis Beam ACI
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Lecture 10Analysis and Design
September 27, 2001
CVEN 444
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Lecture Goals
Pattern Loading
Analysis and Design
Resistance Factors and Loads
Design of Singly Reinforced Rectangular
Beam
Unknown section dimensions
Known section dimensions
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Member Depth
ACI provides minimum member depth and slabthickness requirements that can be used without adeflection calculation (Sec. 9.5)
Useful for selecting preliminary member sizes
ACI 318 - Table 9.5a:
Min. thickness, h
For beams with one end continuous: L/18.5 For beams with both ends continuous: L/21
L is span length in inches
Table 9.5a usually gives a depth too shallow for design,but should be checked as a minimum.
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Member Depth
ACI 318-99: Table 9.5a
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Member Depth Rule of Thumb:
hb (in.) ~ L (ft.)
Ex.) 30 ft. span -> hb ~ 30 in.
May be a little large, but okay as a start to calc. DL Another Rule of Thumb:
wDL (web below slab) ~ 15% (wSDL+ wLL)
Note: For design, start with maximum moment for
beam to finalize depth. Select b as a function of d
b ~ (0.45 to 0.65)*(d)
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Pattern Loads Using influence lines to determine pattern loads
Largest moments in a continuous beam or frame occur
when some spans are loaded and others are not.
Influence lines are used to determine which spans toload and which spans not to load.
Influence Line: graph of variation of shear,
moment, or other effect at one particular point ina structure due to a unit load moving across the
structure.
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Pattern Loads
Quantitative
Influence Lines
Ordinate are
calculated
(exact)
See Fig. 10-7(a-e)
MacGregor (1997)
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Qualitative Influence Lines
The Mueller-Breslau
principle can be stated as
follows:
If a function at a point on
a structure, such as
reaction, or shear, or
moment is allowed to act
without restraint, the
deflected shape of thestructure, to some scale,
represents the influence
line of the function.
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Pattern Loads
Qualitative Influence Lines
Fig. 10-7 (b,f) from MacGregor (1997)
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Pattern
LoadsFrame Example: Maximize +M at point B.
Draw qualitative influence
lines.
Resulting pattern load:checkerboard pattern
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Pattern Loads
ACI 318-99, Sec. 8.9.1:
It shall be permitted to assume that:
The live load is applied only to the floor or roofunder consideration, and
The far ends of columns built integrally with the
structure are considered to be fixed.
** For the project, we will model the entire frame. **
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Pattern Loads
ACI 318-99, Sec. 8.9.2:
It shall be permitted to assume that the
arrangement of live load is limited tocombinations of:
Factored dead load on all spans with full factored
live load on two adjacent spans.
Factored dead load on all spans with full factored
live load on alternate spans.
** For the project, you may use this provision. **
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Project: Factored Load
Combinations for Beam DesignFactored Load Combinations:
U = 1.4 (DL+SDL) + 1.7 (LLa1)
U = 1.4 (DL+SDL) + 1.7 (LLa2)
U = 1.4 (DL+SDL) + 1.7 (LLb)
U = 1.4 (DL+SDL) + 1.7 (LLc1)
U = 1.4 (DL+SDL) + 1.7 (LLc2)
Envelope Load Combinations:
Take maximum forces from all factored loadcombinations
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Moment
Envelopes
Fig. 10-10; MacGregor (1997)
The moment envelope
curve defines the extreme
boundary values of bendingmoment along the beam
due to critical placements
of design live loading.
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F.Approximate Analysis of Continuous
Beam and One-Way Slab Systems
ACI Moment and Shear Coefficients
Approximate moments and shears permittedfor design of continuous beams and one-
way slabs
Section 8.3.3 of ACI Code
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F. Approximate Analysis of Continuous
Beam and One-Way Slab Systems
ACI Moment and Shear Coefficients - Requirements:
Two or more spans
Approximately Equal Spans
Larger of 2 adjacent spans not greater than shorter by > 20%
Uniform Loads
LL/DL 3 (unfactored)
Prismatic members
Same A, I, E throughout member length
Beams must be in braced frame without significant momentsdue to lateral forces
Not state in Code, but necessary for coefficients to apply
** All these requirements must be met to use the coefficients!**
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F. Approximate Analysis of Continuous
Beam and One-Way Slab SystemsACI Moment and ShearCoefficientsMethodology:
2
)(2
nu
vu
numu
lw
CV
lwCM
wu = Total factored dead and liveload per unit length
Cm
= Moment coefficient
Cv = Shear coefficient
ln = Clear span length for span inquestion forMu at interiorface of exterior support, +Muand Vu
ln = Average of clear span lengthfor adjacent spans forMu atinterior supports
See Fig. 10-11, text
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F. Approximate
Analysis of
Continuous Beam
and One-WaySlab Systems
ACI Moment and
ShearCoefficients
See Section 8.3.3of ACI Code
Fig. 10-11, MacGregor (1997)
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Flexural Design of Reinforced
Concrete Beams and Slab SectionsACI Code Requirements for Strength Design
Basic Equation: factored resistance factored load
effect
Ex.un MM
Mu = Moment due to factored loads (required ultimate moment)
Mn = Nominal moment capacity of the cross-section using nominaldimensions and specified material strengths.
= Strength reduction factor (Accounts for variability in dimensions,
material strengths, approximations in strength equations.
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Flexural Design of Reinforced
Concrete Beams and Slab SectionsRequired Strength (ACI 318, sec 9.2)
U = Required Strength to resist factored loads
D = Dead LoadsL = Live loads
W = Wind Loads
E = Earthquake Loads
H = Pressure or Weight Loads due to soil,ground water,etc.F = Pressure or weight Loads due to fluids with well defined
densities and controllable maximum heights.
T = Effect of temperature, creep, shrinkage, differential
settlement, shrinkage compensating.
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Factored Load Combinations
Similar combination for earthquake, lateral pressure,fluid pressure, settlement, etc.
U = 1.05 D + 1.28 L + 1.4 E
U = 0.9 D + 1.43 E
U = 1.4 D + 1.7 L + 1.7 H
U = 0.9 D + 1.7 H
U
=1.4 D + 1.7 L + 1.4 F
U = 0.9 D + 1.4 F
U = 0.75(1.4 D + 1.4 T +1.7 L)
U = 1.4 (D + L)
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Resistance Factors, ACI Sec 9.3.2
Strength Reduction Factors
[1] Flexure w/ or w/o axial tension = 0.90
[2] Axial Tension = 0.90
[3] Axial Compression w or w/o flexure
(a) Member w/ spiral reinforcement = 0.75(b) Other reinforcement members = 0.70
*(may increase for very small axial loads)
[4] Shear and Torsion = 0.85
[5] Bearing on Concrete = 0.70
ACI Sec 9.3.4 factors for regions of high seismic risk
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Background Information for Designing
Beam Sections
1.
2.
Location of Reinforcement
locate reinforcement where cracking occurs
(tension region) Tensile stresses may be due to :a) Flexure
b) Axial Loads
c ) Shrinkage effects
Construction
formwork is expensive -try to reuse at several
floors
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Background Information for Designing
Beam Sections
4. Concrete Cover
Cover = Dimension between the surface of the slab or
beam and the reinforcement
Why is cover needed?
[a] Bonds reinforcement to concrete
[b] Protect reinforcement against corrosion[c] Protect reinforcement from fire (over heating
causes strength loss)
[d] Additional cover used in garages, factories,
etc. to account for abrasion and wear.
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Background Information for Designing
Beam Sections
Minimum Cover Dimensions (ACI 318 Sec 7.7)
Sample values for cast in-place concrete
Concrete cast against & exposed to earth - 3 in.
Concrete (formed) exposed to earth & weather
No. 6 to No. 18 bars - 2 in.
No. 5 and smaller - 1.5 in
Concrete not exposed to earth or weather- Slab, walls, joists
No. 14 and No. 18 bars - 1.5 in
No. 11 bar and smaller - 0.75 in
- Beams, Columns - 1.5 in
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Background Information for Designing
Beam Sections
5. Bar Spacing Limits (ACI 318 Sec. 7.6)
- Minimum spacing of bars
- Maximum spacing of flexural reinforcement in
walls & slabs
Max. space = smaller of
.in18
t3
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Minimum Cover Dimension
Interior beam.
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Minimum Cover Dimension
Reinforcement bar arrangement for two layers.
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Minimum Cover Dimension
ACI 3.3.3
Nominal maximumaggregate size.
3/4 clear space.,
1/3 slab depth,
1/5 narrowestdim.
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Design Procedure for section dimensions
are unknown (singly Reinforced Beams)
1) For design moment
Substitute:
-
-
-
b0.852
bddbd
b0.852
AdA2
adTMM
c
y
y
c
ys
ysnu
f
ff
f
ff
bd
Aand s
c
y
f
f
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Design Procedure for section dimensions
are unknown (singly Reinforced Beams)
Let
59.01bd
d59.0dbdM
d59.0dbdMM
2c
cu
ynu
-
-
-
f
f
f
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Design Procedure for section dimensions
are unknown (singly Reinforced Beams)
Let
R
M
bd
R
59.01bd
M
u
2
c2
u
-
f
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Design Procedure for section dimensions
are unknown (singly Reinforced Beams)
Assume that the material properties, loads, and span length are all known.
Estimate the dimensions of self-weight using the following rules of
thumb:
a. The depth, h, may be taken as approximate 8 to 10 % of the
span (1in deep per foot of span) and estimate the width, b,
as about one-half of h.
b. The weight of a rectangular beam will be about 15 % of thesuperimposed loads (dead, live, etc.). Assume b is about
one-half of h.
Immediate values of h and b from these two procedures should be selected.
Calculate self-weight and Mu.
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
1 Select a reasonable value for based on
experience or start with a value of about 45% to
55 % ofbal.2 Calculate the reinforcement index,
3 Calculate the coefficient
c
y
f
f
59.01R c - f
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
4 Calculate the required value of
5 Select b as a function of d. b ~ (0.45d to 0.65d)
6 Solve for d. Typically round d to nearest 0.5 inch
value to get a whole inch value for h, which is
approximately d = 2.5 in.
R
M
bd
u
2
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
7 Solve for the width, b, using selected d value.
Round b to nearest whole inch value.
8 Re-calculate the beam self-weight and Mu basedon the selected b and h dimensions. Go back to
step 1 only if the new self weight results in
significant change in Mu.
9 Calculate required As = bd. Use the selectedvalue of d from Step 6. And the calculated (not
rounded) value of b from step 7 to avoid errors
from rounding.
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Select steel reinforcing bars to provide
As (As required from step 9). Confirm that
the bars will fit within the cross-section. It may benecessary to change bar sizes to fit the steel in one
layer. If you need to use two layers of steel, the
value of h should be adjusted accordingly.
Calculate the actual Mn for the section dimensions
and reinforcement selected. Check strength,
(keep over-design within 10%)
un
MM
10
11
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Design Procedure for section dimensions
are known (singly Reinforced Beams)
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Design Procedure for section dimensions are
known (singly Reinforced Beams)
1 Calculate controlling value for the design moment,
Mu.2 Calculate d, since h is known.
d h - 2.5in. for one layer of reinforcement.
d h - 3.5in. for two layers of reinforcement.
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Design Procedure for section dimensions are
known (singly Reinforced Beams)
3 Solve for required area of tension reinforcement,
As , based on the following equation.
-
2
adAMM ysnu f
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Design Procedure for section dimensions are
known (singly Reinforced Beams)
Rewrite the equation:
-
2
ad
M
dreq'A
y
u
s
f
Assume (d-a/2) 0.9d to 0.95d and solve for As(reqd)
Note = 0.9 for flexure without axial load
(ACI 318-95, Sec. 9.3)
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Design Procedure for section dimensions are
known (singly Reinforced Beams)
4 Select reinforcing bars so As(provided) As(reqd)
Confirm bars will fit within the cross-section. It
may be necessary to change bar sizes to fit the steel
in one layer or even to go to two layers of steel.
5 Calculate the actual Mn for the section dimensions
and reinforcement selected. Verify .Check strength (keep over-design
with 10%)
un MM ys
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