1. Lecture 9 - Flexure
June 20, 2003
CVEN 444

2. Lecture Goals Load Envelopes Resistance Factors and Loads
Design of Singly Reinforced Rectangular Beam
Unknown section dimensions
Known section dimensions

3. 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)

4. 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

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

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

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

8. Moment Envelopes Example
The shear envelope

9. Moment Envelopes Example
The moment envelope

10. 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.)

11. Flexural Design of Reinforced Concrete Beams and Slab Sections
ACI Code Requirements for Strength Design
Basic Equation: factored resistance factored load effect
Ex.

12. 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.

13. 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

14. 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.

15. 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

16. 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.

17. 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)

18. 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

19. Background Information for Designing Beam Sections1.
Location of Reinforcement locate reinforcement where cracking occurs (tension region) Tensile stresses may be due to : a ) Flexure b ) Axial Loads c ) Shrinkage effects

20. Background Information for Designing Beam Sections2.
Construction
formwork is expensive - try to reuse at several floors

21. Background Information for Designing Beam Sections3.
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.

22. Background Information for Designing Beam Sections4.
Concrete Cover
Cover = Dimension between the surface of the slab or beam and the reinforcement

23. Background Information for Designing Beam Sections4.
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.

24. 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

25. 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

26. Background Information for Designing Beam Sections5.
Bar Spacing Limits (ACI 318 Sec. 7.6)
- Minimum spacing of bars
- Maximum spacing of flexural reinforcement in walls & slabs
Max. space = smaller of

27. Minimum Cover Dimension
Interior beam.

28. Minimum Cover Dimension
Reinforcement bar arrangement for two layers.

29. Minimum Cover Dimension
ACI 3.3.3
Nominal maximum aggregate size.
- 3/4 clear space /3 slab depth /5 narrowest dim.

30. 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.

31. Example - Singly Reinforced Beam
From the calculation of Mn

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

33. Example - Singly Reinforced Beam
Calculate the bd 2

34. Example - Singly Reinforced Beam
Calculate d, if b = 12 in.
Use d =22.5 in., so that h = 25 in.

35. Example - Singly Reinforced Beam
Calculate As for the beam

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

37. Example - Singly Reinforced Beam
Calculate rmin for the beam
The beam is OK for the minimum r

38. 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.

39. Example - Singly Reinforced Beam
Check the height of the beam.
Use h = 25 in.

40. Example - Singly Reinforced Beam
Find a
Find c

41. Example - Singly Reinforced Beam
Check the strain in the steel
Therefore, f is 0.9

42. Example - Singly Reinforced Beam
Compute the Mn for the beam
Calculate Mu

43. 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