note: moment diagram convention
DESCRIPTION
Section 3 design of post-tensioned components for flexure Developed by the pTI EDC-130 Education Committee lead author: trey Hamilton, University of Florida. Note: Moment Diagram Convention. - PowerPoint PPT PresentationTRANSCRIPT
SECTION 3
DESIGN OF POST-TENSIONED COMPONENTS FOR FLEXUREDEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEELEAD AUTHOR: TREY HAMILTON, UNIVERSITY OF FLORIDA
NOTE: MOMENT DIAGRAM CONVENTION
• In PT design, it is preferable to draw moment diagrams to the tensile face of the concrete section. The tensile face indicates what portion of the beam requires reinforcing for strength.
• When moment is drawn on the tension side, the diagram matches the general drape of the tendons. The tendons change their vertical location in the beam to follow the tensile moment diagram. Strands are at the top of the beam over the support and near the bottom at mid span.
• For convenience, the following slides contain moment diagrams drawn on both the tensile and compressive face, denoted by (T) and (C), in the lower left hand corner. Please delete the slides to suit the presenter's convention.
OBJECTIVE 1 hour presentation Flexure design considerations
PRESTRESSED GIRDER BEHAVIOR
k1DL
Lin and Burns, Design of Prestressed Concrete Structures, 3rd Ed., 1981
1.2DL + 1.6LL
DL
LIMIT STATES – AND PRESENTATION OUTLINE
Load Balancing – k1DL Minimal deflection.
Select k1 to balance majority of sustained load. Service – DL + LL
Concrete cracking. Check tension and compressive stresses.
Strength – 1.2DL + 1.6LL +1.0Secondary Ultimate strength.
Check Design Flexural Strength (Mn)
EXAMPLE
Load Balancing > Service Stresses > Design Moment Strength
Given: Cast-in-place, post-tensioned girder supporting parking garage slab with parabolically drapedmulti-strand post-tensioning tendon. See PTI manual Example 7.4 (pg 151).
DIMENSIONS AND PROPERTIES
Geometry and section properties
L 68ft end to end of girder and assumed center-to-center span length
bw 18in web width
h 36in section height
hf 6in flange thickness
s 20ft beam spacing
Tendon data
Aps 28 Aps5 Number of 0.5 in. dia. strands in fully bonded tendon
yh 3.75in Distance from bottom of beam to cgs of tendon at low point of harp.
Material properties
f'c 5000psi Specified 28-day concrete compressive strength.
f'ci 4000psi Specified concrete compressive strength at prestress transfer.
fy 60ksi Specified yield strength of mild steel reinforcement
fpu 270ksi Specified ultimate tensile strength of prestressing strand
SECTION PROPERTIES
I 139118 in4
A 1206 in2
yb 24.94 in
yt 11.06 in
h 36 in
Given: Cast-in-place, post-tensioned girder supporting parking garage slab with parabolically drapedmulti-strand post-tensioning tendon. See PTI manual Example 7.4 (pg 151).
Determine effective flange width according to ACI 8.12.2.
16 hf bw 114 in
s bw( ) 0.5 111 in
0.25 L 204 in Effective flange width is 111 in.
LOAD BALANCING Tendons apply external self-
equilibrating transverse loads to member.
Forces applied through anchorages The angular change in tendon profile
causes a transverse force on the member
Load Balancing > Service Stresses > Design Moment Strength
LOAD BALANCING Transverse forces from tendon
“balances” structural dead loads. Moments caused by the equivalent
loads are equal to internal moments caused by prestressing force
Load Balancing > Service Stresses > Design Moment Strength
EQUIVALENT LOAD
Harped Tendon can be sized and placed such that the upward force exerted by the tendon at midspan exactly balances the
applied concentrated loadLoad Balancing > Service Stresses > Design Moment Strength
EQUIVALENT LOAD
Parabolic Tendon can be sized and placed such that the upward force exerted by the tendon along the length of the member exactly
balances the applied uniformly distributed load
Load Balancing > Service Stresses > Design Moment Strength
FORMULAS (FROM PTI DESIGN MANUAL?)
Load Balancing > Service Stresses > Design Moment Strength
EXAMPLE – LOAD BALANCING
Determine portion of total dead load balanced by prestressing
wDL = 0.20 klfsuperimposed dead load (10psf*20ft)
wsw = 2.06 klfself weight including tributary width of slab
Load Balancing > Service Stresses > Design Moment Strength
EXAMPLE – LOAD BALANCING
wDL 0.20 klf superimposed dead load (10psf x 20ft)
wsw 2.06 klf self weight including tributary width of slab.
fse 175 ksi effective prestress
Aps 4.28 in2 area of 28 0.5 in. dia. strand
ec 21.19 in eccentriciaty of tendon from section centroid
weq8 fse Aps ec
L2 weq 2.29 klf
weq
wDL wsw101 % prestressing balances full dead load
Including all short and long term losses
Load Balancing > Service Stresses > Design Moment Strength
EXAMPLE – LOAD BALANCING
Prestressing in this example balances ~100% of total dead load.
In general, balance 65 to 100% of the self-weight
Balancing in this range does not guarantee that service or strength limit states will be met. These must be checked separately
Load Balancing > Service Stresses > Design Moment Strength
STRESSES Section remains uncracked Stress-strain relationship is linear for
both concrete and steel Use superposition to sum stress effect
of each load. Prestressing is just another load.
Load Balancing > Service Stresses > Design Moment Strength
CALCULATING STRESSES
Load Balancing > Service Stresses > Design Moment Strength
STRESSES Stresses are typically checked at
significant stages The number of stages varies with the
complexity and type of prestressing. Stresses are usually calculated for the
service level loads imposed (i.e. load factors are equal to 1.0). This includes the forces imposed by the prestressing.
Load Balancing > Service Stresses > Design Moment Strength
PLAIN CONCRETE
Load Balancing > Service Stresses > Design Moment Strength(T)
PLAIN CONCRETE
Load Balancing > Service Stresses > Design Moment Strength(C)
CONCENTRIC PRESTRESSING
Load Balancing > Service Stresses > Design Moment Strength(T)
CONCENTRIC PRESTRESSING
Load Balancing > Service Stresses > Design Moment Strength(C)
ECCENTRIC PRESTRESSING
Load Balancing > Service Stresses > Design Moment Strength(T)
ECCENTRIC PRESTRESSING
Load Balancing > Service Stresses > Design Moment Strength(C)
ECCENTRIC PRESTRESSINGSTRESSES AT SUPPORT
Load Balancing > Service Stresses > Design Moment Strength(T)
ECCENTRIC PRESTRESSINGSTRESSES AT SUPPORT
Load Balancing > Service Stresses > Design Moment Strength(C)
VARY TENDON ECCENTRICITY
Harped Tendon follows moment diagram from concentrated load
Parabolic Drape follows moment diagram from uniformly distributed load
Load Balancing > Service Stresses > Design Moment Strength(T)
VARY TENDON ECCENTRICITY
Harped Tendon follows moment diagram from concentrated load
Parabolic Drape follows moment diagram from uniformly distributed load
Load Balancing > Service Stresses > Design Moment Strength(C)
STRESSES AT TRANSFER - MIDSPANfpi 0.7fpu fpi 189 ksi Effective prestress in tendon
Pi fpi Aps Pi 810 kip
ec 21.19 in
Mmaxwsw L2
8 Mmax 1192 kip ft
ftopPi
A
Pi ec
St
Mmax
St ftop 445 psi Compression
fbottPi
A
Pi ec
Sb
Mmax
Sb fbott 1183 psi Compression
6 f'ci psi 379 psi tension at "end" of member
3 f'ci psi 190 psi tension at other locations
0.7 f'ci 2800 psi compression stress limit at "end" of member
0.6 f'ci 2400 psi compression stress limit at otherlocations
OK
Including friction and elastic losses
Load Balancing > Service Stresses > Design Moment Strength
STRESSES AT TRANSFER – FULL
LENGTHfbp1Pi
A
Pi e
Sb
M w0p Sb
0 17 34 51 680
300
600
900
1200
1500Stress in top fiberStress in bott fiber
Location (ft)
Con
cret
e St
ress
(psi
)
Stress in top fiberStress in bottom fiber
Load Balancing > Service Stresses > Design Moment Strength
STRESSES AT SERVICE - MIDSPAN
fse 175 ksi Effective prestress in tendon
Pe fse Aps Pe 750 kip
ec 21.19 in
Mmaxw L2
8 Mmax 1770 kip ft
ftopPe
A
Pe ec
St
Mmax
St ftop 1047 psi Compression
fbottPe
A
Pe ec
Sb
Mmax
Sb fbott 338 psi Tension
7.5 f'c psi 530 psi limit to be considered uncracked (Class U)
12 f'c psi 849 psi limit to be considered transition (Class T) Class U member
0.45 f'c 2250 psi compression stress limit for sustained loads plus prestress
0.6 f'c 3000 psi compression stress limit for total loadplus prestress
OK
Including all short and long term losses
Load Balancing > Service Stresses > Design Moment Strength
STRESSES AT SERVICE – FULL LENGTH
Stress in top fiberStress in bottom fiberTransition (7.5 root f’c)Cracked (12 root f’c)
Load Balancing > Service Stresses > Design Moment Strength
FLEXURAL STRENGTH (MN)ACI 318 indicates that the design moment strength of flexural
members are to be computed by the strength design procedure used for reinforced concrete with fps is substituted for fy
Load Balancing > Service Stresses > Design Moment Strength
ASSUMPTIONS Concrete strain capacity = 0.003 Tension concrete ignored Equivalent stress block for concrete
compression Strain diagram linear Mild steel: elastic perfectly plastic Prestressing steel: strain compatibility, or
empirical Perfect bond (for bonded tendons)
Load Balancing > Service Stresses > Design Moment Strength
fps - STRESS IN PRESTRESSING STEEL AT
NOMINAL FLEXURAL STRENGTH Empirical (bonded and unbonded
tendons) Strain compatibility (bonded only)
Load Balancing > Service Stresses > Design Moment Strength
EMPIRICAL – BONDED TENDONS
270 ksi prestressing strand
Load Balancing > Service Stresses > Design Moment Strength
EMPIRICAL – BONDED TENDONS
NO MILD STEEL
Load Balancing > Service Stresses > Design Moment Strength
BONDED VS. UNBONDED SYSTEMS
• Steel-Concrete force transfer is uniform along the length
• Assume steel strain = concrete strain (i.e. strain compatibility)
• Cracks restrained locally by steel bonded to adjacent concrete
Load Balancing > Service Stresses > Design Moment Strength
BONDED VS UNBONDED SYSTEMS
• Steel-Concrete force transfer occurs at anchor locations
• Strain compatibility cannot be assumed at all sections
• Cracks restrained globally by steel strain over the entire tendon length
• If sufficient mild reinforcement is not provided, large cracks are possible
Load Balancing > Service Stresses > Design Moment Strength
SPAN-TO-DEPTH 35 OR LESS
SPAN-TO-DEPTH > 35
Careful with units for fse (psi)
Load Balancing > Service Stresses > Design Moment Strength
COMBINED PRESTRESSING AND MILD STEEL
Assume mild steel stress = fy Both tension forces contribute to Mn
Load Balancing > Service Stresses > Design Moment Strength
STRENGTH REDUCTION FACTOR
Applied to nominal moment strength (Mn) to obtain design strength ( Mn)
ranges from 0.6 to 0.9 Determined from strain in extreme
tension steel (mild or prestressing) Section is defined as compression
controlled, transition, or tension controlled
Load Balancing > Service Stresses > Design Moment Strength
STRENGTH REDUCTION FACTOR
Load Balancing > Service Stresses > Design Moment Strength
DETERMINE FLEXURAL STRENGTH
Is effective prestress sufficient? Determine fps Use equilibrium to determine:
Depth of stress block a Nominal moment strength Mn
Determine depth of neutral axis and strain in outside layer of steel (et)
Determine Compute Mn
Load Balancing > Service Stresses > Design Moment Strength
fps OF BONDED TENDON
p 0.28 fpy/fpu > 90 for seven-wire prestressing strand
dp 32.3 in
As 0 A's 0 d 0 assume no mild reinforcement present. Usesimplified version of fps equation.
pAps
b dp p 0.00120 no change from unbonded
fps fpu 1p
1p
fpu
f'c
fps 264 ksi more than fps for unbonded
Load Balancing > Service Stresses > Design Moment Strength
a and
afps Aps
0.85 f'c b a 2.40 in hf 6 in Check that the depth of the stress block is less than
the thickness of the hollow core flange. OK.
strain 0.003 1 dp
a1
strain 0.0293 Strain in the prestressing is greater than 0.005 so thephi factor is set equal to 0.9 (ACI 18.8.1)
Load Balancing > Service Stresses > Design Moment Strength
MN – BONDED TENDON
wu 1.2 wDL 1.2 w0p 1.6 wLL wu 3.99 klf Factored uniform load
Mu18
wu L2 Mu 2309 kip ft Design moment strength is OK
Mn 0.9 fps Aps dpa2
sum moments about resultant compressive force
Mn 2633 kip ft
Load Balancing > Service Stresses > Design Moment Strength
REINFORCEMENT LIMITS
Members containing bonded tendons must have sufficient flexural strength to avoid abrupt failure that might be precipitated by cracking.
Members with unbonded tendons are not required to satisfy this provision.
Load Balancing > Service Stresses > Design Moment Strength
REINFORCEMENT LIMITS
fr 7.5 f'c psi Cracking is considereed to have occurred whenthe net tensile stress exceeds the modulus ofrupture. This occurs when the effectiveprecompression and tensile strength are exceeded.Mcr Sb
Pe
A
Pe ec
Sb fr
1.2 Mcr 2231 kip ft Flexural capacity exceeds 1.2 times the crackingmoment. OK.Mn 2633 kip ft
Load Balancing > Service Stresses > Design Moment Strength
fps – UNBONDED TENDON
0.5fpu 135 ksi fse 175 ksi effective prestress is sufficient to allow use of empirical eqn.
1 0.8 rectangular stress block factor. relates depth of stress block to depth of NA
dp yt ec dp 32.3 in effective depth of prestressing steel
pAps
b dp p 0.00120
span-to-depth is less than 35. Use ACI eqn 18-4Lh
22.7
fse 10ksif'c
100 p 227 ksi
fse 60ksi 235 ksi
fpy 0.9 fpu fpy 243 ksi Equation 18-4 controls effective prestress at strength
Load Balancing > Service Stresses > Design Moment Strength
Mn – UNBONDED TENDON
afps Aps
0.85 f'c b a 2.06 in hf 6 in rectangular assumption OK
strain 0.003 1 dp
a1
strain 0.0346 strain > 0.005 phi factor is set equal to 0.9
Mn 0.9 fps Aps dpa2
Mn 2275 kip ft
Mu 2309 kip ft not quite sufficient. Consider minimumbonded steel
Load Balancing > Service Stresses > Design Moment Strength
MIN. BONDED REINF. Members with unbonded tendons must
have a minimum area of bonded reinf. Must be placed as close to the tension
face (precompressed tensile zone) as possible.
As = 0.004 Act
Act – area of section in tension
Load Balancing > Service Stresses > Design Moment Strength
AS MINIMUMAct yb bw Act 448.9 in2
AsMin 0.004 Act AsMin 1.80 in2
SixNo5 6As5 SixNo5 1.86 in2
Add six #5 bars to flexural tension face of beam
d h 1.5in db3 0.5db5 d 33.8 in
Load Balancing > Service Stresses > Design Moment Strength
Mn – UNBONDED TENDONINCORPORATE MILD STEEL
afps Aps fy AsMin
0.85 f'c b a 2.29 in hf 6 in
strain 0.003 1 d
a1
strain 0.0325
Mn 0.9 fps Aps dpa2
fy AsMin da2
Mn 2531 kip ft
Load Balancing > Service Stresses > Design Moment Strength