Upload
todd-mosley
View
229
Download
2
Tags:
Embed Size (px)
Citation preview
PCI 6th EditionPCI 6th Edition
Compression Component Design
Presentation OutlinePresentation Outline
• Interaction diagrams
• Columns example
• Second order effects
• Prestress wall panel example
Compression MembersCompression Members
• Proportioned on the basis of strength design. • Stresses under service conditions, particularly during
handling and erection (especially of wall panels) must also be considered
Design BasisDesign Basis
• The procedures are based on Chapter 10 of the ACI Code
• Recommendations of the PCI Committee on Prestressed Concrete Columns
• Recommendations of the PCI Committee on Sandwich Wall Panel Columns
Design ProcessDesign Process
• The capacity determined by constructing a capacity interaction curve.
• Points on this curve are calculated using the compatibility of strains and solving the equations of equilibrium as prescribed in Chapter 10 of the Code (ACI).
ReinforcementReinforcement
• ACI 318-02 waives the minimum vertical reinforcement requirements for compression members if the concrete is prestressed to at least an average of 225 psi after all losses
• In addition, the PCI Recommended Practice permits the elimination of lateral ties if:– Compression-controlled section– Non-prestressed reinforcement is not considered in the
calculation of Pn
– Non-prestressed reinforcement which is added for tension (e.g., for handling) is not considered in the calculation of Pn
– The nominal capacity is multiplied by 0.85
Development LengthDevelopment Length
• Mild Reinforcement and prestressed development length can play a significant role in capacity
• Additional Mild steel or special termination anchorages may be required
• Mechanical bar termination methods– Threaded ends– Anchored to end plates
Interaction DiagramsInteraction Diagrams
• Separate curves X, Y for none rectangular cross sections
• Most architectural precast column sections are not rectangular, therefore it is necessary to calculate the actual centroid of the compression area
Interaction Diagram StepsInteraction Diagram Steps
Step 1 – Determine Po pure axial capacityStep 2 – Determine maximum moment
Step 3 – Determine Mo for Pn = 0Step 4 – Determine additional points Step 5 – Calculate the maximum factored axial
resistance specified by the Code as:• 0.80Po for tied columns• 0.85Po for spiral columns
Step 1 – Determine Po for Mn = 0Step 1 – Determine Po for Mn = 0
Pn
Mn
Pn, Mn
Pn, Mn
Step 1 – Determine Po for Mn = 0Step 1 – Determine Po for Mn = 0
0
200
400
600
800
1000
1200
0 10 20 30 40
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
• For members with non-prestressed reinforcement, this is the balance point
• For symmetrical prestressed members, it is sufficiently precise to assume that the point occurs when the compression block, a, is one-half the member depth.
• Neutral Axis Location, c
Where:
fy = the yield strength of extreme tension steel
Es = Modulus of elasticity of extreme tension steeld = depth to the extreme tension steel from the
compression face of the member
c 0.003
0.003 fy
Ey
d
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
• Determine the force in steel using strain compatibility
Where:
ds = Depth of steel
sStrain of steel
Es = Modulus of elasticity of reinforcing steel
fs = Force in steel
ss s
c df 0.003 E
c
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
• Maximum Axial Force
Where:Acomp = Compression area A’s = Area of non prestressed compression
reinforcingA’ps = Area of compression prestressing reinforcingAs = area of reinforcing at reinforcement levely’ = distance from top of c.g. to Acomp
n comp s ps c s sP A A' A' 0.85 f ' f A
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
n n
comp s ps t c
s s t s
M P e
A A' A' y y' 0.85 f '
f A y d
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
Pn
Mn
Pn, Mn
Pn, Mn
Step 2 – Determine Maximum MomentStep 2 – Determine Maximum Moment
0
200
400
600
800
1000
1200
0 10 20 30 40
Step 3 – Determine Mo for Pn = 0Step 3 – Determine Mo for Pn = 0
• Same methods used in flexural member design
Where:
and
a
Apsf
ps A
sf
y A'
sf '
y
0.85f 'cb
n ps ps ps s y ps
a aM A f d A f d
2 2
Step 3 – Determine Mo for Pn = 0Step 3 – Determine Mo for Pn = 0
Pn
Mn
Pn, Mn
Pn, Mn
0
200
400
600
800
1000
1200
0 10 20 30 40
0
200
400
600
800
1000
1200
0 10 20 30 40
Pn
Mn
Pn, Mn
Pn, Mn
Compression controlled
( = 0.65 or 0.70)
Tension controlled ( = 0.9)
Step 3 – Determine Mo for Pn = 0Step 3 – Determine Mo for Pn = 0
Step 4 – Additional PointsStep 4 – Additional Points
• Select a value of “c” and calculate a = β1c
• Determine the value of Acomp from the geometry of the section
• Determine the strain in the reinforcement assuming that εc = 0.003 at the compression face of the column. For prestressed reinforcement, add the strain due to the effective prestress εse = fse/Eps
Step 4 – Additional PointsStep 4 – Additional Points
• Determine the stress in the reinforcement. For non-prestressed reinforcement
fsE
s
sf
y
Step 4 – Additional PointsStep 4 – Additional Points
• For prestressed reinforcement, the stress is determined from a nonlinear stress-strain relationship
Step 4 – Additional PointsStep 4 – Additional Points
• If the maximum factored moment occurs near the end of a prestressed element, where the strand is not fully developed, an appropriate reduction in the value of fps should be made
Step 4 – Additional PointsStep 4 – Additional Points
• Calculate Pn and Mn
• For compression controlled sections (without spiral reinforcement) the net tensile strain εt in the extreme tension steel has to be less than or equal to that at the balance point
= 0.65
Step 4 – Additional PointsStep 4 – Additional Points
Pn
Mn
Pn, Mn
Pn, Mn
0
200
400
600
800
1000
1200
0 10 20 30 40
Step 5 – Calculate the Maximum Factored AxialStep 5 – Calculate the Maximum Factored Axial
Pmax –
= 0.80Po for tied columns
= 0.85Po for spiral columns
o c g s s ps ps
ps ps se ps ps ps y
0.8 P 0.8 0.85 f` (A A A A A )
A A f 0.003 E A A f
0
200
400
600
800
1000
1200
0 10 20 30 40
Step 5 – Calculate the Maximum Factored AxialStep 5 – Calculate the Maximum Factored Axial
Pn
Mn
0.80fPo or 0.85fPo
Pn, Mn
Pn, Mn
Example, Find Interaction Diagramfor a Precast Column
Example, Find Interaction Diagramfor a Precast Column
Given:Column cross section shown
Concrete: f′c = 5000 psiReinforcement: Grade 60
fy = 60,000 psi
Es = 29,000 ksiProblem:
Construct interaction curve for bending about x-x axis
Solution StepsSolution Steps
Initial Step: Determine Column Parameters
Step 1 – Determine Po from Strain Diagram
Step 2 – Determine Pnb and Mnb
Step 3 – Determine Mo
Step 4 – Plot and add points as required
Step 5 – Calculate maximum design load
Determine Column ParametersDetermine Column Parameters
β1 = 0.85 – 0.05 = 0.80
d = 20 – 2.5 = 17.5 in
d′ = 2.5 in
0.85f′c = 0.85(5) = 4.25 ksi
Ag = 12(20) = 240 in2
As = As′ = 2.00 in2
yt = 10 in
Step 1 – Determine Po From Strain DiagramStep 1 – Determine Po From Strain Diagram
With no prestressing steel, the equation reduces to:
0.85 ' - ' - '
0.85 5 240- 2- 2 2 2 60 1243
o n c s s s s yP P f A A A A A f
kips
Step 2 – Determine Pnb and MnbStep 2 – Determine Pnb and Mnb
• From Strain Diagram determine Steel Stress
s s s s
s
0.003f ' ' E c d' E
c
0.003 10.5 2.5 29,000 66.3ksi 60ksi
10.5
f ' 60 ksi
Step 2 – Determine Pnb and MnbStep 2 – Determine Pnb and Mnb
• Determine Compression Area
comp
2
A = a b = 0.8 10.5 12
= 100.8 in
Step 2 – Determine Pnb and MnbStep 2 – Determine Pnb and Mnb
• With no prestressing steel
Pnb = (Acomp-A′s)(0.85f′c) +A′s f′s - As fs = (100.8 - 2)(4.25) + 2 (60)- 2(60) = 419.9 kips
Pnb = 0.65(419.9) = 273 kips
Step 2 – Determine Pnb and MnbStep 2 – Determine Pnb and Mnb
• With no prestressing steelMnb = (Acomp − A′s)(yt − y′)(0.85f′c)
+ A′sf′s(yt − d′) + Asfs(d − yt) = (100.8 – 2)(10 – 4.20)(4.25)
+ 2(60)(10 – 2.5) + 2(60)(17.5 – 10) = 2435
+ 900 + 900 = 4235 kip-in.
Mnb = 0.65(4235 kip-in.) = 2752 kip-in. = 229 kip-ft
Step 3 – Determine MoStep 3 – Determine Mo
• Conservative solution neglecting compressive reinforcement
s y
c
A f 2.0 60a 2.35in
0.85 f ' b 4.25 12
o s y
o
a 2.35M A f d- 2.0 60 17- 1959kip- in
2 2
M 163.3kip ft
Step 3 – Determine MoStep 3 – Determine Mo
• Strength Reduction Factor
1
t
t
a 2.35c 2.94in
0.8
0.003 0.003d c 17.5 2.94 0.0149
c 2.940.005 0.9
oM 0.9 163.3 147kip ft
Step 4 – Plot 3 Point InteractionStep 4 – Plot 3 Point Interaction
• From the previous 3 steps, 3 points have been determined. From these 3 points, a conservative 3 point approximation can be determined.
• Add additional points as required
Step 5 – Calculate Maximum Design LoadStep 5 – Calculate Maximum Design Load
Pmax= 0.80 Po = 0.80 (808 kips)
= 646 kips
Wall or ColumnWall or Column
• Effective Width is the Least of – The center-to-center distance between loads– 0.4 times the actual height of the wall– 6 times the wall thickness on either side
Slenderness / Secondary EffectsSlenderness / Secondary Effects
Causes of Slenderness EffectsCauses of Slenderness Effects
• Relative displacement of the ends of the member due to:– Lateral or unbalanced vertical loads in an
unbraced frame, usually labeled “translation” or “sidesway.”
– Manufacturing and erection tolerances
Causes of Slenderness EffectsCauses of Slenderness Effects
• Deflections away from the end of the member due to:– End moment due to eccentricity of the axial load.– End moments due to frame action continuity, fixity
or partial fixity of the ends– Applied lateral loads, such as wind– Thermal bowing from differential temperature– Manufacturing tolerances– Bowing due to prestressing
Calculation of Secondary EffectsCalculation of Secondary Effects
• ACI allows the use of an approximate procedure termed “Moment Magnification.”
• Prestressed compression members usually have less than the minimum 1% vertical reinforcement and higher methods must be used
• The PCI Recommended Practice suggests ways to modify the Code equations used in Moment Magnification, but the second-order, or “P-∆” analysis is preferred
Second-Order (P-∆) AnalysisSecond-Order (P-∆) Analysis
• Elastic type analysis using factored loads.• Deflections are usually only a concern under
service load, the deflections calculated for this purpose are to avoid a stability failure
• The logic is to provide the same safety factor as for strength design
Second-Order (P-∆) AnalysisSecond-Order (P-∆) Analysis
• Iterative approach• Lateral deflection is calculated, and the
moments caused by the axial load acting at that deflection are accumulated
• Convergence is typical after three or four iterations
• If increase in deflection is not negligible the member may be approaching stability failure
Second-Order (P-∆) AnalysisSecond-Order (P-∆) Analysis
• Cracking needs to be taken into account in the deflection calculations
• The stiffness used in the second order analysis should represent the stiffness of the members immediately before failure
• May involve iterations within iterations• Approximations of cracked section properties
are usually satisfactory
Second-Order (P-∆) AnalysisSecond-Order (P-∆) Analysis
• Section 10.11.1 of ACI 318-02 has cracked member properties for different member types for use in second-order analysis of frames
• Lower bound of what can be expected for equivalent moments of inertia of cracked members and include a stiffness reduction factor K to account for variability of second-order deflections
Second-Order (P-∆) AnalysisSecond-Order (P-∆) Analysis
• Effects of creep should also be included. The most common method is to divide the stiffness (EI) by the factor 1 + βd as specified in the ACI moment magnification method
• A good review of second-order analysis, along with an extensive bibliography and an outline of a complete program, is contained in Ref. 24.
Example, Second Order Analysis of Uncracked Member
Example, Second Order Analysis of Uncracked Member
Given:An 8 in. thick, 8 ft wide prestressed wall panel
as shown.Loading assumptions are as follows:
• Axial load eccentricity = 1 in (at one end)• Assume midspan bowing = 1.0 in outward.• Wind load = 30 psf• Pinned top and bottom
Concrete:
f`c = 5000 psi
Ec = 4300 ksi
K Lr
1 360 8 1
12
156
Figure 2.7.5 (page 2-54)Figure 2.7.5 (page 2-54)
AssumptionsAssumptions
• Panel has 250 psi compressive stress due to prestressing after losses
• Simple span
• Maximum moment occur at L/2
Solution StepsSolution Steps
Step 1 – Check P-CriticalStep 2 – Calculate Final Displacement without wind
assuming un-cracked sectionStep 3 – Check moments on panel including secondary
moments without windStep 4 – Check for cracking without windStep 5 – Calculate Final Displacement with wind
assuming un-cracked sectionStep 6 – Check moments on panel including secondary
moments with windStep 7 – Check for cracking with windStep 8 – Compare results against interaction diagram
Step 1 – Check P-CriticalStep 1 – Check P-Critical
Where:
EIeff – Effective Stiffness
P
c=2 EI
eff
l 2
Step 1 – Calculate Effective StiffnessStep 1 – Calculate Effective Stiffness
Where:
1+d = Accounts for sustained loads
EI
eff=
0.70EcI
g
1 d
Step 1 – Calculate Effective StiffnessStep 1 – Calculate Effective Stiffness
c g 6 2eff
d
0.70 E I 0.70 4300 4096EI 7.90 10 kip in
1 1.56
3 34
g
du
bh 96(8 )I 4096 in
12 121.2 101.2DL
0.556P 1.2(10) 1.6 6
Step 1 – Check P-CriticalStep 1 – Check P-Critical
Pc=2 EI
eff
l 22 7.9106 kipsin2
360 in2602 kips
PcP
u
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
• Moments without secondary effect
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
• Calculate displacement for secondary moments due to Axial load
22u
i 6c g
21.6 1 360P e l0.0221 in
16 E I 16 7.90 10
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
• Total mid-span displacement = Axial load displacement + initial mid-span bow
= 1.0 in. + 0.0221 = 1.022 in.
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
• Additional Mid-Span displacement due to P- effect
22u
i 6eff
21.6 e 360P e l0.044 e
8 EI 8 7.90 10
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
Step 2 – Calculate Final Displacement No Wind, Assumed Un-Cracked Section
• 1st Iteration for P- Effect∆ = 0.044(1.022 in) = 0.045 in
• 2nd Iteratione = 1.022 in. + 0.045in. = 1.067 in∆ = 0.044(1.067 in.) = 0.047 in
• 3rd Iteratione = 1.022 in. + 0.047 in.= 1.069 in∆ = 0.044(1.069 in.) = 0.047 in (convergence)
Step 3 – Check Moments onPanel Without Wind
Step 3 – Check Moments onPanel Without Wind
• Conservative Mu at L/2
Mu = 10.8 + 21.6(1.069) = 33.9 kip-in
Step 4 – Check for Cracking Without WindStep 4 – Check for Cracking Without Wind
• Tension Stress at exterior face
u33.9 4M y
33 psi (Ten)I 4096
Step 4 – Check for Cracking Without WindStep 4 – Check for Cracking Without Wind
Bending Stress -33psi
Half Panel weight [100psf(15ft)/(8(12))](1.2)
19 psi
Prestress 250 psi
Dead load 10,000 lbs(1.2)/[8 in.(96 in.)]
16 psi
Total Stress Tension Face 252 psi
Therefore, the analysis is valid.
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Axial Load Displacement
Deflection due to Pue is the same as the previous case
= 0.0221 in
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Determine Effective Stiffness for Wind case
When considering wind, βd = 0
EI
eff=
0.70EcI
g
1 d
0.70 4300 4096in4
1.01.23107 kipin2
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Additive wind load (suction) = 30 psf
wu = 30 psf (8ft)(0.8) = 192 lb/ft
w=
5wul 4
384EIeff
5 192 / 12 3602
384 1.23107 kipin2 0.28in
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Total initial mid-span bow including eccentricity and wind
e = 1.0 + 0.022 + 0.28 = 1.302 in
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
• Additional mid-span displacement due to P- effect
i=
Puel 2
8EcI
g
21.6 e 3602
8 1.23107 kipin2 0.028 e
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
Step 5 – Calculate Final Displacement With Wind, Assumed Un-Cracked Section
• 1st Iteration for P- Effect∆ = 0.028(1.302in.) = 0.036 in
• 2nd Iteratione = 1.302 in. + 0.036 in. = 1.338 in∆ = 0.028(1.338 in.) = 0.037 in
• 3rd Iteratione = 1.302 in. + 0.037 in. = 1.339 in∆ = 0.028(1.339 in.) = 0.037 in. (convergence)
Step 6 – Check Moments on Panel With WindStep 6 – Check Moments on Panel With Wind
Mu
Pue
2 + P
u +
wl 2
8
Mu
21.6 12
+ 21.6 1.339 + 0.192 3602
8299 kipin
Step 7 – Check for Cracking With WindStep 7 – Check for Cracking With Wind
Muyy
I
229 4 4096
292 psi(Tension)
Step 7 – Check for Cracking With WindStep 7 – Check for Cracking With Wind
Bending Stress -292psi
Half Panel weight 19 psi
Prestress 250 psi
Dead load 16 psi
Total Stress Exterior Face -7 psi(T)
Therefore, the analysis is valid.
Step 8 – Check InteractionStep 8 – Check Interaction
• No Wind
• With Wind
Pu =
21.6 kips
8 ft + 1.2(15ft)0.1 kips
ft = 4.50 kips
ft
Mu =
33.9in.-kips
[12in.ft
(8ft)] = 0.35 kip-ft
ft
Mu =
299 in.-kips
[12in.ft
(8ft)] = 3.1 kip-ft
ft
Figure 2.7.5 (pages 2 – 54)Figure 2.7.5 (pages 2 – 54)
Points Are inside Curve
Questions?Questions?