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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 Fig. 10-10; MacGregor (1997) The moment envelope curve defines the extreme boundary values of bending moment along the beam due to critical placements of design live loading.

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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 W u = 1.2w D + 1.6w L Shear Diagram Moment Diagram

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Moment Envelopes Example Use a series of shear and bending moment diagrams W u = 1.2w D + 1.6w L Shear Diagram Moment Diagram (Dead Load Only)

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Moment Envelopes Example Use a series of shear and bending moment diagrams W u = 1.2w D + 1.6w L Shear Diagram Moment 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, f c, reinforcement sizes, location, f y compute resistance or capacity Design:Given factored load effect (such as M u ) select suitable section(dimensions, f c, f y, 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 M u = Moment due to factored loads (required ultimate moment) M n = Nominal moment capacity of the cross-section using nominal dimensions 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 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 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) + 0.5 (L r or S or R) U = 1.2D + 1.6 (L r or S or R) + (L or 0.8W) U = 1.2D + 1.6 W + 1.0L + 0.5(L r 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, ACI Sec 9.3.2 Strength Reduction Factors [1] Flexure w/ or w/o axial tension The strength reduction factor, , will come into the calculation of the strength of the beam.

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Resistance Factors, ACI Sec 9.3.2 Strength Reduction Factors [2] Axial Tension = 0.90 [3] Axial Compression w or w/o flexure (a) Member w/ spiral reinforcement = 0.70 (b) Other reinforcement members = 0.65 *(may increase for very small axial loads)

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Resistance Factors, ACI Sec 9.3.2 Strength Reduction Factors [4] Shear and Torsion = 0.75 [5] Bearing on Concrete = 0.65 ACI Sec 9.3.4 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 318 - Table 9.5(a) min. h based on l (span) (slab & beams) Rule of thumb: h b (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- 1.5 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- 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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Example - Singly Reinforced Beam Design a singly reinforced beam, which has a moment capacity, M u = 225 k-ft, f c = 3 ksi, f y = 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 M n

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Example - Singly Reinforced Beam Select c/d =0.275 so that =0.9. Compute k’ and determine R u

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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 A s for the beam

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Example - Singly Reinforced Beam Chose one layer of 4 #9 bars Compute

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Example - Singly Reinforced Beam Calculate min for the beam The beam is OK for the minimum

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Example - Singly Reinforced Beam Check whether or not the bars will fit into the beam. The diameter of the #9 = 1.128 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, is 0.9

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Example - Singly Reinforced Beam Compute the M n for the beam Calculate M u

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Example - Singly Reinforced Beam Check the beam M u = 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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