The 10 10 Steps Steps in i n Design of Post -Te Tensio nsio ned Floors Tensioned Floors
Dr Bijan O Aalami Professor Emeritus, San Francisco State University Principal, ADAPT Corporation;
[email protected]
www.adaptsoft.com
301 Mission Street San Francisco, California High Seismic Force Region
Column supported multistory building Four Seasons Hotel; Florida High Wind Force Region
Two-way flat slab construction
Post-Tensioning Systems Unbonded System m ) * m " 5 7 . . 2 0 ( 1
WIRE
CORROSION INHIBITING COATING
PLASTIC SHEATHING
NOTE: * NOMINAL DIAMETER
(a) STRAND
Multi-level parking structures One-way beam and slab design
(b) TENDON
GREASE FILLED PLASTIC CAP SHEATH
TUBE
STRAND
(c) ANCHORAGE ASSEMBLY
Example of a Floor System using the Unbonded Post-tensioning System
Post-Tensioning Systems Grouted System
An example of a grouted system hardware with flat duct
Prelimi nary Considerations Desig n of Post-Tensioned Floors
Example of a Floor System Reinforced with Grouted Post-Tensioning System
Dimensions (sizing)
Optimum spans; optimum thickness
Structural system
One-way/two-way; slab band
Boundary conditions; connections
Service performance; strength condition
Load Path; Design strips Design sections; design values
Prelimi nary Considerations Design of Post-Tensi oned Floors Dimensions (sizing)
Optimum spans; optimum thickness
Prelimi nary Considerations Desig n of Post-Tensi oned Floors
Selection of load path for two-way systems – Design Strips
An optimum design is one in which the
reinforcement determined for “service condition” is used in its entirety for “strength condition.”
PT amount in service condition is
governed mostly by:
Hypothetical tensile stresses (USA), or crack width (EC2) Tendon spacing (USA)
Common spans: 25 – 30 ft (8 – 9 m) Span/thickness ratios
40 - 45 for interior 35 for exterior with no overhang
Subdivide the structure into design strips in two orthogonal directions (Nahid slab)
Prelimi nary Considerations Desig n of Post-Tensioned Floors
Subdivide the floor along support lines in design strips
Prelimi nary Considerations Design of Post-Tensi oned Floors Subdivide slab along support lines in design strips in the orthogonal direction 2
1 2
1
3
4
3
4
5
5
A A
B B
C C
D D
E F
E
Y X
F Y X
G
An important aspect of load path selection in a two-way system is that every point of the slab should be assigned to a specific design strip. No portion of the slab should be left unassigned .
Prelimi nary Considerations Desig n of Post-Tensioned Floors Design sections
Design sections extend over the entire design strip and are considered at critical locations, such as face of support and mid-span
Design values
Actions, such as moments at each design section are reduced to a “single” representative value to be used for design
559 is the area (total) value of bending moment at face of support
10 Steps Design of Post Tensioned Floors
Step 1 Geometry and Struc tural Syst em Select design strip and Idealize
1.
Geometry and Structural System
Extract; straighten the support line; square the boundary
2. Material Properties 3. Loads 4. Design Parameters 5. Actions due to Dead and Live Loads 6. Post-Tensioning 7. Code Check for Serviceability 8. Code Check for Strength 9. Check for Transfer of Prestressing 10. Detailing
Step 1 Geometry and Stru ctu ral System
Step 2 Material Properties
Select design strip and Idealize
Extract; straighten the support line; square the boundary Model the slab frame with a row of supports above and below. This represents an upper level of multi-story concrete frame. Assume rotational fixity at the far ends; Assume roller support at the far ends
Concrete
Weight 24 kN/m3 40 MPa 28 day cylinder Elastic modulus 35,220 MPa Long-term deflection factor 2
Non-Prestressed reinforcement fy Elastic modulus
460 MPa 200,000 MPa
Prestressing
Strand diameter Strand area Ultimate strength Effective stress Elastic modulus
View of idealized slab-frame
13 mm 99 mm2 1,860 MPa 1,200 MPa 193,000 MPa
Step 3 Loads
Step 4 Design Parameters
Selfweight
Applicable code
Based on member volume
ACI 318-11; EC2 EN 1992-1-1:2004 IBC 2012 Local codes, such as California Building Code (CBC 2011); or otherwise
Superimposed dead load Min (partitions)
1 kN/m2
Live load
Residential Office Shopping mall Parking structure
Cover for protection against corrosion
2.0 kN/m2 2.5 kN/m2 3.5 kN/m2 2.0 kN/m2
Cover to rebar
Not exposed to weather 20 mm Exposed to weather
Lateral loads Wind Earthquake
Cover to tendon
Not exposed to weather 20 mm 25 mm Exposed to weather
Example assumes Superimposed DL Live load
SDL= 2 kN/m2 LL = 3 kN/m2
Step 4 Design Parameters
Step 4 Design Parameters
Cover for fire resistivity
Cover for fire resistivity
Identify “restrained” and “unrestrained panels.”
Restrained or Unrestrained
Restrained
Cover Thickness, mm. for Fire Endurance of
Aggregate Type
Unrestrained
1 hr
1.5 hr
2 hr
3 hr
4 hr
Carbonate Siliceous Lightweight
-
-
40 40 40
50 50 50
-
Carbonate Siliceous Lightweight
-
-
20 20 20
25 25 25
30 30 30
For 2-hour fire resistivity Restrained Unrestrained
50 mm
20 mm. 40 mm
Identify “restrained” and “unrestrained panels.”
Step 4 Design Parameters Allowable Stresses or Crack Width
Selection for Two-Way System Based on ACI Total load case Tension 0.5√f’c Compression 0.60 f’c Sustained load case Tension 0.5√f’c Compression 0.60 f’c
Based on EC2
Frequent load case
Tension Ft = 0.30 f ck (2/3) Compression 0.60f ck Quasi permanent load case Tension Ft = 0.30 f ck (2/3) Compression 0.45f ck Crack width normal exposure (?) Unbonded Bonded
Step 5 Actions due to Dead and Live Loads Analyze the design strip as a single
level frame structure with one row of supports above and below, using
In-house simple frame program
(Simple Frame Method; SFM); or in-house Equivalent Frame Program (EFM); Specialty commercial software
All the three options yield safe designs.
But, each will give a different amount of reinforcement. The EFM is suggested by ACI-318. To some extent, it accounts for biaxial action of the prototype structure in the frame model. Accuracy and ability of commercial software for optimization varies
Step 4 Design Parameters Allowable deflections (EC2/ approx. ACI)
For visual impact use total deflection
Span/250 Use camber, if necessary
Total deflection subsequent to installation of members that are likely to be damaged Span/350
Immediate deflection due to live load Span/500 Long-term deflection magnifier 2. This brings
the total long-term deflection to 3,
Step 5 Actions due to Dead and Live Loads Analyze the design strip as a single level frame structure with one row of supports above and below.
Step 6 Post-Tensioning
Selection of design parameters Selection of PT force and profile
Step 6 Post-Tensioning Selection of PT force and profile Two entry value assumptions must be made to initiate the computations. Select precompression and % of DL to balance
Effective force/tendon selection option
- force selection
Calculation of balanced loads;
adjustments for percentage of load balanced
Calculation of actions due to balanced
loads
Step 6 Post-Tensioning Selection of design parameters
Select average precompression 1 MPa Target to balance 60% of DL
Selection of PT force and profile
Assume simple parabola mapped within the bounds of top and bottom covers
Force diagram of simple parabola
Step 6 Post-Tensioning Assume
simple parabola for hand calculation
STEP 6 Post-Tensioning
STEP 6 Post-Tensioning Calculation of balanced loads;
adjustment of % of DL balanced F121_ACI_PT_2_way_082012
Calculation of balanced loads; adjustment of % of DL balanced
Assume P/A =150psi[1MPa]
F121_ACI_2-way_PT_force_082012
1
Select critical span
Select max drap e using tendons from critical span
2
Select max drape
Calculate %of DL balanced (%DL)
Calculate %of DL balanced (%DL) Yes
No
%DL < 50%?
No
P /A<300ps i [2MP a]?
No
%DL > 80%?
No
Yes
Yes %DL > 80%?
Yes
Increase P/A P/A>125psi [0.8MPa]? Yes
No Reduce drape
Is it practical to reduce P/A or tendons?
Reduce P/A
No
Yes
Raise tendon to reach %DL ~ 60%
Go to next span
Reduce P/A or tendons to %DL balanced ~ 60% ; P/A >= 125 psi [0.8 MPa]
Move to next span
Exit after last span
Member with widely different spans
STEP 6 Post-Tensioning Calculation of balanced loads
Lateral forced from continuous tendons Lateral force from terminated tendons Moments from change in centroid of member
STEP 6 Post-Tensioning Calculation of balanced loads
Lateral forced from continuous tendons Lateral force from terminated tendons Moments from change in centroid of member
Force from terminated tendon
Example of force from continuous tendon
P = 500 k a = 93 mm
; b = 186 mm ; L = 9 m ;
c = {[93/186]0.5/[1 + (93/186) 0.5]} * 9.00 = 3.73 m
L = 10 m ; a = 93 mm ; P = 119 kN; c =0.20*10 = 2.00 m
Wb/tendon = 2 P*a/c 2 = 119.0 kN * (2*93/1000)/3.73 2 = 119.0 kN / tendon * 0.013 / m = 1.59 kN/m / tendon
Wb = (3 * 119.0 * 2 * 93 / 1000) / 2.0 2 16.60 kN/m Concentrated force at dead end = 2*16.60 = 33.20 k
STEP 6 Post-Tensioning Calculation of balanced loads
Lateral forced from continuous tendons Lateral force from terminated tendons Moments from change in centroid of member
STEP 6 Post-Tensioning Calculation of actions due to balanced loads
Check balanced loads for static equilibrium Determine moments/shears from balanced loads applied the frame used for dead and live loads Note down reactions from balanced loads
Example of force from change in member centroid
Moment at face of dr op = M M = P * shi ft in c entroid =P * (Yt-Left – Yt-Right ) P = 23*119 kN; 146 mm
Yt-Left
= 120 mm ; Yt-Right =
M= 23*119(120 – 146)/1000 = -71.16 kNm
STEP 6 Post-Tensioning Calculation of actions due to balanced loads
Obtain moments at face-of-supports and mid-spans Note the reactions. The reactions are hyperstatic actions.
Comments: Moments and precompression will be used for serviceability check. Reactions will be used for Strength check.
STEP 7 Code Check fo r Serviceabil ity Code requirements for serviceability
Load combinations Stress/crack width check Minimum reinforcement Deflection check. Load combination Frequent (Total) load condition 1.00DL + 0.50LL + 1.00PT
Quasi Permanent (Sustained) load condition 1.00DL + 0.30LL + 1.00PT
Stress check
Using engineering judgment, select the locations th likely to be cr itical. Typically, these are at the face o and for hand calculation at mid-span At each sec ti on sel ect ed for ch eck , us e the desig n applicable to the entire design section and apply th the entire cross -section of the design secti on to arri the hypothetical stresses used in code check. = (MD + 0.5ML + MPT) / S + P/A S = I/Y c ; I = second moment of area of ; Yc = distance to farthest tension fib er
STEP 7 Code Check for Serviceabil ity
STEP 7 Code Check for Serviceabil ity
ACI Minimum Reinforcement
ACI 318-11 Minimum Reinforcement F114_041112
ACI Minimum Rebar for two-way systems `
1
Rebar over support is function of geometry of the design strip and the strip in the orthogonal direction Rebar in span is a function of the magnitude of the hypothetical tensile stress
PT system? 2
9 Unbonded
3
At supports As = 0.0075Acf
Bonded
Calculate the cracking moment Mcr at supports and spans
`
In span calculate hypothetical tension stress ft
4
Does Mcr exceeed 1.2xmoment capacity?
10
11
No
Yes
5 ft ? tension stress
12 No added rebar required
6 7
8
ft > 2 root 'c [ft > 0.17 root f'c ]
Add rebar to resist force in tensile zone
Add rebar to increase Moment capacity to 1.2 Mcr
ft =< 2 root 'c [ ft =< 0.17 root f'c ]
No added rebar required
As = 0.00075 * A cf EXIT
STEP 7 Code Check for Servic eabil ity
As = Ar ea of s teel req ui red A cf = Larger of cross -sectional area of the strip in direction of analysis and orthogonal to
STEP 7 Code Check for Serviceabil ity
ACI 318-11 Minimum Reinforcement
Rebar in span is a function of the magnitude of the hypothetical tensile stress
In span, provide rebar if the hypothetical tensile stress exceeds 0.166 f’ c The amount of reinfor cement A s is given by: A s = N / (0.5*f y) where N is the tensile force in tension ne
h = member thickness; b = design section width
EC2 Minimum Reinforcement
Minimum based on cross-sectional area Minimum based on hypothetical tensile stress
•
Based on cross-sectional “area” of section
A smin *d
≥
(0.26* f ctm *b t *d / f yk )
≥
0.0013* b t
•Based on value of hypothetical tensile stresses for crack control Check probable crack width Add rebar based using the code
STEP 7 Deflect ion Check Read deflections from the frame analysis of the
STEP 8 Strength Check Steps in strength check
design strip for dead, live and PT; ( ΔDL , ΔLL , and ΔPT ). . Make the following load combinations and check against the allowable values for each case
Total Deflection (1 + 2)( ΔDL + ΔPT + 0.3 ΔLL ) + 0.7 ΔLL < span/250 This is on the premise of sustained load being 0.3 time the design live load. It is for visual effects; Provide camber to reduce value, where needed and practical Immediate deflection from live load Δimmediate = 1.00 ΔL < span/500 This check is applicable, where non-structural members are likely to be damaged. Otherwise, span/240 applies
Presence of members likely to be damaged from sustained deflection (1+ 2)(0.3 ΔLL ) + 0.7 ΔLL < span/350
Determination of Hyperstatic actions Direct Method – based on reactions from balanced loads Indirect Method – Using primary and post-tensioning moments
STEP 8 Strength Check
Determination of Hyperstatic actions Direct Method – based on reactions from balanced loads
Load combinations (EC2) U1 = 1.35DL + 1.50LL + 1.0HYP where, HYP is moment due to hyperstatic actions from prestressing
STEP 8 Strength Check
Load combinations Determination of hyperstatic actions Calculation of design moments (Mu) Calculate capacity/rebar for design moment Mu Check for punching shear Check/detail for unbalanced moment at support
A comment on capacity versus demand Post-tensioned members possess both a positive and negative moment capacity along the member length Rebar needs to be added, where capacity falls short of demand First, find the capacity and compare it with demand
STEP 8 Strength Check
STEP 8 Strength Check
The figure below shows the forces on a PT member. In calculating the force from PT tendons, use either the code formulas or the following simplified procedure, based on parametric study of common building structures can be sued.
Check for adequate ductility ACI Ductility is deemed adequate, if c/dt <= 0.345 EC2
Ductility is deemed adequate, if c/h<= 0.43 h = member thickness
This condition guarantees that steel will yield, before concrete in compression crushes.
USING EC2
As su me t end on st res s u nd er s erv ic e co nd it io n 1,200 M
As su me t end on st res s at ul ti mat e li mi t s tat e 1300 MPa
USING ACI Tendon Length 38 m for singl e end stressin g; ; length 35 m length 75 m double end stressin g f ps is con servatively 1,480 MPa if span is less than 11 f ps is conservatively 1,340 MPa if span is greater than
Punching Shear Design
At stressing:
PUNCHED OUT COLUMN REGION
SHEAR STRESS DUE TO k M u
Mu
D 1 0 8 / S L I D E S / 0 6 0 5 9 1
STEP 9 Check for Transf er of Prestressi ng
Tendon has its maximum force; Concrete is at its weakest strength; and Live load to counteract prestressing is absent
SHEAR STRESS DUE TO Vu
Vu
Hence the member is likely to experience stresses more severe than when in service Add
rebar when “representative” (hypothetical) tension stresses exceed a threshold
CRITICAL SURFACE
TWO-WAY SLAB
ILLUSTRATION OF CRITICAL SURFACE FOR THE EVALUATION OF PUNCHING SHEAR STRESSES
Refer to ACI 318-11 section 11.11
Do not exceed “representative” hypothetical compressive stresses
STEP 9 Check fo r Transfer of Prestressi ng
Load combination
STEP 10 Detailing
SUPPORT
U = 1.00*Selfweight + 1.15*PT
EQ. EQ.
Check for allowable stresses
Tension stress
Compression stress
If tension exceeds, provide rebar in tensile zone to resist Nc
Position of rebar MID-SPAN
SUPPORT
SEE PLAN EQ.
EQ. EQ.
EQ.
STAGGER
R E G G A T S
TOP REBAR AT SUPPORT TYP. WALL
DROP CAP COLUMN
BOTTOM
PLAN
If compression exceeds, wait until concrete gains adequate strength
Lc/6
Lc/6 POST-TENSIONED SLAB
* Lc/3 DROP COLUMN
Lc SUPPORT LINE
ELEVATION
Thank you for listening.
www.adaptsoft.com;
[email protected]
Ld
