1. Flexural Component Design
PCI 6th Edition
Flexural Component Design

2. Presentation Outline What’s new to ACI 318 Gravity Loads Load Effects
Concrete Stress Distribution
Nominal Flexural Strength
Flexural Strength Reduction Factors
Shear Strength
Torsion
Serviceability Requirements

3. New to ACI 318 – 02 Load Combinations Stress limits
Member Classification
Strength Reduction factor is a function of reinforcement strain
Minimum shear reinforcement requirements
Torsion Design Method
Torsion will be discussed in detail later in the presentation

4. Load Combinations U = 1.4 (D + F)
U = 1.2 (D + F + T) (L + H) (Lr or S or R)
U = 1.2D (Lr or S or R) + (1.0L or 0.8W)
U = 1.2D + 1.6W + 1.0L + 0.5(Lr or S or R)
U = 1.2D + 1.0E + 1.0L + 0.2S
U= 0.9D + 1.6W + 1.6H
U= 0.9D + 1.0E + 1.6H
D = dead load
L = live load
R = rain load
S = snow load
Lr = roof live load
H = soil load
Lr = roof live load W = wind load
E = earthquake
T = effect of temperature, creep and shrinkage

5. Comparison of Load Combinations
U=1.2D L 2002
U= 1.4D + 1.7L 1999
If L=.75D
i.e. a 10% reduction in required strength

6. Classifications No Bottom Tensile Stress Limits
Classify Members Strength Reduction Factor
Tension-Controlled
Transition
Compression Controlled
Three Tensile Stress Classifications
Class U – Un-cracked
Class T – Transition
Class C – Cracked

7. Copied from ACI 318 2002, ACI 318-02 table R18.3.3

8. Class C Members Stress Analysis Based on Cracked Section Properties
No Compression Stress limit
No Tension Stress limit
Increase awareness on serviceability
Crack Control
Displacements
Side Skin Reinforcement

9. Minimum Shear Reinforcing
1999
2002

10. System Loads Gravity Load Systems Tributary Area Load distribution
Beams
Columns
Floor Member – Double Tees, Hollow Core
Spandrels
Tributary Area
Floor members, actual top area
Beams and spandrels
Load distribution
Load path
Floor members  spandrels or beams  Columns

11. Live Load Reduction Live Loads can be reduced based on: Where: KLL = 1
Lo = Unreduced live load and
At = tributary area

12. Live Load Reduction
Or the alternative floor reduction shall not exceed or
Where:
R = % reduction ≤ 40%
r = .08

13. Member Shear and Moment
Shear and moments on members can be found using statics methods and beam tables from Chapter 11
Refer to chapter 11 beam tables design aid

14. Strength Design
Strength design is based using the rectangular stress block
The stress in the prestressing steel at nominal strength, fps, can be determined by strain compatibility or by an approximate empirical equation
For elements with compression reinforcement, the nominal strength can be calculated by assuming that the compression reinforcement yields. Then verified.
The designer will normally choose a section and reinforcement and then determine if it meets the basic design strength requirement:

15. Concrete Stress Distribution
Parabolic distribution
Equivalent rectangular distribution

16. Stress Block Theory Stress-Strain relationship is not constant
f’c=6,000 psi
f’c=3,000 psi
1000 psi incraments.

17. Where :strain at max. stress
Stress Block Theory
Stress-Strain relationship
Stress-strain can be modeled by:
Where :strain at max. stress
and :max stress

18. Stress Block Theory
The Whitney stress block is a simplified stress distribution that shares the same centroid and total force as the real stress distribution =

19. Equivalent Stress Block – b1 Definition
when f’c < 3,000 psi
b1 = 0.65
when f’c > 8,000 psi

20. Design Strength Mild Reinforcement – Non - Prestressed
Prestress Reinforcement

21. Strength Design Flowchart
Figure page 4-9
Non-Prestressed Path
Prestressed Path

22. Non-Prestressed Members
Find depth of compression block

23. Depth of Compression Block
Where:
As is the area of tension steel
A’s is the area of compression steel
fy is the mild steel yield strength
Assumes compression steel yields

24. Flanged Sections
Checked to verify that the compression block is truly rectangular

25. Compression Block Area
If compression block is rectangular, the flanged section can be designed as a rectangular beam = =

26. Compression Block Area
If the compression block is not rectangular (a> hf), =
To find “a”

27. Determine Neutral Axis
From statics and strain compatibility

28. Check Compression Steel
Verify that compression steel has reached yield using strain compatibility

29. Compression Comments
By strain compatibility, compression steel yields if:
If compression steel has not yielded, calculation for “a” must be revised by substituting actual stress for yield stress
Non prestressed members should always be tension controlled, therefore c / dt < 0.375
Add compression reinforcement to create tesnion controlled secions

30. Moment Capacity 2 equations
rectangular stress block in the flange section
rectangular stress block in flange and stem section

31. Strength Design Flowchart
Figure page 4-9
Non- Prestressed Path
Prestressed Path

32. This portion of the flowchart is dedicated to determining the stress in the prestress reinforcement

33. Stress in Strand fse - stress in the strand after losses
fpu - is the ultimate strength of the strand
fps - stress in the strand at nominal strength

34. Stress in Strand Typically the jacking force is 65% or greater
The short term losses at midspan are about 10% or less
The long term losses at midspan are about 20% or less

35. Stress in Strand
Nearly all prestressed concrete is bonded

36. Stress in Strand Prestressed Bonded reinforcement
gp = factor for type of prestressing strand, see ACI 18.0
= .55 for fpy/fpu not less than .80
= .45 for fpy/fpu not less than .85
= .28 for fpy/fpu not less than .90 (Low Relaxation Strand)
rp = prestressing reinforcement ratio

37. Determine Compression Block

38. Compression Block Height
Assumes compression steel yields
Prestress component
Where
Aps - area of prestressing steel
fps - prestressing steel strength
This additional term is prestress component and just as in the non prestressed path the initial assumption for the compression steel is it is at yield

39. Flange Sections Check

40. Compression Steel Check
Verify that compression steel has reached yield using strain compatibility

41. Moment Capacity 2 Equations rectangular stress block in flange section
rectangular stress block in flange and stem section

42. Flexural Strength Reduction Factor
Based on primary reinforcement strain
Strain is an indication of failure mechanism
Three Regions

43. Member Classification
On figure
The member classification and strength reduction factor determination is in this region.

44. Compression Controlled
e < at extreme steel tension fiber or
c/dt > 0.600
= 0.70 with spiral ties
= 0.65 with stirrups

45. Tension Controlled e > 0.005 at extreme steel tension fiber, or
c/dt < 0.375
f = 0.90 with spiral ties or stirrups

46. Transition Zone
0.002 < e < at extreme steel tension fiber, or
0.375 < c/dt < 0.6
f = (e) or
f = (e) with spiral ties
f = /(c/dt) or
f = /(c/dt) with stirrups

47. Strand Slip Regions ACI Section 9.3.2.7
‘where the strand embedment length is less than the development length’
f =0.75

48. Limits of Reinforcement
To prevent failure immediately upon cracking, Minimum As is determined by:
As,min is allowed to be waived if tensile reinforcement is 1/3 greater than required by analysis

49. Limits of Reinforcement
The flexural member must also have adequate reinforcement to resist the cracking moment
Where
Correction for initial stresses on non-composite, prior to topping placement
Section after composite has been applied, including prestress forces

50. Critical Sections
For drapped sections there are two significant critical sections and for straight strand there is one critical point. If debonding is added additional sections form at their respective end of debonding point.

51. Horizontal Shear
ACI requires that the interface between the composite and non-composite, be intentionally roughened, clean and free of laitance
Experience and tests have shown that normal methods used for finishing precast components qualifies as “intentionally roughened”

52. Horizontal Shear, Fh Positive Moment Region
Based on the force transferred in topping (page 4-53)

53. Horizontal Shear, Fh Negative Moment Region
Based on the force transferred in topping (page 4-53)

54. Unreinforced Horizontal Shear
Where
f – 0.75
bv – width of shear area
lvh - length of the member subject to shear, 1/2 the span for simply supported members

55. Reinforced Horizontal Shear
Where
f – 0.75
rv - shear reinforcement ratio
Acs - Area of shear reinforcement
me - Effective shear friction coefficient

56. Shear Friction Coefficient
For horizontal shear the denominator, Vu is replace with the horizontal shear force that needs to be transfered. As previously discussed most finishes meet the Hardened concrete with roughened surface.

57. Shear Resistance by Non-Prestressed Concrete
Shear strength for non-prestressed sections
Simpified form for shear thrength

58. Prestress Concrete Shear Capacity
Where:
ACI Eq 11-9
Effective prestress must be 0.4fpu
Accounts for shear combined with moment
May be used unless more detail is required

59. Prestress Concrete Shear Capacity
Concrete shear strength is minimum is
Maximum allowed shear resistance from concrete is:

60. Shear Capacity, Prestressed
Resistance by concrete when diagonal cracking is a result of combined shear and moment
Where:
Vi and Mmax - factored externally applied loads e.g. no self weight
Vd - is un-factored dead load shear

61. Shear Capacity, Prestressed
Resistance by concrete when diagonal cracking is a result of principal tensile stress in the web is in excess of cracking stress.
Where:
Vp = the vertical component of effective prestress force (harped or draped strand only)
Tends to be a controlling factor at the end of prestressed members where the strand is undeveloped

62. Vcmax
Shear capacity is the minimum of Vc, or if a detailed analysis is used the minimum of Vci or Vcw

63. Shear Steel
If:
Then:

64. Shear Steel Minimum Requirements
Non-prestressed members
Prestressed members
Remember
both legs of a stirrup count for Av

65. Torsion Current ACI Provision for alternate solution
Based on compact sections
Greater degree of fixity than PC can provide
Provision for alternate solution
Zia, Paul and Hsu, T.C., “Design for Torsion and Shear in Prestressed Concrete,” Preprint 3424, American Society of Civil Engineers, October, Reprinted in revised form in PCI JOURNAL, V. 49, No. 3, May-June 2004.

66. Torsion
For members loaded two sides, such as inverted tee beams, find the worst case condition with full load on one side, and dead load on the other
1.0D
1.2D+1.6L

67. Torsion In order to neglect Torsion Where:
Tu(min) – minimum torsional strength provided by concrete

68. Minimum Torsional Strength
Where:
x and y - are short and long side, respectively of a component rectangle
g - is the prestress factor

69. Prestress Factor, g For Prestressed Members Where:
fpc – level of prestress after losses
For non prestress member gamma reduces to 1

70. Maximum Torsional Strength
Avoid compression failures due to over reinforcing
Where:

71. Maximum Shear Strength
Avoid compression failures due to over reinforcing

72. Torsion/Shear Relationship
Determine the torsion carried by the concrete
Where:
T’c and V’c - concrete resistance under pure torsion and shear respectively
Tc and Vc - portions of the concrete resistance of torsion and shear

73. Torsion/Shear Relationship
Determine the shear carried by the concrete

74. Torsion Steel Design
Provide stirrups for torsion moment - in addition to shear
Where
x and y - short and long dimensions of the closed stirrup

75. Torsion Steel Design
Minimum area of closed stirrups is limited by

76. Longitudinal Torsion Steel
Provide longitudinal steel for torsion based on equation or
Whichever greater

77. Longitudinal Steel limits
The factor in
the second equation need not exceed

78. Detailing Requirements, Stirrups
135 degree hooks are required unless sufficient cover is supplied
The 135 degree stirrup hooks are to be anchored around a longitudinal bar
Torsion steel is in addition to shear steel

79. Detailing Requirements, Longitudinal Steel
Placement of the bars should be around the perimeter
Spacing should spaced at no more than 12 inches
Longitudinal torsion steel must be in addition to required flexural steel (note at ends flexural demand reduces)
Prestressing strand is permitted 60ksi)
The critical section is at the end of simply supported members, therefore U-bars may be required to meet bar development requirements

80. Serviceability Requirements
Three classifications for prestressed components
Class U: Uncracked
Class T: Transition
Class C: Cracked
Stress

81. Uncracked Section Table 4.2.2.1 (Page 4.24) Easiest computation
Use traditional mechanics of materials methods to determine stresses, gross section and deflection.
No crack control or side skin reinforcement requirements

82. Transition Section Table 4.2.2.1 (Page 4.24)
Use traditional mechanics of materials methods to determine stresses only.
Use bilinear cracked section to determine deflection
No crack control or side skin reinforcement requirements

83. Cracked Section Table 4.2.2.1 (Page 4.24) Iterative process
Use bilinear cracked section to determine deflection and to determine member stresses
Must use crack control steel per ACI modified by ACI and ACI

84. Cracked Section Stress Calculation
Class C member require stress to be check using a Cracked Transformed Section
The reinforcement spacing requirements must be adhered to

85. Cracked Transformed Section Property Calculation Steps
Step 1 – Determine if section is cracked
Step 2 – Estimate Decompression Force in Strand
Step 3 – Estimate Decompression Force in mild reinforcement (if any)
Step 4 – Create an equivalent force in topping if present
Step 5 – Calculate transformed section of all elements and modular ratios
Step 6 – Iterate the location of the neutral axis until the normal stress at this level is zero
Step 7 – Check Results with a a moment and force equilibrium set of equations

86. Steel Stress fdc – decompression stress
stress in the strand when the surrounding concrete stress is zero – Conservative to use, fse (stress after losses) when no additional mild steel is present.

87. Simple Example
Page 4-31

92. Deflection Calculation – Bilinear Cracked Section
Deflection before the member has cracked is calculated using the gross (uncracked) moment of inertia, Ig
Additional deflection after cracking is calculated using the moment of inertia of the cracked section Icr

93. Effective Moment of Inertia
Alternative method
Where:
ftl = final stress
fl = stress due to live load
fr = modulus of rupture

94. Prestress Losses Prestressing losses
Sources of total prestress loss (TL)
TL = ES + CR + SH + RE
Elastic Shortening (SH)
Creep (CR)
Shrinkage (SH)
Relaxation of tendons (RE)

95. Elastic Shortening
Caused by the prestressed force in the precast member
Where:
Kes = 1.0 for pre-tensioned members
Eps = modulus of elasticity of prestressing tendons (about 28,500 ksi)
Eci = modulus of elasticity of concrete at time prestress is applied
fcir = net compressive stress in concrete at center of gravity of prestressing force immediately after the prestress has been applied to the concrete

96. fcir
Where:
Pi = initial prestress force (after anchorage seating loss)
e = eccentricity of center of gravity of tendons with respect to center of gravity of concrete at the cross section considered
Mg = bending moment due to dead weight of prestressed member and any other permanent loads in place at time of prestressing
Kcir = 0.9 for pretensioned members

97. Creep Creep (CR) Caused by stress in the concrete Where:
Kcr = 2.0 normal weight concrete
= 1.6 sand-lightweight concrete
fcds = stress in concrete at center of gravity of prestressing force due to all uperimposed permanent dead loads that are applied to the member after it has been prestressed

98. fcds
Where:
Msd = moment due to all superimposed permanent dead and sustained loads applied after prestressing

99. Shrinkage Volume change determined by section and environment Where:
Ksh = 1.0 for pretensioned members
V/S = volume-to-surface ratio
R.H. = average ambient relative humidity from map

100. Relative Humidity Page 3-114 Figure 3.10.12
Error in book on page The Relative Humidity definition refers to Figure and should refer figure on page 3-114/

101. Relaxation
Relaxation of prestressing tendons is based on the strand properties
Where:
Kre and J - Tabulated in the PCI handbook
C - Tabulated or by empirical equations in the PCI handbook

102. Relaxation Table Values for Kre and J for given strand
Table page 4-85

103. Relaxation Table Values for C
fpi = initial stress in prestress strand
fpu = ultimate stress for prestress strand
Table (Page 4-86)

104. Prestress Transfer Length
Transfer length – Length when the stress in the strand is applied to the concrete
Transfer length is not used to calculate capacity

105. Prestress Development Length
Development length - length required to develop ultimate strand capacity
Development length is not used to calculate stresses in the member

106. Beam Ledge Geometry

107. Beam Ledge Design
For Concentrated loads where s > bt + hl, find the lesser of:

108. Beam Ledge Design
For Concentrated loads where s < bt + hl, find the lesser of:

109. Beam Ledge Reinforcement
For continuous loads or closely spaced concentrated loads:
Ledge reinforcement should be provided by 3 checks
As, cantilevered bending of ledge
Al, longitudinal bending of ledge
Ash, shear of ledge

110. Beam Ledge Reinforcement
Transverse (cantilever) bending reinforcement, As
Uniformly spaced over width of 6hl on either side of the bearing
Not to exceed half the distance to the next load
Bar spacing should not exceed the ledge depth, hl, or 18 in

111. Longitudinal Ledge Reinforcement
Placed in both the top and bottom of the ledge portion of the beam:
Where:
dl - is the depth of steel
U-bars or hooked bars may
be required to develop
reinforcement at the end
of the ledge

112. Hanger Reinforcement Required for attachment of the ledge to the web
Distribution and spacing of Ash reinforcement should follow the same guidelines as for As

113. Hanger (Shear) Ledge Reinforcement
Ash is not additive to shear and torsion reinforcement
“m” is a modification factor which can be derived, and is dependent on beam section geometry. PCI 6th edition has design aids on table

114. Dap Design
(1) Flexure (cantilever bending) and axial tension in the extended end. Provide flexural reinforcement, Af, plus axial tension reinforcement, An.
Design based on failure modes

115. Dap Design
(2) Direct shear at the junction of the dap and the main body of the member. Provide shear friction steel, composed of Avf + Ah, plus axial tension reinforcement, An

116. Dap Design
(3) Diagonal tension emanating from the re-entrant corner. Provide shear reinforcement, Ash

117. Dap Design
(4) Diagonal tension in the extended end. Provide shear reinforcement composed of Ah and Av

118. Dap Design
(5) Diagonal tension in the undapped portion. This is resisted by providing a full development length for As beyond the potential crack.

119. Dap Reinforcement 5 Main Areas of Steel Tension - As Shear steel - Ah
Diagonal cracking – Ash, A’sh
Dap Shear Steel - Av

120. Tension Steel – As
The horizontal reinforcement is determined in a manner similar to that for column corbels:

121. Shear Steel – Ah
The potential vertical crack (2) is resisted by a combination of As and Ah

122. Shear Steel – Ah
Note the development ld of Ah beyond the assumed crack plane. Ah is usually a U-bar such that the bar is developed in the dap

123. Diagonal Cracking Steel – Ash
The reinforcement required to resist diagonal tension cracking starting from the re-entrant corner (3) can be calculated from:

124. Dap Shear Steel – Av
Additional reinforcement for Crack (4) is required in the extended end, such that:

125. Dap Shear Steel – Av
At least one-half of the reinforcement required in this area should be placed vertically. Thus:

126. Dap Limitations and Considerations
Design Condition as a dap if any of the following apply
The depth of the recess exceeds 0.2H or 8 in.
The width of the recess (lp) exceeds 12 in.
For members less than 8 in. wide, less than one-half of the main flexural reinforcement extends to the end of the member above the dap
For members 8 in. or more wide, less than one-third of the main flexural reinforcement extends to the end of the member above the dap

127. Questions?