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!