cven9806 lecture 2 - serviceability 2009
TRANSCRIPT
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
1/26
1
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
2/26
2
CVEN9806 Prestressed Concrete 3
Prestressing force P and the cable profile are based on
Serviceability Requirements:
- Crack Free Design limit stresses
- Control Deflections
- Camber
- Axial Shortening
For Crack Control need to specify Stress Limits.
CVEN9806 Prestressed Concrete 4
Short Term: sG Q P + +
Long Term: lG Q P + +
- Dead Load - Live Load
- Short Term Live Load Factor
- Long Term Live Load Factor
s
L
G Q
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
3/26
3
CVEN9806 Prestressed Concrete 5
time
loading
G
sQ
lQ
CVEN9806 Prestressed Concrete 6
Type of Live Load Short-Term Factors Long-Term Factor l
Floors
Domestic 0.7 0.4
Offices 0.7 0.4Parking 0.7 0.4
Retail 0.7 0.4
Storage 1.0 0.6
Other As per storage, unless
Assessed otherwise
As per storage, unless
Assessed otherwise
Roofs
Trafficable 0.7 0.4
Non-Trafficable 0.7 0.0
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
4/26
4
CVEN9806 Prestressed Concrete 7
Table 2.4.2 AS3600
Notes:
1. In flat slabs, the deflection to which the above limits apply is the theoretical deflection of the line diagram representing the
idealized frame defined in Clause 7.5.2
1. Deflection limits given may not safeguard against ponding
2. For cantilevers, the values of/Lefgiven in this table apply only if the rotation at the support is included in the calculation of .
CVEN9806 Prestressed Concrete 8
0.2% proofstress
True curve
Approx. curve
Strain
Stress
fpy
fpu
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
5/26
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
6/26
6
0.8 fpu pretensioned tendons
0.85 fpu stress-relieved post-
tensioned tendons
0.75 fpu not stress-relieved post-tensioned tendons
CVEN9806 Prestressed Concrete 11
CVEN9806 Prestressed Concrete 12
During the stressing operation, immediate losses can
occur by:
elastic contraction of the concrete, friction along the ducts and slip and deformation in the end anchors.
Estimate ~ 8% Pi ~ 0.92 Pj
Pj
jacking prestress
Pi
initial prestress
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
7/26
7
CVEN9806 Prestressed Concrete 13
inelastic creep and shrinkage strains at the level of thebounded steel.
stress relaxation in the tendons.
Pi
initial prestress
Pe
effective prestress
Estimate ~ 15-20% Pe ~ 0.8 Pi
CVEN9806 Prestressed Concrete 14
The initial stress level in prestressing steel after transfer is usually high,
often in the range 60-75% of the tensile strength of the material.
If a tendon is stretched and held at a constant length (constant strain), the
development of creep strain in the steel is exhibited as a loss of elastic strain,
and hence a loss of stress.
Relaxation in steel is highly dependent on the stress level
and increases at an increasing rate as the stress level increases.
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
8/26
8
CVEN9806 Prestressed Concrete 15
-
+
(a)
-
+
(b) (c)
-
(d)
-
(e)
-
Transfer Full Service Load Condition Ultimate Load
Condition
CVEN9806 Prestressed Concrete 16
-
+
Prestress Cable
Immediately after application of Prestress
Pi - initial prestress
Pi
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
9/26
9
CVEN9806 Prestressed Concrete 17
-
+ -
+
-
Prestress Cable
After full service loads and Time Dependent
Losses
Pe - effective prestress
Pe
CVEN9806 Prestressed Concrete 18
Concrete Compressive Stress Limit: Fci= 0.5 f ci Large nonlinear creep strains.
Large prestress losses Safety against brittle failure (should check strength at
transfer)
Tensile Stress Limit: Fti =
advisable especially in unreinforced zones
cracks may not close completely (local spalling)
if cracking allowed add conventional steel
'0.25 cif
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
10/26
10
CVEN9806 Prestressed Concrete 19
Concrete Compressive Stress Limit: Fc= 0.45 f c usually not enforced.
prevents large creep strains
Tensile Stress Limit: Ft =
upper limit some cracking due to shrinkage, may need
added reo.
' '0.25 to 0.5c cf f
No Limits needed unless Cracking
To be avoided
CVEN9806 Prestressed Concrete 20
Fti tensile stress limit at transfer
Ft tensile stress limit under fullload
Fci compressive stress limit at transfer
Fc compressive stress limit under fullload
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
11/26
11
CVEN9806 Prestressed Concrete 21
TRANSFER
FULL
SERVICELOADS
These limits are not explicitly suggested in AS3600
but are generally used for most prestressed designs.
cc fF'45.0==== compression
citi fF'25.0==== tension
ct fF'5.0==== tension
'0.5c i ciF f= compression
AS3600 Clause 8.1.4.2
CVEN9806 Prestressed Concrete 22
If Cracking Permitted - CHECK CRACK WIDTHS
Increment in Steel Stress near the Tension Face< 200 MPa as the load increases from its valuewhen the extreme tensile fibre is at zero stressto the full short-term service load
c/c spacing 200 mm for beams; 500 mm for slabs
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
12/26
12
CVEN9806 Prestressed Concrete 23
C A
ye
yt
yb
-
+
SECTION
STRESSES
ResultantDue to Pi(prestress force)
++++
I
eyP
A
P ii
Due to Mo(w, applied load)
I
yMo
8
2wLMo ====
-
+ t
b
=+
Mo
-
+
CVEN9806 Prestressed Concrete 24
(((( ))))ti
t
oii FZ
MeP
A
P
++++====
ti
totii
t
FI
yM
I
eyP
A
P++++====
Axial
stress
Bending
(due to Pi)
Bending
(due to Mo)
tensile
stress limit
at transfer
t
o
t
iti
Z
M
Z
Ae
A
PF
1
8
2wLMo ====
TENSILE STRESS LIMIT AT TRANSFER
(1)
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
13/26
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
14/26
14
CVEN9806 Prestressed Concrete 27
tbTbii
b FI
yM
I
eyRP
A
RP++++====
Axial
stress
Bending
(due to Pi)
Bending
(due to MT)
Bottomfibrestress
b
T
b
it
Z
M
Z
Ae
A
RPF ++++
++++ 1
TENSILE STRESS LIMIT
(3)b
by
IZ ====
CVEN9806 Prestressed Concrete 28
i i t T t
t c
R P R P e y M y
FA I I = +
Axial
stress
Bending
(due to Pi)
Bending
(due to MT)
Topfibrestress
1i Tct t
R P MA eF
A Z Z
tt
y
IZ ====
COMPRESSIVE STRESS LIMIT
(4)
due to Pi due to w
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
15/26
15
CVEN9806 Prestressed Concrete 29
Equations (1) (4) can be rearranged to express 1/Pi
as a linear function ofe.
Equation (1) gives:
++++
t
i
t
oti
Z
AeP
Z
MFA 1
or(((( ))))toti
t
i ZMFA
ZAe
P /
/11
++++
++++
t
ty
IZ ====
b
by
IZ ====
CVEN9806 Prestressed Concrete 30
Ift
tZ
A====
b
bZ
A====
(((( ))))
ttc
t
i MAF
eR
P
++++
11(4)
(((( ))))tbt
b
i MAFeR
P
++++++++ 11(3)
obci
b
i MAF
e
P
++++
++++
11(2)
otti
t
i MAFe
P
++++ 11(1)
Fti , Fci - TRANSFER Ft , Fc FULL LOADING
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
16/26
16
CVEN9806 Prestressed Concrete 31
-1/b 1/t Eccentricity, e
1/Pi Equation 1
Equation 3
Equation 2
Equation 4
acceptable region
cF
ciF
tiF
tFminimumiP
maxe
In order to minimise prestressing costs,
the smallest possible value ofPi would generally be selected.
eLimit
CVEN9806 Prestressed Concrete 32
b b
c i b o t b t
R
A F M A F M
=
+ +
( ) t obt c i
M R MZ
F R F
If section too small line 2 will lie above line 3
When slopes are equal:
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
17/26
17
CVEN9806 Prestressed Concrete 33
otti
t
i MAF
e
P
++++
11(1)
(((( ))))
tbt
b
i MAF
eR
P
++++
++++
11(3)=
1L i m i t
t b
ae
a
+=
( )ti t o
t b t
R A F M a
A F M
+=
+
Equation 1 = 3
CVEN9806 Prestressed Concrete 34
When the prestressing force and eccentricity are determined for the
critical section, the location of the cable at every section
along the member must be specified.
For a member which has been designed to be uncracked throughout,
the tendons must be located so that
the stress limits are observed on every section.
At any section, Equations (1) (4) may be used to establish a range of
values for eccentricity which satisfy the selected stress limits.
IfMo and MT are the moments caused by the external loads at transfer and
under full service loads, respectively, and Pi and Pe are the corresponding
prestressing forces at the same section, the extreme fibre stresses must satisfy
Equations (1) - (4).
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
18/26
18
CVEN9806 Prestressed Concrete 35
After Pi and Pe have been determined at the critical sections,
the friction losses along the member are estimated and the corresponding
prestressing forces at intermediate sections are calculated.
At each intermediate section, the maximum eccentricity that will satisfy
both stress limits at transfer is obtained from either Equation (1) or (2).
The minimum eccentricity required to satisfy the tensile and compressive
stress limits under full loads is obtained from either Equation (3) or (4).
A permissible zone is thus established in which the line of action of the
resulting prestressing force must be located.
CVEN9806 Prestressed Concrete 36
Permissible zone
Equation (3) and (4)Equation (1) and (2)
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
19/26
19
CVEN9806 Prestressed Concrete 37
When the prestress and eccentricity at the critical sections are
selected using the load-balancing approach,
the cable profile should match, as closely as practicable,
the bending moment diagram caused by the balanced load.
In this way, deflection will be minimised.
For cracked, partially prestressed members, Equations (1) and (2)are usually applicable and fix the maximum eccentrictity.
The cable profile should then be selected according to the loading
type and moment diagram.
CVEN9806 Prestressed Concrete 38
A one-way slab is simply supported over a span of 12 m and is
to be designed to carry a service load of 7 kPa (kN/m2) in
addition to its own self-weight. The slab is post-tensioned by
regularly spaced tendons with parabolic profiles.The slab thickness D = 300 mm.
The material properties are:
MPa25' ====cif
MPa25300====ciE
MPa32' ====cf
MPa28600====cE
MPa1840====pf
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
20/26
20
CVEN9806 Prestressed Concrete 39
The prestressing force and eccentricity are to be determined to
satisfy the following concrete stress limits:
MPa25.12525.0 ========tiF
MPa5.12255.0 ========ciF
MPa41.13225.0 ========tF
MPa0.16325.0 ========cF
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
CVEN9806 Prestressed Concrete 40
At mid-span, the instantaneous and time-dependent losses are
taken to be 8% and 16%, respectively.
Slab self-weight(which is the only load other than the prestress at transfer):
kN/m2.73.024 ========sww (1 m wide strip)
and the moments at mid-span
both at transfer and under the full service load are:
kNm/m6.1298
122.7 2====
====oM
(((( ))))kNm/m6.255
8
122.70.7 2====
++++====TM
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
0.94R =
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
21/26
21
CVEN9806 Prestressed Concrete 41
Cross-section properties:
/mmm10300 23====A /mmm10225046
====I
/mmm101536
============ bt ZZZ
02.0/ ============ ZAbt
mm5011
========bt
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
CVEN9806 Prestressed Concrete 42
( ) t obt c i
M R MZ
F R F
( )( )
( )
6 3 6
6 3 6 3
255.6 0.94 129.615 10 mm /m 10
1.41 0.94 12.5
15 10 mm /m 10.2 10 mm /m
b
b
Z
Z
=
=
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
22/26
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
23/26
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
24/26
24
CVEN9806 Prestressed Concrete 47
If 12.7 mm diameter strand is used with 30 mm concrete
cover, then
mm11436150max ====e
And from the Design Diagram, or Equation (3), the
corresponding minimum permissible value ofPi is found to be
610588.0
1 ====iP
kN/m1700==== iPand
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
CVEN9806 Prestressed Concrete 48
At the jacking point, the required prestressing force is
kN/m185092.0
1700========jP
(8% instantaneous losses)
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
25/26
25
CVEN9806 Prestressed Concrete 49
From table 2.1, a 12.7 diameter 7-wire,
low-relaxation strand has a cross-sectional area of 100 mm2
and a minimum breaking load of 184 kN.
A flat duct containing four 12.7 mm strands
can therefore be stressed with a maximum jacking force of
kN626184485.0 ====For design purposes, the yield strength of stress-relieved wires
may be taken as 0.85 times the minimum tensile strength
(i.e. 0.85fp)
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
CVEN9806 Prestressed Concrete 50
The minimum number of cables
required in each metre width of slab is therefore:
96.2626/1850 ====
and the maximum spacing between cables is
mm33896.2/1000 ====
Therefore use 1-4 strand tendon every 330 mm.
Example 1Example 1
(Gilbert & Mickleborough Ex. 3.1)(Gilbert & Mickleborough Ex. 3.1)
-
8/7/2019 CVEN9806 Lecture 2 - Serviceability 2009
26/26
CVEN9806 Prestressed Concrete 51
Use the following design strategy:
Set the tensile stress to zero at top & the compressive stress to 0.6 fcp at btmat mid-span (include SW).
i i swtop
t t
i i swbtm
tm btm
P Pe M
A Z Z
P Pe M
A Z Z
= +
= +
Solve the two stress equations for P & e. Determine no. of cables and adjust the jacking load. Determine the Cable profile: e at midspan & zero at ends. (use straight cables) Distribute the cables along the span to maintain e
Show the cable layout through the span.
Pretensioned Beam Design Strategy