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Load bearing capacity of roof trusses

John Dalsgaard SørensenAalborg University, Aalborg, Denmark

• Stochastic model for strength of timber beam• Load bearing capacity of roof truss• Statistical characteristics• Reliability aspects

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Stochastic model

• Bending strength of timber beam – model 1

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Stochastic modelLength:• weak sections : a length = 0.15 m• strong sections : Lj length of cross-section: Gamma (0.494 m, 0.310 m)

L1, L2, L3 ,… : independentBending strength:• weak sections: fij = bending strength of weak cross-section j in beam I• strengths in different beams : independent• strong sections: bending strengths of all strong sections = strength of

the strongest weak sectionModulus of elasticity:• Ejj = c fij , c = 3.8 103

• Correlation with bending strength: correlation coefficient = 0.8

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Stochastic modelBending strength:• Lognormal distributed: fij = τi εij COV = 0.25

τi = mean strength of beam no i: Lognormalεij = difference between mean strength of beam i and

strength in cross-section j: Lognormal• 40% and 60% of the variance of fij are related to τi and εijCompression strength:• Lognormal distributed COV = 0.15• Correlation with bending strengths: correlation coeff. =0.9Tension strength:• Lognormal distributed COV = 0.30• Correlation with bending strengths: correlation coeff. =0.8

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Stochastic modelCharacteristic values (MPa) and coefficient of variation (COV):

Stochastic model: calibrated to standard test specimens

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Stochastic model

• Bending strength of timber beam – model 2

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Example 1 – roof truss

Load: permanent + snow

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Example 1 – roof trussStochastic model 1: L = 12.45 m:

Stochastic model 1: L = 8.85 m:

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Example 1 – roof truss

Lognormal Weibull

0 1 2 3 4 5 6 7 8

0.00

0.20

0.40

0.60

0.80

1.00

0 1 2 3 4 5 6 7 8

0.00

0.20

0.40

0.60

0.80

1.00

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Example 1 – roof trussStochastic model 1: L = 8.85 m:

Stochastic model 2: L = 8.85 m:

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Example 2 – collar tie roof truss

Load: permanent + snow + imposed

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Example 2 – collar tie roof truss

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Reliability aspects

Limit state function:

Danish code: VR = 20% for structural timber

( )QGzRXg R αα +−−= )1(

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Reliability aspects

Design equations:LC 2.1:LC 2.3:

Danish code: = 1.65 for structural timber

( ) 0)1(/111 =+−− cQcGRc QGRz αγγαγ

( ) 0)1(/333 =+−− cQcGRc QGRz αγγαγ

),max( 31 zzz =

Rγ

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Reliability aspects – reliability index

Imposed load Environmental load

0.0 0.2 0.4 0.6 0.8 1.0α

3.0

3.5

4.0

4.5

5.0

5.5

β

VR=0.20

VR=0.15

0.0 0.2 0.4 0.6 0.8 1.0α

3.0

3.5

4.0

4.5

5.0

5.5

β

VR=0.20VR=0.15

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Reliability aspects- partial factor for strength

Imposed load Environmental load

0.0 0.2 0.4 0.6 0.8 1.0α

1.0

1.2

1.4

1.6

1.8

2.0

γ

VR=0.20VR=0.15VR=0.10

VR=0.25

0.0 0.2 0.4 0.6 0.8 1.0α

1.0

1.2

1.4

1.6

1.8

2.0

γ

VR=0.20VR=0.15VR=0.10

VR=0.25

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Summary

• Stochastic model for strength of timber beams• Load-bearing capacity: COV is approximately 15% →

system factor = 1.1• Compared to the DK code based values: characteristic

values are at least 10% higher → system factor = 1.1• System factor = 1.2 for design load bearing capacity