chapter 4 flexural design of beam 2021.pdf
- 1. Flexural Analysis and Design of Beams Chapter 4
- 2. 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 5, 6, 7.
- 3. Bending of Homogeneous beam • Steel, timber • Internal forces-normal and tangential • Normal-bending/flexural stress-bending moment • Tangential-shear stress-shear force
- 4. 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
- 6. 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
- 7. Reinforced Concrete Beam Behaviour
- 8. • Read the article
- 9. Video • See video clips
- 10. • Tensile stress in concrete is smaller than modulus of rupture Transformed section can be used Stresses elastic, section uncracked
- 16. 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
- 20. Try this
- 23. Find I cracked Try the same problem using My/I Find allowable moment M, if allowable stresses are fc=0.45fc’ fs= 24 ksi
- 25. • 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
- 28. Failure initiated by yielding
- 29. Failure by concrete crushing Quadratic equation for c
- 30. Balanced reinforcement ratio ρb
- 33. Design of Tension-reinforced Rectangular Beams • Demand < Capacity • USD method of design- ACI 2008, BNBC 2020 • Limit states Design –Europe • ULS, SLS limit states • Hypothetical overload stage/demand with load factor • Reduced capacity with strength reduction factor
- 34. Equivalent Rectangular Stress Distribution C S Whitney
- 38. Underreinforced beam • Compression failure is abrupt • Tensile failure gradual • ρ should be less than ρb
- 39. ρ should be less than ρb, why?
- 40. ACI provisions for underreinforced beam • ACI establishes some safe limits • Net tensile strain Єt at farthest from comp face • Strength reduction factor φ
- 44. 𝜑 = 0.483 + 83.3𝜀𝑡
- 48. Review problem
- 55. 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. • 𝑀𝑛 < 𝑀𝑐𝑟 • To ensure against this type of failure, a minimum amount of reinforcement is provided
- 59. Review problem 𝜑 = 0.483 + 83.3𝜀𝑡
- 63. Design Problem
- 70. • Infinite number of solution is possible • Economic 0.5ρ0.005 to 0.75ρ0.005
- 71. Determination of steel area
- 75. 𝜑 = 0.483 + 83.3𝜀𝑡
- 82. Design Aids: Find Mn
- 86. Design Aids: Concrete dimensions and steel
- 87. Design Aids: find steel area
- 88. 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
- 90. • 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
- 91. 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
- 92. 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
- 93. 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
- 96. 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
- 97. Tension and compression steel both at yields Effective depth Compression steel is always denoted as As’
- 100. Compression steel below yield stress
- 105. Example 4.12
- 107. Nadim Hassoun
- 110. Find ϕMn
- 117. Design of Doubly Reinforced Beam
- 126. T-beam
- 127. T-beam • 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
- 130. 1. Symmetrical T beams bf < 16hf+bw bf < Span/4+bw Changed bf < c/c beam spacing 2. Beam having slab on one side bf < span/12+bw bf < 6hf+bw bf < Half the clear span +bw 3. Isolated T beam hf > bw/2 bf < 4bw Effective flange width
- 131. Strength Analysis Two possibilities • Just like rectangular beam • T-beam analysis required
- 132. If a>hf T-beam else Rectangular beam with b=bf bf
- 155. hf 20’X12=240”
- 159. At interior support
- 160. End of Chapter 4: Flexural Analysis and Design of Beams