Models for resistance Uncertainties originate from • • •

Variability in material properties f Variability in dimensions a Model uncertainties C

R = C ⋅ f ⋅a where C, a and f usually are assumed to be lognormally distributed

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Konstruktionsteknik, LTH

The assumption of log-normal distribution means that •The mean is

μ R = μC μ a μ f •The coefficient of variation VR is

VR = VC2 + Va2 + V f2

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Konstruktionsteknik, LTH

Simulation of structural resistance Reinforced concrete beam-columns

Basic variables fc, fs, As, As´, b, d, h … 3

Konstruktionsteknik, LTH

Model uncertainties

Y = f ( X 1 , X 2 ..... X n ) Y ′ = θ ⋅ f ( X 1 , X 2 ..... X n )

Model ”Reality”

θ is a random variable describing model uncertainty

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Konstruktionsteknik, LTH

Fig 3.9.1 i JCSS

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Konstruktionsteknik, LTH

Modelling uncertainties - shear of concrete beams

Hedman & Losberg 1975 Tests

Verification of empirical shear capacity model

6

Model LTH Konstruktionsteknik,

Statistiska data för modellosäkerhet enligt JCSS

7

Konstruktionsteknik, LTH

Uncertainties in dimensions • A number of investigations can be found in the literature Basic format

Y = X − X nom where Y is a random variable describing deviations from nominal dimensions

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Konstruktionsteknik, LTH

Geometrisk variation, stålprofiler

9

Konstruktionsteknik, LTH COV≈ 5%

Deviations Y in external dimensions for concrete elements Mirza & MacGregor, 1979

Type of dimension

In situ

μY, mm

Precast

σY, mm

μY, mm

σY, mm

Slab thickness

+1

12

0

5

Beam depth

-3

6

+3

4

Beam width

+2.5

5

0

5

Rectangular column, width

+1.5

6

+1

3

Circular column, diameter

0

5

0

2.5

10

Konstruktionsteknik, LTH

Deviations Y in concrete cover and effective depth for reinforcement in concrete elements, Mirza & MacGregor, 1979 Type of dimension

In-situ

Precast

μY, mm

σY, mm

μY, mm

σY, mm

Slabs, top reinforcement •Concrete cover •Effective depth

20 -20

20 15

0 0

5.5 2.5

Slabs, bottom reinforcement •Concrete cover •Effective depth

9 -8

10 16

0 0

5.5 2.5

Beams, top reinforcement •Concrete cover •Effective depth

3 -6

16 17.5

0 3

8 9

Beams, bottom reinforcem. •Concrete cover •Effective depth

1.5 -5

11 12.5

0 3

8 8

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Konstruktionsteknik, LTH

Material properties – steel (JCSS) Property

Mean value, E[⋅]

COV

fy

fy,nom⋅α⋅exp(k⋅COVfy) - C

0.07

fu

B⋅E[fy]

0.04

E

Enom

0.03

εu

εu,nom

0.06

Within-batch COV:s can be taken as ¼ of the values in the table. k depends on the statistical definition of nominal value (1,64 if 5th percentile) α and B depends on steel type and product type C difference between test in mill and static yield strength ≅ 20 MPa 12

Konstruktionsteknik, LTH

Correlation matrix –structural steel

13

Konstruktionsteknik, LTH

Reinforcing steel The yield stress X can be seen as the sum of three independent normal distributed variables

X = X1 + X2 + X2 where

X 1 ∈ N ( μ1 (d ), σ 1 )

global mean

X2 ∈N(0,σ2)

variation between batches in a mill

X 3 ∈ N (0, σ 3 )

variation within a batch

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Konstruktionsteknik, LTH

Variability of reinforcing steel σ1 = 19 Mpa (global variation of mean) σ2 = 22 Mpa

(between batch variation)

σ3 = 8 MPa (within batch variation) σtot = 30 Mpa

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Konstruktionsteknik, LTH

Effect of bar diameter, Degerman, 1981 Mean yield strength for reinforcement 800

700 Ks 60S Ks 60

MPa

600

Ks40 500

Ks40S 400

300 5

10

15

20

25

Bar diameter, mm 16

30

Konstruktionsteknik, LTH

35

Prestressing steels

Prestressing steels are usually defined through ultimate tensile strength fp 17

Konstruktionsteknik, LTH

Mechanical properties for prestressing steel, JCSS Variable

Mean

Standard dev.

COV

Reference

fp

1.04 fpk or fpk + 66 [MPa]

40 MPa

0.025 -

Mirza et al (1980) FIP (1976)

Ep

200 GPa (wires) 195 GPa (strands) 200 GPa (bars)

-

0.02

Mirza et al (1980)

εpu

0.05

0.0035

-

Mirza et al (1980)

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Konstruktionsteknik, LTH

Concrete properties Material description of concrete is complex • A number of different strength parameters are needed • Material properties change with age • Material properties depend on size and type of test specimen • Properties in-situ differ from those of standard test specimens • In-situ properties depends on position in structure

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Konstruktionsteknik, LTH

Definitions • Standard compressive strength is the measured compressive strength of a standard test specimen which is sampled, made, cured and tested in accordance with standardised methods • Core compressive strength is the measured compressive strength of a core taken from the structure • In-situ compressive strength in a structure is expressed as the strength of a standard test specimen. 20

Konstruktionsteknik, LTH

Standard testing • Cylinders with length 300 mm and diameter 150 mm (strength fc,cyl) • 150 mm cubes (strength fc,cube)

f c,cyl ≈ 0.8 ⋅ f c,cube Further conversion factors with respect to size and shape can be found in Betonghandbok Material

21

Konstruktionsteknik, LTH

Interpretation of tests from different size and shape of specimens has been added into the background document. Example:

d fc 150 fc =

β (d )

d (mm)

50

100

150

200

β(d)

1.10

1.05

1.0

0.95

22

Konstruktionsteknik, LTH

In-situ strength < standard strength • In-situ strength can vary within a member both randomly and in an ordered fashion. • The variations of in-situ strength within structural members can vary from one member to another. • In-situ strength decreases towards the top of the pour, even for slabs, and can be up to 25 % lower at the top than in the body of the member. In-situ strength (top pour) ≈ 0.85 x standard strength

23

Konstruktionsteknik, LTH

Illustrative example

24

Konstruktionsteknik, LTH

Design value for compressive strength f cc =

f cck

ηγ mγ n

ηγ m = 1.5 η accounts for the difference between insitu strength and standard strength (≈ 1.2) γm accounts for variability in standard strength

25

Konstruktionsteknik, LTH

Statistical parameters for compressive strength If fck is given through strength class (Eurocode 2)

f ck = f cm − 8( MPa) i.e. σfc ≈ 5 Mpa but for modern concrete production the variability is usually smaller

26

Konstruktionsteknik, LTH

Other concrete properties •Tensile strength •Elastic modulus •Ultimate strain •Long term response •Properties under dynamic loading Can be estimated by transformation from compressive strength by empirical relations EC2, CIB-FIP Model Code 27

Konstruktionsteknik, LTH

Empirical relations between uniaxial tensile strengt and compressive strength 7

6

Mean tensile strength, MPa

JCSS

EC2

5

4

3

Degerman 2

1

0 10

30

50

70

90

Compressive strength, mean value, MPa

28

Konstruktionsteknik, LTH

Empirical relations between elastic modulus and compressive strength 60000

50000

Mean elastic modulus, MPa

JCSS

40000

EC2

30000 ACI 318 20000

10000

0 10

30

50

70

90

Compressive strength, mean value, MPa

29

Konstruktionsteknik, LTH

Age dependence for concrete strength Eurocode 2/CEB FIP Model Code 1990:

f cm (t ) = β cc (t ) ⋅ f cm ( 28 days )

[(

β cc (t ) = exp s 1 − {

}

28 1 / 2 t

)]

s depends on type of cement

30

Konstruktionsteknik, LTH

Age dependence of strength 1,6

A B

Relative strength

1,4

1,2 1

0,8

0,6

0,4 0,2

0 0

10

20

30

40

Age, Years

50

60

70

A: Slow hardening B: Normal cement

31

Konstruktionsteknik, LTH

A more reliable method to determine change of properties due to aging should be developed. The Eurocode 2 relation is not reliable for ages higher than a year

32

Konstruktionsteknik, LTH

Strength at 28 d., 2.5 y. and 30 y.

from Walz, 1976

900

800

30 years

Strength, kp/cm2

700

2.5 y.

600

500

Serie1 Serie2 Serie3

400

300

28 d.

200

100

0 0,2

0,4

0,6

0,8

1

1,2

vct

Walz (1976), Betontechnische Berichte, 4, 1976, pp. 57-78 33

Konstruktionsteknik, LTH

1,4

Relative increase in strength from 28 d. to 30 years 5 4,5 4

f(30y)/f(28 d)

3,5 3

Portland

2,5

Hoch

2 1,5 1 0,5 0 0,2

0,4

0,6

0,8

1

1,2

1,4

VCT

The relative increase of strength is larger for concrete of low quality

34

Konstruktionsteknik, LTH

Timber properties • Tests are made on full size beams to account for defects, knots etc. • Bending strength fm is used as a base parameter • Other strength properties are related to bending strength • Typical COV for fm is 25-40 % depending on species etc. Scandinavian timber has low variability (≈25 %). 35

Konstruktionsteknik, LTH

Tord Isaksson will tell you more about the intricate issue of strength for timber with size effects, load configuration effects etc. The strength of timber is also affected significantly by duration of load. 36

Konstruktionsteknik, LTH