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Lecture 10 Analysis and Design

- September 27, 2001
- CVEN 444

Lecture Goals

- Pattern Loading
- Analysis and Design
- Resistance Factors and Loads
- Design of Singly Reinforced Rectangular Beam
- Unknown section dimensions
- Known section dimensions

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.

Member Depth

ACI 318-99 Table 9.5a

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)

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.

Pattern Loads

- Quantitative Influence Lines
- Ordinate are calculated (exact)
- See Fig. 10-7(a-e)

MacGregor (1997)

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)

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.

Pattern Loads

Qualitative Influence Lines

Fig. 10-7 (b,f) from MacGregor (1997)

Pattern Loads

- Frame Example
- Maximize M at point B.
- Draw qualitative influence lines.
- Resulting pattern load
- checkerboard pattern

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.

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.

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)

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

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)

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

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!

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

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)

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.)

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.

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.

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

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)

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

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

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.

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.

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

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

Minimum Cover Dimension

Interior beam.

Minimum Cover Dimension

Reinforcement bar arrangement for two layers.

Minimum Cover Dimension

ACI 3.3.3 Nominal maximum aggregate size. 3/4

clear space., 1/3 slab depth, 1/5 narrowest dim.

Design Procedure for section dimensions are

unknown (singly Reinforced Beams)

1) For design moment Substitute

Design Procedure for section dimensions are

unknown (singly Reinforced Beams)

Let

Design Procedure for section dimensions are

unknown (singly Reinforced Beams)

Let

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.

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

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.

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.

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

Design Procedure for section dimensions are known

(singly Reinforced Beams)

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.

Design Procedure for section dimensions are known

(singly Reinforced Beams)

- Solve for required area of tension reinforcement,

As , based on the following equation.

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)

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)

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)