Probabilistic Modelling of Duration of Load Effects in Timber Structures

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1 Proailistic Modelling of Duration of Load Effects in Timer Structures Jochen Köhler Institute of Structural Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland. Summary In present code formats the descried load duration effect on the strength of timer material is represented y a modification factor k mod. In the present report this modification factor is calirated for snow load. The modeling of the load duration effect has een descried with particular emphasis on the Damaged Viscoelastic Material (DVM) model. The parameters of the model have een estimated as stochastic variales using the Maximum Likelihood method. The snow load process has een modeled y a Poissonprocess with triangular load pulses. Keywords: timer, duration of load, code caliration, snow load.. Introduction Reliaility analysis of timer structures for the purpose of code caliration or reliaility verification of specific timer structures and components requires that the relevant failure modes are represented in terms of limit state functions. The special ehavior of structural timer in regard to long-term load situations, exclude modeling load and resistance as independent variales in the limit state function. That the strength of timer material is influenced y oth the intensity and the duration of loads, often referred to as the duration of load (DOL) effect, has een an area of great interest over the last decades in the timer research community. Numerous experimental programs have focused on duration of load effects in full size timer components and a numer of different models, mostly ased on experimental evidence, have een proposed to descrie the phenomenon. The so-called Madison curve [] is one of those models uilt on results from tests on clear wood specimens i.e. specimen, with no visile defects, loaded with constant loads. The empirical models have the disadvantage that oservale quantities, like load history and environmental conditions, can not directly e utilized for caliration and updating of the parameters in these models. L. Fuglsang Nielsen suggests a model ased on analytical physical considerations in e.g. [] where the oservale and measurale quantities, such as the strength and stiffness properties of structural timer, the creep ehavior and load variations, are taken into account. This fracture mechanical model considers structural timer as a damaged viscoelastic material. In the present report this model is applied in the context of reliaility analysis where the essential uncertain model parameters are represented y random variales. For the propose of illustration and comparison with other models such as reported in [3] appropriate load duration modification factors k mod for use in load and resistance factor design (LRFD) are calirated.. Resistance models. Duration of load One mechanism leading to the decrease in load carrying capacity of timer components sujected to longterm load is creep-rupture. Specifically, creep rupture arises from the propagation of small damages in the microstructure of the timer at a load level lower than the short-term load carrying capacity. A numer of models of creep-rupture, commonly referred to as damage accumulation models, have een proposed y e.g. [4], [5], [6] and [7] to assess the damage accumulation in timer sujected to random load histories. These models have the general form: ( σ ( t), α ( t), σ, σ?) d α dt = F () S 0

2 where t is time, a is the damage state variale which ranges from 0 (no damage) to (failure), s(t) is the applied stress, s S is the short-term strength, s 0 is the threshold stress elow which damage does not accumulate and? is a vector of parameters descriing the model. The parameters of the model? have to e fitted to experimental data and have no further physical meaning. The damage accumulation models are ale to consider loading and unloading in immediate, stepwise and ramp wise form. However, the main assumption is that load histories may e modeled as a sum of increment periods of constant load and that the damage accumulates as a consequence of these only, i.e. no other failure mechanisms esides creep-rupture are considered. Damage accumulation models may e seen entirely as suitale curve fitting tools to experimental data. Such models do not necessarily reflect the physical failure mechanism. A different approach is to try to model the mechanisms in line with the experimental evidence that for the most loading modes structural timer fails as a result of crack growth.. Timer as a Damaged Viscoelastic Material A model ased on viscoelastic fracture mechanics was proposed y Nielsen in [8]. The main idea ehind the Damaged Viscoelastic Material (DVM) -model is that structural timer may e seen as an initially damaged material, where the damage is represented y cracks along the fiers. The time dependent ehavior of timer under load is modeled y a single crack under stress perpendicular to the crack plane. The crack is modeled as a Dugdale crack with a time dependent modulus of elasticity. The time dependence of the modulus of elasticity is descried y a power law expression, i.e. creep is modeled y the normalized creep function C(t) as: C ( t) = + ( t τ ) where t and are material parameters depending e.g. on loading mode and moisture history. The DVM-Theory for constant load is modelling creep-rupture y considering initiation of crack development at time t s and crack development until the crack has grown to a critical length l crit at time t cat corresponding to the applied constant load. For the case of a constant load level, a damage accumulation law can e formulated from the DVM-theory as: dκ = dt ( π FL ) qτ κ SL ( ) ) κ SL 8 where SL is the stress level (or load level) defined y the ratio s/ s cr etween applied stress s (load) and shortterm strength s cr. The latter measured in a very fast ramp test. FL is the strength level defined as the ratio s cr / s l etween short-term strength s cr as defined aove and the intrinsic strength of the (hypothetical) non-cracked material s l.? = l/l o is the damage and defined as the ratio etween the actual crack length l and the initial crack length l 0.? = corresponds to no damage and? = SL - to full damage. q is given as a function of the creep exponent as q = ( 0.5( + )( + ) ) and considers a paraolic increasing crack opening progression. The time to failure t f can e expressed y the time etween?(t = 0) = and?(t f ) = SL -, as t f = 8qτ ( π FL SL) ( ϕ ) SL ϕ dϕ Fig. : Crack propagation under constant load. d is the crack front opening and l is the crack length. where f is a damage state variale. The model of crack propagation under constant load y the DVM-theory is of a format similar to the damage accumulation laws mentioned aove, with the difference that no stress threshold s 0 is introduced and the damage? is defined as the physical status of the crack length. Based on the solution for constant load the DVM-model was developed to consider rectangular pulsed load histories y () (3) (4)

3 introducing crack closure and contact effects due to unloading. It has een shown that the failure mechanism in timer is not only due to creep rupture. Frequency dependent fatigue effects in timer have een shown theoretically y Nielsen in [4] and experimentally e.g. in [9] y Clorius. The DVM-theory as presented in [8] is restricted to model harmonic and weakly non-harmonic rectangular pulsed load histories. The representation of damage in timer suject to a random load process of any load pulse shape is not possile with the theory at present state. It can e shown that the solution of the DVM-Model for constant loads is also applicale for the modeling of damage in timer suject to load histories with moderately varying loads which means that if the rate (velocity) of unloading is sufficiently small, the effect of crack closure might e counteracted y the relaxation of the viscoelastic material in the crack front zone. Assuming that the rate of unloading is small, equation 3 can e used to model the increment damage over the time, with κ π FL κ SL κ n = ( tn+ 8q τ n n+ ( κ nsln+ ) ) t ) n+ n (5).3 Model parameters and their estimation The parameters required to estimate time-to-failure, the intrinsic strength of a non-cracked material, s l and the creep parameters t and are all physical presented in nature. But none of these parameters is readily otainale y measurements. The creep modeled in the DVM-theory is a very local phenomenon of the material in front of the crack tip and thus depends on the condition of grain orientation and morphology around the crack tip. The strength of the un-cracked material can only e assessed y theoretical considerations. For these reasons the model parameters are fitted to data otained from duration of load tests, where constant load is applied until ultimate failure. On the asis of equation 4 and introducing a normal distriuted error term e with zero mean and unknown standard deviation, the parameter t can e estimated using the Maximum Likelihood method. The parameters and FL can not e estimated with the same freedom; only values of e.g. = /3, /4, /5,.. will lead to closed solutions of the integral in equation 4. Furthermore, the parameter FL is dependent on t and can therefore not e estimated individually. Based on test results from Hoffmeyer[0], the parameter t and the standard deviation of the error term are estimated as normal distriuted random variales; and FL are estimated as constants. All parameters are given in tale. 3. Load model The solutions of the DVM-theory for constant load can e used to model load histories with randomly varying loads if the rate of the unloading is small. Snow on roofs of uildings is considered as such a load and a stochastic load model is formulated in accordance with the load model presented in [3]. Snow load on the ground is modeled y triangularly shaped load pulses (snow packs). This is illustrated in figure. Based on Danish snow data over 3 years from five locations the following load process parameters have een estimated. The occurrence of a snow package at times X, X, is modeled y a Poisson- P SG (t) process. The duration etween snow packages is therefore exponential distriuted with expected value /?, where? is annually the expected numer of snow packages. The magnitude of each snow package P m is assumed to e Gumel distriuted. The duration of a snow package T is assumed to e related to the snow package magnitude with T = X T P m where the factor X T is exponential distriuted. The load history on the ground P(t) can e simulated and transformed to the roof load Q(t) y introducing a Gumel distriuted shape factor C with Q(t) = C P(t). The parameters for the stochastic models are found in tale. P m P m X X +T X X +T t Fig. : Snowload at the ground modeled y tritangular pulses

4 Tale : Overview of the stochastic Models used in this report. Stochastic Variale Distriution Type Mean Value Standard Deviation Short term LS Resistance model Load model Short-term strength R 0 Log-normal 0. Snow load (50 y max) Q Gumel max 0. Model uncertainty X R Log-normal 0.05 Relaxation time [h] t Normal stdevia. Error term log[d] s e Normal Strength level FL constant Creep power constant 0. - Duration etween snow pack [y] d Exponential /.75 - Max. per snow pack [kn/m ] P m Gumel max Duration pack factor [d/kn/m ] X T Exponential 75 - Roof shape factor [-] C Gumel max Caliration of the modification factor k mod In present code formats as the EC5 the descried load duration effect of on the load carrying capacity of timer components is represented y a modification factor k mod. The following design equation can e found in the codes. zrk kmod γ QQk = 0 γ m where z is the design variale, R k and Q k are the characteristic values of short-term strength (5% quantile) and variale load (98% quantile of one year maximum distriution).? m and? Q are the partial safety factors for material parameter, and variale load. k mod is the modification factor. The limit state function for the long-term load carrying capacity can e formulated as g = κ ult ( SL, t, z) κ ( SL, t, FL, τ,, z) (7) where? ult is the ultimate damage at time t and load level SL. For the caliration of the modification factor k mod the following limit state function for short-term load carrying capacity is used. g = zr X 0 R CQ (8) where z is the design parameter, R 0 the short-term strength, X R is the model uncertainty, Q is the snow load on the ground and C is the shape factor. The modification k mod can e calirated as follows. The partial safety factor for the short-term strength? M,S is calirated to a target reliaility index, using the limit state function (8) and the design equation (6) with k mod = and fixed? Q. The partial safety factor for the long-term strength? M,L is calirated using the design equation (6) with k mod = and fixed? Q. A snow load history of 50 years and the herey-resulted damage in the timer is simulated and the time to failure T F is determined. After a sufficient numer of simulations the proaility of failure P F is estimated y the numer of failures during the design life of 50 years and the total numer of realizations. The corresponding reliaility index is determined from ß = - F - (P F ).? M.L has to e modified until the target reliaility index is reached. k mod is then estimated y γ M, S ( β ) kmod = γ M, L( β ) (9) (6)

5 5. Oservations and Discussion The modification factor k mod is calirated as previously descried. A value of k mod = 0.8 was found. This result is consistent with the results presented in Sorensen et al [3], where the same snow load model ut different damage accumulation models were used in the caliration procedure. To illustrate the effect of damage? to the strength the timer material, the residual strength SR is defined y SR = κ In figure 3, the maximum load level of each snow package and the residual strength of the timer is plotted over lifetime. Failure occurs when the residual strength drops as consequence of an extreme snow load, i.e. when SR < SL. In the case illustrated in figure 3 failure occurs after a time of approximately 3 years. The load history prior to failure seems not to e of significance to the damage and the residual strength respectively. A similar oservation was made in [] where other damage accumulation models were used. In figure 4, a snow load process is used where the standard deviation of the maximum load level of each snow package is lower compared to the example shown in figure 3. It is seen that the residual strength drops as a consequence of major snow loads several times until failure occurs. The damage accumulates with the numer of major snow events. Depending on the load level as well as the temporal characteristics of the underlying stochastic load process failure may e considered time independent. Rather than simulating complete load histories and incrementally accumulating damage, a time independent limit state function can e formulated. Only the load pulse that cause significant damage, may e considered for reliaility analysis with the considered snow load characteristics. If the standard deviation of the maximum load level of a snow package is getting smaller, damage accumulates several times (figure 4) and the time to failure has to e evaluated y simulation as descried in this report. The DVM-model does not include a threshold value s 0 elow which no damage accumulates as some other damage accumulation models. However, through the highly non-linear properties of the model an effective threshold can e oserved y analyzing the results of the simulations. 6. Conclusion and prospect for further studies SL and SR The Damaged Viscoelastic Material model (DVM) has een used for the caliration of the modification factor k mod, which considers the duration of load effect in load and resistance factor design (LRFD). Snow load is considered and a stochastic model is formulated. The results correspond with the results from other investigations where the same load model ut different resistance models were used. It has een shown that for the considered load situation the load history efore failure is not significant and that failure occurs as a consequence of a comination of an extreme snow load event and the resulting damage. (0) time [years] Fig.3: Maximum magnitude of each snow package and the residual strength of the timer specimen is plotted over lifetime. SL and SR time [years] Fig. 4: Maximum magnitude of each snow package with lower standard deviation and the residual strength of the timer specimen is plotted over lifetime.

6 In further studies the comination of different loads should e considered and a classification of load situations should e undertaken for taking damage accumulation effects into account. Further the DVMmodel should e developed to also account for fatigue-effects due to stochastic load histories. The model as is at the present restricted to harmonic and weakly non harmonic load processes. 7. Acknowledgements The work descried in the present report was conducted as part of the European research project,reliaility design of timer structures under the EC Action COST E4. The financial support from the Federal Office of education and science, Switzerland is gratefully acknowledged. 8. References [] WOOD, L.W. Relation of strength of wood to duration of load. Report no. 96. Madison, USDA Forest service, Forest products laoratory; 95. [] NIELSEN, L.F., Lifetime and Residual Stregth of wood sujected to static and variale load Part I +II., Holz als Roh- und Werkstoff 58: 8-90 and 4-5. Springer, 000. [3] SORENSEN, J.D., STANG, B.D., SVENSSON, S., Caliration of duration factor k mod.(draft) to e pulished, 00. [4] GERHARDS, C.C., Time-related effects of loads of wood strength. A linear cumulative damage theory. Wood science 979, 9(): [5] GERHARDS, CC., LINK, C.L., A cumulative damage model to predict load duration characteristics in lumer. Wood and Fier Science 987, 9(): [6] BARRETT, J.D., FOSCHI, R.O., Duration of load and proaility of failure in wood. Part I + II, Canadian Journal of Civil Engineering 978, 5(4): [7] FOSCHI, R.O., YAO, Z.C., Another look at three duration of load models. Proc. 9 th CIB/W8 Meeting, Florence, Italy, 986. [8] NIELSEN, L.F., Crack failure of dead-, ramp- and comined loaded viscoelastic materials. Proceedings First International Conference on Wood Fracture, Banff, Alerta, Canada, 979. [9] CLORIUS, C.O., Fatigue in Wood An Investigation in Tension Perpendicular to the Grain. PhD Thesis, Department of Structural Mechanics and Materials, Technical University of Denmark, 00. [0] HOFFMEYER, P., Failure of Wood as Influenced y Moisture and Duration of Load. PhD Thesis, College of Environmental Science and Forestry Syracuse, State University of New York, 990. [] ROSOWSKI, D. V., BULLEIT, W.M., Load duration effects in wood memers and connections: order statistics and critical loads. Structural Safety 00, 4: