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DurabilityDurability--design of predesign of pre--cast cast concrete structural membersconcrete structural members
PhD thesisPhD thesis
KKáálmlmáán KORISn KORISBME Faculty of Civil EngineeringBME Faculty of Civil Engineering
Department of Structural EngineeringDepartment of Structural Engineering
Budapest University of Technology and EconomicsDepartment of Structural Engineering
Research objectivesResearch objectives
The importance of durability of structures is continuously increasing due to economical reasons.
Main objective of the research was to provide a calculation method for the durability-design of pre-cast concrete members, that is:
11//2222
Based on state of the art probabilistic approach with arbitrary number of stochastic parametersConsidering the loads as a function of timeConsidering the decrease of load carrying capacity (Fu)in time due to slow deformations, degradation of structural geometry and aging of materialsFast and accurate enough for practical applicationResults are easy to use for practicing designers
Why preWhy pre--cast concrete members?cast concrete members?
The research was focusing on the analysis of prefabricated prestressed concrete girders because:
2/2/2222
They are widely used for the construction of residential houses and industrial buildings
The stochastic characteristics of process parameters (mean values and standard deviations of structural geometry and material properties) can be obtained from the results of quality control
Manufacturing conditions of the members can be more precisely controlled and modified if necessary
Probabilistic approach for the durabilityProbabilistic approach for the durability--designdesign
Durability: The structure is durable enough (functionsproperly) if the desired level for the probability of failure is not exceeded during its life-span.
3/3/2222
Probability of failure: pG = prob[(R-S)<0]The probability that the load effect (S) is exceeding the structural resistance (R).
Changes of the stochastic distribution of external loads (S) and the structural resistance (R) in time causing the increase of failure probability.
Applied calculation methodApplied calculation method
Mean value of structural resistance (Fu,m) was calculated by Finite Element Method (FEM).
4/4/2222
1 ni Beam elements (compression and bending)
Mean values of structural dimensions and material properties were used for the calculationNon-linear material model was used to describe the post-cracking behavior of concreteThe applied load was single-parameter load.The value of the load-intensity (Φ) was increased in steps until structural failure (crushing of concrete or splitting of steel bars) occurred.
F - load-intensityΦ - load distribution vector F
Fi = F·ΦiCurvature
Form
ing
of c
rack
s
Ben
din
g m
om
ent
chord stiffness
Mx
κx
Applied calculation methodApplied calculation method
Standard deviation of structural resistance (sFu,νFu) was calculated by Stochastic Finite Element Method (SFEM).
5/5/2222
The variation of stiffness matrix (δK) was approximately expressed by first order partial derivative of K with respect to an “x” random input variable:
δK= sx where sx is the standard deviation of x.
The standard deviation of structural resistance was evaluated from:
Cq = δqδqT = K-1 uδx CrδxTuT K-T
q – load vector including load intensity (F) u – nodal displacementsK – stiffness matrix Cr – correlation matrixK – stiffness matrix including F Cq – covariance of load vectorδx – matrix including standard deviations of random input variables
∂x
∂K
∂α∂K
∂α∂KT
~ ~ ~ ~
~
~
Previous studies on the field of probabilistic designPrevious studies on the field of probabilistic design
Ran
do
m s
tru
ctu
ral
geo
met
ry
Ran
do
m m
ater
ial
pro
per
ties
Ran
do
m l
oad
eff
ect
No
n-l
inea
r m
ater
ial
beh
avio
r
Ch
ang
e o
f g
eom
etry
and
mat
eria
l p
rop
erti
es i
n t
ime
Ch
ang
e o
f lo
ad e
ffec
t in
tim
e
Car
bo
nat
ion
in
du
ced
co
rro
sio
n o
f st
eel
bar
s
Handa & Andersson (1975) SFEM1
Almási (1978) FEM + MCS
Liu, Besterfield & Belytschko (1988) SFEM1
Dasgupta & Yip (1989) SFEM1
Besterfield, Liu & Lawrence (1990) SFEM1
Deodatis (1990) SFEM1
Teigen, Frangopol, Sture, Felippa (1991) SFEM1
Ruiz & Aguilar (1994) MCS
Eibl &Schmidt-Hurtienne (1995) SFEM1
Bergmeister, Novák & Pukl (2004) FEM + MCS
Krätzig & Petryna (2004) SFEM1
Koris (2008) SFEM1
AppliedmethodAuthor
Effects considered
1 Different authors used different mathematical approach and formulation.FEM – Finite Element Method, SFEM – Stochastic Finite Element Method, MCS – Monte-Carlo Simulation
6/6/2222
Initial values of process parametersInitial values of process parameters
Concrete strength:Compression tests on 150x150x150 mm cubesAltogether 732 specimens tested at age of 28 days5 different concrete classes
Steel bar strength:Tensile tests on altogether 291 specimens3 different classes, 9 different diameters
Prestressing tendon strength:Tensile tests on 20 specimens2 different types
Geometry of the cross-sections was measured on manufactured beams.
7/7/2222
Strength of concrete at 28 days
52,5 56,0
0
20
40
60
C40/50 C50/60
Expected concrete class
Mea
n va
lue
[N/m
m2 ]
Strength of concrete at 28 days
6,4
4,6
0
2
4
6
C40/50 C50/60
Expected concrete class
Rel
ativ
e st
anda
rd d
evia
tion
[%]
Products of 7 different Hungarian companies were considered during the determination of material properties.
Evaluation of process parameters as a function of timeEvaluation of process parameters as a function of time
Effects considered:Loss of initial prestress σp0 (Eurocode 2: EN 1992-1-1)
Increase of standard deviation of geometrical sizes (Mistéth, 2001)
Decrease of mean value of material strength (concrete, steel bars, prestressing tendons) (Mistéth, 2001)
Increase of standard deviation of material strength (Mistéth, 2001)
Carbonation of concrete ( fib bulletin 34. “Model Code for Service Life Design”, 2006)
Carbonation induced corrosion of steel bars and tendons (D. Zao & L. Fan, 2007)
Change of mean value and standard deviation of loads (Mistéth, 2001)
8/8/2222
New result #1.New result #1.
I developed a method based on probabilistic approach for the durability-design of prefabricated concrete structural members. This method can predict the probability of failure of the members at any given point of time. The deterioration of material strengths and geometrical sizes, the effect of carbonation induced corrosion as well as the change of load effect in time can be taken into account during the analysis. Random input parameters that are considered by the developed method are the strength of concrete, steel bars and prestressing tendons, the height and width of cross section, effective height of steel bars and tendons and the load effect.
9/9/2222
Results of analyses / Verification Results of analyses / Verification
Cross-section (EE-42)bf
ba
ha
hfap
b
hdp
Concrete: C40/50Prestressing wires: d = 5mm 1770/1540
L
L/5 L/5L/5L/5L/5
F/4 F/4 F/4 F/4
Test arrangement
Lb
Bending tests on prestressed EE beams:
4 different beam types (different size and reinforcement)
Altogether 26 beams were tested
Type of beamL
[m]Number of wires
Lbm
[m]hm
[mm]bf,m
[mm]ba,m
[mm]ap,m
[mm]dp,m
[mm]Fu,m
[kN]
EE-42 4,27 1+4 4,40 189,8 80,8 144,1 37,8 168,5 49,28EE-48 4,87 1+6 5,01 195,2 80,6 145,4 30,8 175,4 53,93EE-54 5,47 1+6 5,64 196,1 81,7 144,4 39,5 176,5 51,60EE-66 6,67 1+6 6,85 197,1 79,8 142,3 45,7 175,3 47,50
Type of beam L[m]
Number of wires
νLb
[%]νh
[%]νbf
[%]νba
[%]νap
[%]νdp
[%]νFu
[%]
EE-42 4,27 1+4 0,171 1,60 3,13 1,59 18,21 1,73 6,92EE-48 4,87 1+6 0,224 1,86 1,30 1,49 19,48 0,86 3,86EE-54 5,47 1+6 0,119 2,72 1,96 1,98 11,32 2,07 6,32EE-66 6,67 1+6 0,084 2,27 0,48 0,23 4,42 0,92 6,93
Average of measured values:
Relative standard deviation of measured values:
10/10/2222
Bending test result
[%]
SFEM analysis
[%]
Difference between test and calculation [%]
EE-42 6,92 6,81 -1,6EE-48 3,86 4,20 8,8EE-54 6,32 6,08 -3,8EE-66 6,93 7,25 4,6
νFu
Type of beam
Comparing the relative standard deviation of ultimate load (νFu) derived from bending tests and from numerical analysis:
Results of analyses / Verification II.Results of analyses / Verification II.
6,32
6,92
3,86
6,936,81
4,20
6,08
7,25
3,5
4,5
5,5
6,5
7,5
EE-42 EE-48 EE-54 EE-66Type of beam
Rel
ativ
e st
and
ard
dev
iati
on
of
ult
imat
e lo
ad [
%]
Bending test result SFEM analysis
11/11/2222
Results of analyses / Verification III.Results of analyses / Verification III.
The effect of the standard deviation of different input parameters on the standard deviation of ultimate load (νFu).
Beam type EE-42
6,0
6,5
7,0
7,5
8,0
8,5
9,0
9,5
10,0
10,5
11,0
0,25·s 0,5·s 0,75·s s 1,5·s 2·s 3·s
⎯F
u [
%]
width of cross section
height of cross section
effective depth of wires
strength of concrete
strength of wires
Beam type EE-48
3,5
4,0
4,5
5,0
5,5
6,0
6,5
7,0
7,5
0,25·s 0,5·s 0,75·s s 1,5·s 2·s 3·s
⎯F
u [
%]
width of cross section
height of cross section
effective depth of wires
strength of concrete
strength of wires
12/12/2222
New result #2.New result #2.I compared the standard deviation of load carrying capacity of prestressed concrete beams derived from bending test results to the results of numerical analyses. I proved by this comparison that the developed method is appropriate for the analysis of the standard deviation of load carrying capacity in case of pre-cast concrete structural members. I demonstrated that results ofthe numerical analysis can be used for practical purposes if the values of input parameters are derived from material test results and geometry measurements on the corresponding members.
13/13/2222
New result #3.New result #3.
I performed parametric numerical analyses on pre-cast, prestressed concrete beams with the following results:
a.) I determined the effect of standard deviations of different input parameters (height and width of cross-section, effective depth of tendons, strength of concrete and tendons) on the standard deviation of load carrying capacity of examined beams. The influence of the standard deviations of effective depth and concrete strength is the most significant, while change of the standard deviation of tendon strength has the least influence. (parts b,c and d will follow on page 21)
Results of analyses / Durability of longResults of analyses / Durability of long--span girdersspan girders
The durability of two pre-cast members was analyzed by the implemented method.
14/14/2222
60
30
22
14
6
145
8
5
51
3×4
5 3
ID. Nr. “4000”Length: 28,782 mHeight: 1,45 mFunction: main girder
35
7,5
74,9
53
12,5
2
67,4
3×4
ID. Nr. “4700”Length: 6,06 mHeight: 0,749 mFunction: supporting main girders
Reinforcement of beam “4000”
Results of analyses / Durability of longResults of analyses / Durability of long--span girders II.span girders II.
Beam “4000” Beam “4700”
Values measured on 10 beams:Number of
strands L hm bm
Mean value[mm] 6060 752 351
Standard deviation [%] 0,070 0,435 0,4512+12
L
F=g+q
Values measured on 11 beams:Number of
strands L hm bf,m
Mean value[mm] 27691 1442 604
Standard deviation [%] 0,028 0,533 1,1772+16
L
F=g+q
Initial values of input parameters:
Concrete: C40/50Steel bars: 4Ø16+2Ø16 B60.50Prestressing strands: 8+2 Fp-100/1770-R2
Concrete: C40/50Steel bars: 4Ø20+2Ø20 B60.50Prestressing strands: 16+2 Fp-100/1770-R2
15/15/2222
Results of analyses / Durability of longResults of analyses / Durability of long--span girders III.span girders III.
To perform the calculations, I developed a computer software (PFEM2008) using the Matlab® mathematical software package.
Conditions that were changed during the analysis:
Time: t = 10, 25, 50, 75 and 100 yearsRelative ambient humidity:
– RH = 50%, 65% and 80%Initial value of imposed load (t=0):
– q0 = 16, 18, 20, 22 and 24 kN/m (beam “4000”)– q0 = 115, 120, 125, 130 and 135 kN/m (beam “4700”)
Number of runs was 5×3×5 = 75 for each beam.
16/16/2222
Results of analyses / Durability of longResults of analyses / Durability of long--span girders IV.span girders IV.
Change of mean value and relative standard deviation of material strength in case of beam “4000”:
17/17/2222
Relative standard deviation of material strength
1
2
3
4
5
6
7
8
10 25 50 75 100
Time [years]
f [%
]concrete
reinforcing steel
prestressing tendons
Change of concrete strength
51,4
51,6
51,8
52,0
52,2
52,4
52,6
10 25 50 75 100
Time [years]
fp,m
[N
/mm
2 ]
Change of tendon strength
1890
1900
1910
1920
1930
1940
10 25 50 75 100
Time [years]
fp,m
[N
/mm
2 ]
1100
1150
1200
1250
1300
10 30 50 70 90
Time [years]
As
[mm
2 ]RH=50%
RH=65%
RH=80%
0,30
0,45
0,60
0,75
0,90
1,05
1,20
1,35
1,50
10 25 50 75 100
Time [years]
b ,
h ,
d [
%]
height of cross sectionwidth of cross sectioneffective height
Results of analyses / Durability of longResults of analyses / Durability of long--span girders V.span girders V.
Change of relative standard deviation of structural sizes (height, width, effective height) in case of beam “4000”
Change of cross-sectional area of steel bars due to corrosion in time at different relative humidity levels in case of beam “4000”
18/18/2222
4
5
6
7
8
9
10
10 20 30 40 50 60 70 80 90 100
Time [years]
⎝p
u [
%]
RH=50%
RH=65%
RH=80%
19/19/2222
51,5
52,0
52,5
53,0
53,5
54,0
54,5
55,0
10 20 30 40 50 60 70 80 90 100
Time [years]
Fu
,m [
kN/m
]
RH=50%
RH=65%
RH=80%
Change of mean value of structural resistance in time at different relative humidity (RH) levels in case of beam “4000”
Change of relative standard deviation of structural resis-tance in time at different RH levels in case of beam “4000”
Results of analyses / Durability of longResults of analyses / Durability of long--span girders Vspan girders VII..
Results of analyses / Durability of longResults of analyses / Durability of long--span girders VII.span girders VII.
Change of the probability of failure of beam type “4000” in case of different initial imposed loads (q0) and relative humidity (RH) levels.The probability of failure is:
• increasing as time is passing by,• increasing as the level of relative humidity
(RH) is increasing,• increasing as the initial value of imposed load
(q0) is increasing.
20/20/2222
1,00E-16
1,00E-14
1,00E-12
1,00E-10
1,00E-08
1,00E-06
1,00E-04
1,00E-02
1,00E+00
10 20 30 40 50 60 70 80 90 100
Time [years]
Pro
bab
ility
of
failu
re
q0=16 kN/mq0=18 kN/mq0=20 kN/mq0=22 kN/mq0=24 kN/m
1,00E-18
1,00E-16
1,00E-14
1,00E-12
1,00E-10
1,00E-08
1,00E-06
1,00E-04
1,00E-02
10 20 30 40 50 60 70 80 90 100
Time [years]
Pro
bab
ility
of
failu
re
q0=16 kN/mq0=18 kN/mq0=20 kN/mq0=22 kN/mq0=24 kN/m
RH=50%
1,00E-12
1,00E-10
1,00E-08
1,00E-06
1,00E-04
1,00E-02
1,00E+00
10 20 30 40 50 60 70 80 90 100
Time [years]
Pro
bab
ility
of
failu
re
q0=16 kN/mq0=18 kN/mq0=20 kN/mq0=22 kN/mq0=24 kN/m
RH=65% RH=80%
New result #3. (continued)New result #3. (continued)
b.) I proved that the failure probability of pre-cast, prestressed concrete beams is increasing as time is passing by; it is increasing as the level of relative humidity is increasing and it is increasing as the initial value of imposed load is increasing. I demonstrated the increase rate of failure probability as a function of different parameters graphically. Durability-design of examined girders can be performed by the presented charts.
c.) I proved that the application of the presented method results in a more economic design (higher load carrying capacity or smaller member sizes) of the examined pre-cast, prestressed concrete beams than the use of the relevant Eurocode 2 standard.
d.) I proved that the presented method can be efficiently applied for the durability analysis of existing pre-cast, prestressed concrete members using geometry measurements and material tests on the examined members.
21/21/2222
AcknowledgementsAcknowledgements
I would like to thank prof. Kálmán Szalai for aiming my attention to this research topic, my supervisor prof. IstvánBódi for his help, prof. György Farkas, Head of Department of Structural Engineering and all my colleagues at the Department who supported my work, prof. Josef Eibl at Technical University Karlsruhe and prof. StanisławMajewski at Silesian University who helped me while working abroad, the manufacturing companies who provided data for the analyses and finally my family for being patient with me during the long period of research.
22/22/2222
Thank you for your attention!Thank you for your attention!