flexural analysis and design of beams - · fundamental assumptions relating to flexure and shear...
TRANSCRIPT
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Flexural Analysis and Design of Beams
Chapter 3
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Introduction
• Fundamental Assumptions
• Simple case of axial loading
• Same assumptions and ideal concept apply
• This chapter includes analysis and design for flexure, dimensioning cross section and reinforcement
• Shear design, bond anchorage, serviceability in chapters 4, 5, 6.
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Bending of Homogeneous beam
• Steel, timber
• Internal forces-normal and tangential
• Normal-bending/flexural stress-bending moment
• Tangential-shear stress-shear force
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Fundamental assumptions relating to flexure and shear
1. Plane cross section remain plane
2. Bending stress f at any point depends on the strain at that point
3. Shear stress also depends on cross section and stress-strain diagram. Maximum at neutral axis and zero at extreme fibre. Same horizontal and vertical.
4. The intensity of principal stresses
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5. At neutral axis, only horizontal and vertical shear present-pure shear condition
6. When stress are smaller than proportional limit
a. Neutral axis = cg
b. f=My/I
c. v=VQ/It
d. Shear distribution parabolic, max at na, zero at outer fibre. For rectangular max=1.5V/bh
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Reinforced Concrete Beam Behaviour
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Video
• See video clips
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• Tensile stress in concrete is smaller than modulus of rupture Transformed section can be used
Stresses elastic, section uncracked
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Stresses Elastic, Section cracked
• Concrete tensile stress exceeds mod of rupture
• Concrete compressive stress is less than fc’ /2
• Steel stress less than yield
• Assume tension crack up to neutral axis
• Transformed section can still be used
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Flexural Strength
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• Yielding of steel fs=fy
• Crushing of concrete εu= 0.003-0.004
• Either can reach first
• Exact shape not necessary
• Necessary –Total compressive force and location
• βc- location from comp face
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Failure initiated by yielding
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Failure by concrete crushing
Quadratic equation for c
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Balanced reinforcement ratio ρb
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Design of Tension-reinforced Rectangular Beams
• Demand < Capacity
• USD method of design- ACI 2008, BNBC 2017 • Limit states Design –Europe • ULS, SLS limit states
• Hypothetical overload stage/demand with load factor
• Reduced capacity with strength reduction factor
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Equivalent Rectangular Stress Distribution
C S Whitney
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Balanced Strain condition
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Underreinforced beam
• Compression failure is abrupt
• Tensile failure gradual
• ρ should be less than ρb
Read points why?
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ρ should be less than ρb, why?
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ACI provisions for underreinforced beam
• ACI establishes some safe limits
• Net tensile strain Єt at farthest from comp face
• Strength reduction factor φ
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Minimum Reinforcement Ratio
• If the flexural strength (of cracked section) is less than the moment that produced cracking of the previously uncracked section, the beam fails immediately upon formation of first flexural crack.
• To ensure against this type of failure, a minimum amount of reinforcement is provided
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Review problem
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Design Problem
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• Infinite number of solution is possible
• Economic 0.5ρ0.005 to 0.75ρ0.005
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Determination of steel area
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Overreinforced beam
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Design Aids: Find Mn
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Design Aids: Concrete dimensions and steel
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Design Aids: find steel area
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Practical considerations in the design of Beams: Concrete Protection for reinforcement
• Protection for steel against fire and corrosion
• Concrete cover depends on member and exposure
• Surfaces not exposed to ground or weather – Not less than ¾ in for slab
– Not less than 1.5 in for beams and columns
• Surfaces exposed to weather or in contact with ground – At least 2in
• Cast against ground with no form work – Min 3 in cover
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• b and h are rounded to 1 or 2 inch
• Slab rounded to ¼ or ½ inch (greater than 6 inch)
• Proportions- d 2-3 times of b
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Selection of bar and spacing
• No 3 to No 11 for beams
• No 14 and No 18 for columns
• Mixing of sizes allowed with 2 bar sizes
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Gap between bars
• Clear distance between bars not less than bar dia or 1 inch (for columns 1.5d or 1.5inch)
• Two or more layers- min 1 inch
• Upper bar directly above
• Clear distance and cover not less than 1.33times maximum aggregate size
• Vibrator nozzle
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Reinforcements –usual sizes
• Slab- No 3, 4, 5 (10mm, 12mm, 16mm)
• Beam- No 5,6, 7, 8 (16 20 22 25mm)
• Stirrup/tie- No, 3 4 (10 12mm)
• Column –No 5, 6 7 8 9 10 11 14 18 (16 20 22 25 28 32 ….)
• Mat- No 4,5,6,8 (12 16 20 25 mm)
• Smaller sizes preferred as long as there is no congestion
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DOUBLY REINFORCED BEAM
• Beams with tension and compression reinforcement
• Cross section is limited
• Compression steel is used for other reasons-long term deflection, reversal of moment, hanger bar for stirrup
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Tension and compression steel both at yields
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Compression steel below yield stress
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Example 3.12
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Nadim Hassoun
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Find ϕMn
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dt=d+1.0 if d=h-3.5
dt=d+1.5 if d=h-4.0
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Design of Doubly Reinforced Beam
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Use actual area provided
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T-beam
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• RC beam and slab are monolithically cast
• Beam stirrups and bent bars extend into the slab
• A part of slab act along with beam top to take longitudinal compression
• Slab forms the beam flange
• Part of beam below slab is called web/stem
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Effective flange width
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Effective flange width
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1. Symmetrical T beams
b < 16hf+bw
b < Span/4
b < c/c beam spacing
2. Beam having slab on one side
b < span/12+bw
b < 6hf+bw
b < Half the clear span +bw
3. Isolated T beam
hf > bw/2
b< 4bw
Effective flange width
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Strength Analysis
Two possibilities • Just like rectangular beam • T-beam analysis required
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If a>hf T-beam
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bw
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