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Lecture 9 - Flexure June 20, 2003 CVEN 444

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**Lecture Goals Load Envelopes Resistance Factors and Loads**

Design of Singly Reinforced Rectangular Beam Unknown section dimensions Known section dimensions

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Moment Envelopes The moment envelope curve defines the extreme boundary values of bending moment along the beam due to critical placements of design live loading. Fig ; MacGregor (1997)

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**Moment Envelopes Example**

Given following beam with a dead load of 1 k/ft and live load 2 k/ft obtain the shear and bending moment envelopes

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**Moment Envelopes Example**

Use a series of shear and bending moment diagrams Wu = 1.2wD + 1.6wL Moment Diagram Shear Diagram

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**Moment Envelopes Example**

Use a series of shear and bending moment diagrams Wu = 1.2wD + 1.6wL (Dead Load Only) Moment Diagram Shear Diagram

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**Moment Envelopes Example**

Use a series of shear and bending moment diagrams Wu = 1.2wD + 1.6wL Moment Diagram Shear Diagram

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**Moment Envelopes Example**

The shear envelope

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**Moment Envelopes Example**

The moment envelope

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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.

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**ACI Code Requirements for Strength Design**

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

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**Flexural Design of Reinforced Concrete Beams and Slab Sections**

Required Strength (ACI 318, sec 9.2) 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 = 1.2 D +1.6 L Always check even if other load types are present. U = 1.2(D + F + T) + 1.6(L + H) (Lr or S or R) U = 1.2D (Lr or S or R) + (L or 0.8W) U = 1.2D W + 1.0L + 0.5(Lr or S or R) U = 0.9 D + 1.6W +1.6H U = 0.9 D + 1.0E +1.6H

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**Resistance Factors, f - ACI Sec 9.3.2 Strength Reduction Factors**

[1] Flexure w/ or w/o axial tension The strength reduction factor, f, will come into the calculation of the strength of the beam.

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**Resistance Factors, f - ACI Sec 9.3.2 Strength Reduction Factors**

[2] Axial Tension f = 0.90 [3] Axial Compression w or w/o flexure (a) Member w/ spiral reinforcement f = 0.70 (b) Other reinforcement members f = 0.65 *(may increase for very small axial loads)

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**Resistance Factors, f - ACI Sec 9.3.2 Strength Reduction Factors**

[4] Shear and Torsion f = 0.75 [5] Bearing on Concrete f = 0.65 ACI Sec f factors for regions of high seismic risk

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**Background Information for Designing Beam Sections**

1. Location of Reinforcement locate reinforcement where cracking occurs (tension region) Tensile stresses may be due to : a ) Flexure b ) Axial Loads c ) Shrinkage effects

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**Background Information for Designing Beam Sections**

2. Construction formwork is expensive - try to reuse at several floors

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**Background Information for Designing Beam Sections**

3. Beam Depths ACI 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

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**Background Information for Designing Beam Sections**

4. Concrete Cover 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 in

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**Background Information for Designing Beam Sections**

Minimum Cover Dimensions (ACI 318 Sec 7.7) Concrete not exposed to earth or weather - Slab, walls, joists No. 14 and No. 18 bars in No. 11 bar and smaller in - Beams, Columns 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 /3 slab depth /5 narrowest dim.

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**Example - Singly Reinforced Beam**

Design a singly reinforced beam, which has a moment capacity, Mu = 225 k-ft, fc = 3 ksi, fy = 40 ksi and c/d = 0.275 Use a b = 12 in. and determine whether or not it is sufficient space for the chosen tension steel.

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**Example - Singly Reinforced Beam**

From the calculation of Mn

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**Example - Singly Reinforced Beam**

Select c/d =0.275 so that f =0.9. Compute k’ and determine Ru

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**Example - Singly Reinforced Beam**

Calculate the bd 2

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**Example - Singly Reinforced Beam**

Calculate d, if b = 12 in. Use d =22.5 in., so that h = 25 in.

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**Example - Singly Reinforced Beam**

Calculate As for the beam

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**Example - Singly Reinforced Beam**

Chose one layer of 4 #9 bars Compute r

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**Example - Singly Reinforced Beam**

Calculate rmin for the beam The beam is OK for the minimum r

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**Example - Singly Reinforced Beam**

Check whether or not the bars will fit into the beam. The diameter of the #9 = in. So b =12 in. works.

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**Example - Singly Reinforced Beam**

Check the height of the beam. Use h = 25 in.

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**Example - Singly Reinforced Beam**

Find a Find c

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**Example - Singly Reinforced Beam**

Check the strain in the steel Therefore, f is 0.9

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**Example - Singly Reinforced Beam**

Compute the Mn for the beam Calculate Mu

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**Example - Singly Reinforced Beam**

Check the beam Mu = 225 k-ft*12 in/ft =2700 k-in Over-designed the beam by 6% Use a smaller c/d ratio

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