Structural Reliability Analysis of the bending strength of steel beams

7/30/2019 Structural Reliability Analysis of the bending strength of steel beams

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Steel Innovations Conference 2013, Christchurch, New Zealand, 21-22 February 2013 1

Structural Reliability Analysis

Dr Stephen Hicks & Prof. Br ian Uy




Structural Reliability according to AS 5104 (ISO 2394

identical, which is referenced in AS/NZS 1170)

• Design life = reference period(normally taken as 50 years)

• Probability of failure for resistance at ULS over a 50-year reference period (lowrelative costs for safety

measures and moderateconsequences of failure)p f = 7.2×10-5 ≡ =3.8

• E and R are vectors of thelimit state function, whichaccording to AS 5104 (ISO

2394) may be taken as -0.7and 0.8, respectively for FORM when 0.16 < E / R < 7.6

p f 10-6 10-5 10-4 10-3 10-2 10-1

4.75 4.27 3.72 3.09 2.32 1.28

ULS SLS

Probability of failure vs. reliability index

Design point and reliability index according

to the first order reliability method (FORM)




Basis of Load and Strength capacity reduction

factors

• Normal distribution usuallytaken

• Characteristic value• Mean value if variability small• Upper value (normally 95%

fractile) if variability is not small

• Log-normal distribution usuallytaken (resistance doesn’t havenegative values)

• Characteristic (or nominal) valuefor resistance For large sampleof data and a normal distribution

• 5% fractile = 1.64 σR • Design value = 0.8x3.8σR = 3.04σR

(≡ 1 in 845 = 0.12% probability of

observing lower value)

Loads Material and product properties

S S k S d

S β σS

RR d R k

- R β σR

=1 / RS

R k / R d =R = 1/




For the design fractile factor k d,n as n , k d, = R = 3,04

For the characteristic fractile factor k n

as n , k n, = 1,64

If log-normal distribution is taken (which is desirable as it falls to zero at

the origin, so there are no negative resistances)

where rt and is the weighting factor for Q rt, Q rt is a coefficient for variation

of the variables in the resistance function, is the weighting factor for Q ,

Q is a coefficient for variation of the error term and Q is a coefficient for

variation of the resistance.

Corrected partial factor

where r n is the nominal resistance evaluated from the theoretical

resistance equation using nominal values for the basic variables and k c =

r n / r k

25,0exp QQk Qk X bg r nrt rt mrt k

2

,, 5,0exp QQk Qk X bg r nd rt rt d mrt d

Evaluation of partial safety factor M

d

k c M c M r

r k k */1




Design model for beams in bending

• Design section moment capacity• M s = f y Z e• where is the strength reduction factor, f y is

the yield strength used in design (i.e. the

nominal value) and Z e is the effective section

modulus, which is dependant on whether the

section is compact, non-compact or slender

• Design model which contains the

basic variables

• r t = g rt (X m ) = f y,m Z e,m

• where f y,m and Z e,m are the mean measured

basic variables that are included in a report

from a laboratory test

• Correction factor • bi = r ei/r ti• where r ei is the experimental resistance for

specimen i and r ti is the theoretical resistance

on specimen i • For beams in bending, typically b =

1.14 and 1.19 for partially laterally

restrained and fully laterally

restrained, respectively

Failure deemed to occur at endrotation of 6 degrees




Negative cross-sectional tolerances for beams supplied to

different product standards

Parameter EN10034: 1993 JIS G 3192AS 5100.6

AS/NZS 1365

Depth (h) (mm)

h≤180

180<h≤400

400<h≤700

h>700

-2

-2

-3

-5

h<400

400≤h≤600

h≥600

-2

-3

-4

-h/50

Width (b) (mm)

b ≤110110<b≤210

210<b≤325

b>325

-1-2

-4

-5

b<100

100≤b<200

b≥200

-2

-2.5

-3

-b/100

Web thickness

(tw) (mm)

tw<7

7≤ tw<10

10≤ tw<20

20≤ tw<40

40≤ tw<60

tw>60

-0.7

-1

-1.5

-2

-2.5

-3

tw<16

16≤ tw<25

25≤ tw<40

tw≥40

-0.7

-1.0

-1.5

-2.0

4.5 < t ≤ 6

6 < t ≤ 10

-0.85

-0.9

Flange thickness

(tf ) (mm)

tf <6.5

6.5≤ tf <10

10≤ tf <20

20≤ tf <30

30≤ tf <40

40≤ tf <60

tf >60

-0.5

-1

-1.5

-2

-2.5

-3

-4

tf <16

16≤ tf <25

25≤ tf <40

tf ≥40

-1.0

-1.5

-1.7

-2.0

Mass (%) -4 tf < 10tf ≥ 10 -5-4 -4




Frequency of tensile tests

• According to AS/NZS 3679.1,

for tensile tests, samples

representative of the batch

shall be taken as follows:• One sample for a batch not

exceeding 50 t.

• One additional sample for thebalance of the batch.

• Mill tests are invariably

conducted at a higher

strain rate than

laboratory testingtypically mean mill f y /

mean specified f y = 1.21

(CoV = 7%)




Stress-strain curves for steel and definition of yield

stress

With the exception of NZ only appendix for steel in seismic and fracture critical

applications given in AS/NZS 3678 and 3679.1, all product standards such asEN 10025 define f y = R eH




Full distribution of tensile test results

Lower End Upper End Frequency

265 0

265 270 1

270 275 1

275 280 2

280 285 2

285 290 3

290 295 6

295 300 14

300 305 23305 310 28

310 315 25

315 320 19

320 325 15

325 330 6

330 335 4

335 340 3

340 345 1345 350 1

350 1

Mean 310

SD 13.2

0

5

10

15

20

25

30

270 280 290 300 310 320 330 340 350

Yield Strength, fy (Mpa)




Full distribution of results

Lower End Upper End Frequency

265 0

265 270 1

270 275 1

275 280 2

280 285 2

285 290 3

290 295 6

295 300 14

300 305 23

305 310 28

310 315 25

315 320 19

320 325 15

325 330 6

330 335 4

335 340 3

340 345 1

345 350 1

350 1

Mean 310

SD 13.2

0

5

10

15

20

25

30

270 280 290 300 310 320 330 340 350

Yield Strength, fy (Mpa)

250 grade steel

Real test results a 250 Grade Steel




Test Certificates




Strength Reduction Factor

n 1 2 3 4 5 6 8 10 20 30

k n 2.31 2.01 1.89 1.83 1.80 1.77 1.74 1.72 1.68 1.67 1.64

k d,n 4.36 3.77 3.56 3.44 3.37 3.33 3.27 3.23 3.16 3.13 3.04

2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.20.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Reliability index ()

C a p

a c i t y f a c t o r

Steel ()

25,0exp QQk Qk X bg r nrt rt mrt k

2

,,

5,0exp QQk Qk X bg r nd rt rt d mrt d

Compact sections manufactured to EN 10034

manufacturing tolerances ϕ=0.94 at β = 3.04

ck

d

k r

r 1




Conclusions

• For beams in bending, reliability analyses show 0.9 is appropriate when conservatively assuming

that all negative geometrical tolerances occur and

no mass tolerance is used, f y = R eH and CoV = 7%.

• Yield strengths on mill certificates are given for conformity assessment purposes.

• It is dangerous to use yield strengths from mill

certificates as the results represent a small fraction

of the steel used in design and allowances for theactual yield strength being larger than the nominal

yield strength have been allowed for in the

evaluation of




Where can I get further information?

[email protected]

http://www.hera.org.nz/

Structural Reliability Analysis of the bending strength of steel beams

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