On time invariant probabilistic modelling of duration of load effects for timber

1. On time invariant probabilistic modelling of durationof load effects for timber Sven Thelandersson Structural Engineering Lund University

2. DOL effects in the probabilistic model code The following limit state function is proposed by Larsen g = 1-(S(t),R,p) where  is the damage, S(t) is the load history, R is the resistance and p is a vector with model parameters. The design condition is that g>0 during the whole lifetime of the structure This makes problem time variant.

3. Problem • Design of timber structures almost always involves DOL effects • Probabilistic design will therefore imply time variant analysis in most cases • An alternative approach allowing for time invariant reliability based design of timber structures is needed • A method for this is proposed here

4. Basic input • Random load model describing the load history S(t) • Damage model describing the effect of a given S(t) on the resistance • Life time T of the structure

5. Load model For a load combination with n loads where Si(t) is the history of random load i which can be modelled e.g. according to JCSS model code

6. Damage model A general form for the damage model is where Ro is the initial (”short term”) resistance Any of the damage models available in literature can be used

7. Proposed method For a given life time T: Perform Monte Carlo simulation to generate N random load sequences S(t) during the period T Determine for each simulated load sequence

8. For each simulated load sequence Si(t) • Assume a value for the initial resistance Ro • Calculate the damage (t) and in particular the damage (T) at the end of the prescribed life time • Adjust Ro until (T) = 1 • This value of Ro is denoted RoDOL • RoDOL is the required initial resistance to survive load sequence Si(t)

9. Duration of load effect The initial resistance RoST required to survive load sequence Si(t) if no DOL effect were present is RoST=Smax < RoDOL A random duration of load factor  may be defined as = RoST/ RoDOL

10. Simulated load sequence for 50 year snow load+permanent load Load Time, years

11. Example: 100 % snow load- Foschi´s damage model - T= 50 years Required resistance P Short term With DOL Relative strength

12. Time invariant description of DOL The strength of timber segment j in board i can be determined by with notations according to Larsen.

13. The properties of the random variable  will depend on: • Type of load • Combination of loads • Type of damage model • Assumed life time of the structure • …….? • Simulations for relevant cases can be performed and the results can be included in the model code

14. Conclusions • A time invariant description of DOL-effects should be included in the model code as an alternative • A methodology to do this has been proposed • Further work is needed before a complete set of data can be presented