John Dalsgaard Sørensen Aalborg University,...
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1 Load bearing capacity of roof trusses John Dalsgaard Sørensen Aalborg University, Aalborg, Denmark • Stochastic model for strength of timber beam • Load bearing capacity of roof truss • Statistical characteristics • Reliability aspects
Transcript of John Dalsgaard Sørensen Aalborg University,...
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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
