Probability calibration of deformation factor for timber roofs in the Polish mountain zones

The resistance parameters of timber material structures decrease with time, depending on the type of load and timber classes. Strength and Modulus of Elasticity reduction effects, referred to as creeprupture effects, due to long term loading at high stress ratio levels are known for many materials. Timber materials are highly affected by this reduction in strength and deflection with duration of load. Characteristic values of load duration and deformation factors are calibrated by means of using probabilistic methods. The reliability is estimated by means of using representative shortand long-term limit states. Time variant reliability aspects are taken into account using a simple representative limit state with time variant strength and simulation of whole life time load processes. The parameters in these models are fitted by the Maximum Likelihood Methods using the data relevant for Polish structural timber . Based on Polish snow data over 45 years from mountain zones in: Zakopane – Tatra , Świeradów – Karkonosze, Lesko – Bieszczady, the snow load process parameters have been estimated. The reliability is evaluated using representative short – and long –term limit states. The deformation factor kdef is obtained using the probabilistic model.


Introduction
Timber is a general name which comprises some of the elements of timber materials presented in Fig.1. ( Where: T a is the total deflection -in millimeters at time t minutes, 1 a is the deflection in millimeters at 1 minute, 0 a is the deflection in millimeters of unloaded test. The illustration of the creep factor for two load scenarios is presented in Fig. 2.   Fig. 2. Relative deflection factor for gluam beam for permanent load and for combination with a random snow load [4,5]. (4) The deformations are calculated parallel to the grain, a linear strain distribution and Bernoulli hypothesis are assumed.
The elastic strain depends on the elastic compliance, which is the function of the current moisture content.
The normal creep strain, which is due to the deformation induced by the duration of load is given in a Kelvin series form.
The mechano-sorptive creep strain, which is due to additional creep deformation caused by the coupled effect of variable moisture and load duration. In constant stress case this takes the following form: Where: ¥ J is the limit mechano-sorptive compliance, u is the moisture content and c is the material parameter. The shrinkage strain, the reversible moisture induced strain, is assumed to be affected by the total strain. This strain component does not cause creep, but in cyclic humidity conditions it accounts for the oscillation of the creep curve.
Where: s a is the shrinkage coefficient, b is a material parameter and ( ) t e is the current total strain excluding the shrinkage strain.
The output from the analysis consists of the following distributions at time t : Service class 2 will be considered in this paper. Strain history will be dependant on the stress history.

Stress history
The effect of sustained stress on timber bending strength and timber deflections has been recognized since at least 1850. Wood developed a time strength curve that was incorporated into the wood design procedure. This curve, often referred to as the "Madison curve", is still in use. [6]. During the last decades structural reliability methods have been further developed, improved and adopted and are now at a level, where they are being applied in practical engineering problems.
This knowledge has now reached a level where it enables designers to take into account uncertainties in material properties and actions in assessing the load carrying capacity, serviceability and service life of timber structures and connections. Nowadays, most building codes are based on a probabilistic safety approach. The code formats are deterministic with connections to reliability design achieved by failure probability, partial safety factors and characteristic values. Partial safety factors are calibrated for standard cases against probabilistic analysis for similar cases. The condition for calibration that is the probabilistic analysis and deterministic code should be fulfilled with the same safety requirements. Therefore, the required safety is usually not accomplished by using probabilistic theories in everyday designing process. Timber is a rather complex building material. The timber material characteristics depend on the specific wood species, the geographical location where the wood has been grown and furthermore on the local growing conditions. Timber is an orthotropic material and it consists of high strength grains which are predominantly orientated along the longitudinal axis of a tree. Material characteristics -the ultimate bending stress and the bending stiffness depend on the orientation of the moment axis to the grain direction. Irregularities in regard to grain direction, knots and fissures become highly decisive for load bearing capacity of a timber structural elementIt.It is possible to avoid inregulaties with small test specimen only, with so-called clear wood specimens.
The Gerhards [6] damage accumulation model is taken into consideration. The mechanism leading to the reduction of strength of a timber member under sustained load is a creep rupture. This could arise from propagation of voids in the microstructure of the timber at a stress level, lower than the short-term strength. A number of models of creep rupture, involving a damage state variable (similar to that used in the analysis of metal fatigue), have been proposed to assess damage accumulation in wood structural members subject to loading histories, typically modeled as renewal pulse process. The basic model [7][8][9][10][11] has the following form Eq. (7) where t -time, a -the damage state variable which ranges from 0 (no damage) to 1 (failure),the function F(.) has two constants that must be determined from test data, and ( ) t s -the ratio of the applied stress to the failure stress under short-term ramp loading.
Damage models are used for mathematical description of the long term strength reduction as a function of stress level and duration of loading.. The characteristics of damage models are that a is defined as the degree of damage, i.e. a = 0 stands for no damage and a = 1 stands for total damage or failure.

The damage accumulation model presented by Gerhard, [6] is
Where A and B are constant, s is the stress and fo is short term strength of member.
Solution of equation (8) is : where; end d models the uncertainty related to the formula (7). d is assumed to be Normal distributed with expected value equal 0 and standard deviation 1. Assuming constant load and considering f as the residual strength the solution to (9) is as follows: This expression is used in simulating the damage due to load duration.
A stochastic model of snow load in the mountain zone in Zakopane -Tatra , Świeradów -Karkonosze, Lesko -Bieszczady is established on the basis of meteorological data [12,13]. Examples of season snow loads in Zakopane , Świeradów, Lesko are shown in Figs 3, 4, 5.   A model calibrated against direct measurements of snow load is issued to transform the meteorological data into snow loads model. The load model is illustrated in Fig 6. rectangular and triangular [14]. The snow load on terrain S(t), and duration T of snow packages (snow pulses) are modelled as follows: -The occurrence of snow packages at times X1, X2,… is modeled by a Poisson process.
-The duration between snow packages is exponential distributed with expected value l / 1 , where l is expected number of snow packages per year.
-The magnitude of the maximum snow load P m in one snow package (snow pulse) is assumed to be Gumbel [15] distributed with expected value p µ and standard deviation p s .
The Table 1. shows probabilistic parameters of snow packets in Zakopane (Z), Świeradów (S) and Lesko (L).

Deformation factor estimation
In the code format EC5 load duration effect and deformation factor are represented by the modification factor k mod and k def The following design equation for long-term situation with one variable load can be found in the code [16,17].
where ) ( s S m b g is the short term partial safety factor as function of b and ) ( L L m b g is the short term partial safety factor as function of b. The results of calculations of deformation factor for timber materials such as: Solid Timber, Medium Density Fibreboard (MDF), Oriented Strand Board (OSB) and Plywood are presented in Fig. 7.

Conclusions
The probabilistic models for the graded timber material properties have been formulated so that they my be readily applied in structural reliability analysis. It has been noted that a significant effect of the time variation of snow impulses-packages on the accumulated damage model has been found. Therefore the observed snow packages are quite different and the triangular and rectangular time variations are included in the present probabilistic calibration of deformation factor. The deformation factor k def value depends signifcantly on the ratio between permanent and variable (snow) loads. More research is needed on the probabilistic resistance parameters of wood planks and another types of timber boards to describe timber deformaction factors.