Lecture 10 Analysis and Design

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Lecture 10 Analysis 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 slab
  thickness requirements that can be used without a
  deflection 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 -gt 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
  to load and which spans not to load.
• Influence Line graph of variation of shear,
  moment, or other effect at one particular point
  in a 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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Pattern Loads

• Qualitative Influence Lines
• Mueller-Breslau Principle
• Figs. 10-7(f), 10-8, 10-9
• Used to provide a qualitative guide to the shape
  of the influence line
• For moments
• Insert pin at location of interest
• Twist beam on either side of pin
• Other supports are unyielding, so distorted shape
  may be easily drawn.
• For frames, joints are assumed free to rotate,
  assume members are rigidly connected (angle
  between members does not change)

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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
the structure, 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 Loads

• Frame 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
  roof under 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 to
  combinations 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 Load Cases for Beam Design

• DL Member dead load (self wt. of slab, beams,
  etc.)
• SDL Superimposed dead load on floors
• LLa1 Case a1 LL
• (maximize Mu/-Mu in 1st exterior beam)
• LLa2 Case a2 LL (optional)
• (maximize Mu/-Mu in 2nd exterior beam
  symmetric to 1st exterior beam)
• LLb Case b LL
• (maximize Mu in interior beams)
• LLc1 Case c1 LL
• (maximize -Mu in beams 1st interior support)
• LLc2 Case c2 LL (optional)
• (maximize -Mu in beams at 2nd interior support
  symmetric to LLc)

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Project Factored Load Combinations for Beam
Design

• Factored Load Combinations
• U 1.4 (DLSDL) 1.7 (LLa1)
• U 1.4 (DLSDL) 1.7 (LLa2)
• U 1.4 (DLSDL) 1.7 (LLb)
• U 1.4 (DLSDL) 1.7 (LLc1)
• U 1.4 (DLSDL) 1.7 (LLc2)
• Envelope Load Combinations
• Take maximum forces from all factored load
  combinations

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MomentEnvelopes
The moment envelope curve defines the extreme
boundary values of bending moment along the beam
due to critical placements of design live loading.
Fig. 10-10 MacGregor (1997)
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F.Approximate Analysis of Continuous Beam and
One-Way Slab Systems

• ACI Moment and Shear Coefficients
• Approximate moments and shears permitted for
  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 gt 20
• Uniform Loads
• LL/DL 3 (unfactored)
• Prismatic members
• Same A, I, E throughout member length
• Beams must be in braced frame without significant
  moments due 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 Systems

• ACI Moment and Shear
• Coefficients Methodology

wu Total factored dead and live load per unit
length Cm Moment coefficient Cv Shear
coefficient ln Clear span length for span in
question for Mu at interior face of exterior
support, Mu and Vu ln Average of clear span
length for adjacent spans for Mu at interior
supports See Fig. 10-11, text
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F. Approximate Analysis of Continuous Beam and
One-Way Slab Systems

• ACI Moment and Shear Coefficients
• See Section 8.3.3 of ACI Code

Fig. 10-11, MacGregor (1997)
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
Analysis Versus Design
Analysis Given a cross-section,fc ,
reinforcement sizes, location,fy compute
resistance or capacity Design Given
factored load effect (such as Mu) select
suitable section(dimensions, fc, fy,
reinforcement, etc.)
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
ACI Code Requirements for Strength Design
Basic Equation factored resistance
factored load effect
Ex.
Mu Moment due to factored loads (required
ultimate moment) Mn Nominal moment capacity of
the cross-section using nominal dimensions and
specified material strengths. f 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 Sections
Required Strength (ACI 318, sec 9.2)
U Required Strength to resist factored
loads D Dead Loads L 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
U 1.4 D 1.7 L Always check even if other
load types are present. U 0.75( 1.4 D
1.7 L 1.7 W) U 0.75( 1.4 D 1.7 L) U 0.9
D 1.3 W
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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, f - ACI Sec 9.3.2 Strength
Reduction Factors
1 Flexure w/ or w/o axial tension f 0.90 2
Axial Tension f 0.90 3 Axial Compression
w or w/o flexure (a) Member w/ spiral
reinforcement f 0.75 (b) Other reinforcement
members f 0.70 (may increase for very small
axial loads) 4 Shear and Torsion f
0.85 5 Bearing on Concrete f 0.70 ACI Sec
9.3.4 f 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
3.

• Beam Depths
• ACI 318 - Table 9.5(a) min. h based on
  l(span) (slab beams)
• Rule of thumb hb (in) l (ft)
• Design for max. moment over a support to set
  depth of a continuous beam.

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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
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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 maximum aggregate size. 3/4
clear space., 1/3 slab depth, 1/5 narrowest dim.
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
1) For design moment Substitute
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Let
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Let
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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 the
superimposed 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)

• Select a reasonable value for r based on
  experience or start with a value of about 45 to
  55 of rbal.
• Calculate the reinforcement index,
• Calculate the coefficient

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Calculate the required value of
• Select b as a function of d. b (0.45d to
  0.65d)
• 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.

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Solve for the width, b, using selected d value.
  Round b to nearest whole inch value.
• Re-calculate the beam self-weight and Mu based on
  the selected b and h dimensions. Go back to step
  1 only if the new self weight results in
  significant change in Mu.
• Calculate required As rbd. Use the selected
  value 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
  be necessary 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)

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)

• Calculate controlling value for the design
  moment, Mu.
• 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)

• Solve for required area of tension reinforcement,
  As , based on the following equation.
  

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Rewrite the equation

Assume (d-a/2) 0.9d to 0.95d and solve for
As(reqd) Note f 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)

• 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.
• Calculate the actual Mn for the section
  dimensions and reinforcement selected. Verify
  . Check strength
  (keep over-design with 10)

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Check whether As(provided) is within the
  allowable limits. As(provided)
  As(max) 0.75 As(bal) As(provided)
  As(min)