Wood creep data collection and unbiased parameter identification of compliance functions

1.1 Wood creep

To develop a database and prediction model, it is first necessary to understand how various factors influence the long-term shortening of wood. First consider that wood is an anisotropic material, generally assumed orthotropic for engineering purposes. Indeed, wood is typically stiffer and stronger in the direction parallel to the grain, verses perpendicular to the grain, even in creep. In the majority of studies it is relevant to test parallel to the grain, though some such as Ranta-Maunus (1993) and more recently Massaro and Malo (2019a), also conducted tests perpendicular to the grain. In the present study it is assumed that the elastic properties are time-independent and the reported experimental values are given parallel to the grain.

Three types of creep are identified in wood. The first, occuring in service conditions and constant relative humidity, is most studied and of most interest. The second type of creep is associated with transient MC changes and is generally known as mechano-sorptive creep. The source of mechano-sorptive creep is not fully known, but its effects can be quite large. The third type consists of pseudo creep and recovery phenomenon. In this paper only the first type of creep is considered.

The evolution of creep deformations is divided into three stages. In the initial stage, or in primary creep, the strain rate is relatively high. As secondary creep begins, the strain rate is reduced to a minimum and becomes nearly constant. In tertiary creep, the strain rate increases exponentially with stress, and is associated with damage. Under relative small stress, creep increases linearly as stress levels increase and stays within the primary and secondary region. According to Schaffer (1972), “Wood behaves non-linearly over the whole stress-level range, with linear behavior being a good approximation at low stress. Because of this nearly linear responses at low levels of stress, Boltzmann's superposition principle applies to stress-strain behavior for stresses up to 40% of short time behavior.” The experiments of Schaffer's indicate that within certain limits of stress and at constant MC and temperature wood can be treated as a linear viscoelastic material. This finding was later verified by other studies (Bažant 1985; Foudjet and Bremond 1989; Hering and Niemz 2012; Massaro and Malo 2019a; Nakai and Grossman 1983). Youngs et al. (1957) conducted compression creep tests perpendicular to grain of red oak under various stress levels at 28 °C (82 °F). Hoyle et al. (1985) tested Douglas-fir in bending at constant room temperature and low MC at different levels of applied stress, below 50% of clear wood ultimate bending strength. Jong and Clancy (2004) compared wood creep in an hour under two different load levels, 6 and 8 MPa (870 and 1160 psi). Hunt (2004) analyzed the data of Gressel's 8 years wood creep bending test, concluding that there can be no creep limit in a constant environment (Gressel 1984). Creep can reach tertiary stages due to stress levels and load durations which are over certain limit values. Liu (1993) analyzed bending samples under different applied stress level ranges from 4 to 60% of the rupture strength of Scots pine. The experimental results show tertiary creep deflections and the associated failure points.

Temperature is also a contributing factor to material behavior. Higher temperature decreases the elastic modulus of wood (Bach 1968). It is also found that increasing of temperature accelerates the creep deformation and creep rate. King (1961) compared creep curves of Hoop pine from 21 to 27 °C (75 to 80 °F) under different levels of stress, and confirmed this effect. Davidson (1962) conducted bending tests to study the influence of temperature of wood by using six species subjected to temperatures ranging from 20 to 60 °C (68 to 140 °F). Similarly, Kingston and Budgen (1972) concluded from bending and compression tests in the range of 20 to 50 °C (68 to 122 °F) that temperature had a obvious effect on wood creep. Experiments by Jong and Clancy (2004) show that an increase in temperature increases the creep with the temperature ranges from 20 to 100 °C (68 to 212 °F).

One of the most important factors in wood behavior is MC and changes thereof. As mentioned previously, shrinkage in wood occurs when the moisture level drops. However MC also affects creep behavior. Water in wood acts as a plasticizer, in effect decreasing the viscosity; increases in MC will lead to increases in creep compliance (Schniewind 1968). For example, in bending, total deformation and creep rate have been found to increase with increasing moisture content (Bodig and Jayne 1982; Fridley et al. 1991; Hering and Niemz 2012; Schniewind 1968). Armstrong and Kingston (1960) found that the increase in deformation of initially saturated wood is much greater than in initially dry wood (Armstrong 1962). It is also shown that MC exhibits larger influence on wood creep than temperature (Hsieh and Chang 2018). The larger effect of MC on creep was also observed in tensile tests of Norway spruce (Engelund and Salmén 2012).

As mentioned, mechano-sorptive creep defines the response produced by the combination of mechanical influence and moisture adsorption, which cannot be predicted from each effect separately (Grossman 1976). It was proposed that changes in the MC of the wood while under load markedly influenced both creep-rate and total creep, which indicates that mechano-sorptive effects will results in failure under lower stresses or in shorter time (Armstrong and Kingston 1960; Armstrong 1972). This phenomenon is supported by a large number of experiments (Armstrong and Christensen 1961; Bodig and Jayne 1982; Fridley et al. 1992; Mårtensson 1994; Toratti 1991; Zhou et al. 1999). Experimental results also indicate that the deflection of a loaded beam which is taken through one or more cycles of humidity increases far beyond the deflection of beams loaded after they have been conditioned to either of the extreme moisture levels (Hunt 1999). Additional results indicate that mechano-sorptive creep is exacerbated in directions perpendicular to the grain compared to the longitudinal direction (Ranta-Maunus 1993).

In addition to stress and environmental conditions, other factors such as species and material structure play a role in creep. These factors are more difficult to isolate. For example, in construction, often only a species group such as Douglas-Fir-Larch is specified, for which general properties are provided. However due to variation in the microstructure of different tree species the instantaneous deformation and total creep may vary, and thus cannot be considered independently of species. Foudjet and Bremond (1989) conducted 14 day bending tests of four tropical hardwoods under room temperature, at less than 35% of stress at failure and with constant MC. The creep function curve shows similar trends between species, but different absolute values. Similar results are found in other studies (Kingston and Budgen 1972; Tissaoui 1996).

Wood is also divided into early- and late wood, according to the time within the life cycle during which the cells were grown. The main factor which determines the different properties of the early and late wood is the structure within the cell wall, defined by the microfibril angle (MFA). This angle is a consequence of the need for higher flexibility in young saplings, compared to the stiff trunks of grown trees. Experiments suggest that in the region of small MFA, the creep deformation becomes much smaller (Kojima and Yamamoto 2004; Roszyk et al. 2010). As the tree grows, the MFA decreases, which means that early wood creep is larger than that of late wood (Hunt 1999).

2.1 Data collection

As noted above, variable environmental conditions were not considered for this study. Thus, to build the database, only experiments which were conducted under constant environmental conditions were considered viable. They were then filtered based on realistic engineering requirements. Humidity, temperature, and loading should reflect values typical in mass timber projects. Thus, the requirements for data include: room temperature (20–25 °C or 68–78 °F), a small range of equilibrium moisture content (8–15% is the typical range for structures in service), and relatively low stress levels (under 40% of ultimate strength or modulus of rupture). MFA was not taken into account as it is generally not measured in experimental creep studies.

The database includes 12 different tree species (from multiple continents), loaded in bending, tension, and compression in the longitudinal direction. Time spans ranged from 4 h to a maximum of 450 days. One major drawback of available data was the lack of long-term studies, as previously mentioned. Though one study was found which included tests over 10 years, the data was unavailable (Gressel 1984). This is a serious limitation on the predictive capabilities of the compliance function.

Nakai and Grossman (1983) tested bending specimen with the size 15×15×800 mm (0.6 × 0.6 × 7.8 in) under four different load levels corresponding to 17, 33, 50, and 67% of the modulus of rupture, as tested by the authors with matching material. (The modulus was not provided by the authors; table values for the species are Jarrah: 110 MPa (16 ksi), Mountain Ash: 98 MPa (14.1 ksi), and Hoop Pine: 85 MPa (12.3 ksi).) However, only the data at 17 and 33% of the modulus of rupture were taken into the database, to comply with the initial assumption of linear behavior. Hoyle et al. (1985) conducted bending tests of Douglas-fir beams under stress levels of 8.61 MPa (1250 psi), 13.1 MPa (1900 psi), 17.9 MPa (2600 psi), and 21.7 MPa (3150 psi) with 10 × 10 × 488 cm (4 in × 4 in × 16 ft) specimens. Foudjet and Bremond (1989) performed bending creep tests of four different species for a 600 mm (23.5 in) span with various width and depth to resist shear force. Fridley et al. (1992) showed one typical compression test results with specimens 5 × 5 × 365 cm (2 in × 4 in × 12 ft) in size under a stress level 5% of the static strength. Bond (1993) conducted tension and compression tests under stress levels equal to 20–25% ultimate strength, with a specimen size of 1.25 × 1.25 × 30.5 cm (1/2×1/4×12 in.) and 1.25 × 1.25 × 10 cm (1/2×1/2×4 in.) respectively. Liu (1993) grouped bending tests according to elastic modulus. Experimental results with load levels under 45% of the ultimate tensile strength were used in the database. Tissaoui (1996) tested tension and compression creep for a specimen size of 1.25 × 1.25 × 30.5 cm (1/2×1/4×12 in.) and 1.25 × 1.25 × 10 cm (1/2×1/2×4 in.) respectively for southern pine and yellow poplar. Hering and Niemz (2012) conducted three groups of bending tests under MCs at 8.1, 15.5, and 23.2%. The two low MC tests were averaged and used in the database. Note also that different papers plotted experimental results either averaged or as individual test curves. For consistency all curves of identical temperature, stress, etc. per study were averaged within the database. A summary of the papers collected into the database, as well as experimental conditions and test methods therein, is presented in Table 1.

After the studies were filtered, the data was converted to a consistent format. Researchers generally focused on different aspects of the results, and so data might be presented as deflection, relative deflection, strain, relative strain, compliance, or relative compliance. All data was converted into compliance with the assumption J0=1/E0, where J0 is the instantaneous elastic strain caused by σ=1, and E0 is the elastic modulus. The specimen are all slender beams and only parallel-to-grain experiments are taken into account. It is therefore reasonable to ignore Poisson's effect and assume a one-dimensional problem. Some papers also do not provide elastic modulus; in this case elastic modulus values were assumed according to the species. Figure 1 displays compliance vs. log of time for the experimental data.

Figure 1:

Creep data with instantaneous compliance.

2.2 Creep models

Whilst any empirical equation may be used, some have been found to be more relevant than others for creep compliance, particularly in wood. Three equations were considered in this study, based on a review of the literature:

where t0 is 1 h, the same units as t, and a0, a1, a2, a, c and b are empirical constants (Holzer et al. 1989). For Eq. (1), (2), and (3), t is the time of loading, and the constant c corresponds to the instantaneous compliance J0=1/E0, as previously mentioned. Note that these equations are only applicable to constant-load histories (Fridley et al. 1992).

King (1961) developed Eq. (2) to fit tension experimental result for domestic and tropical species. Bach (1968) conducted tension tests of hard maple for 1,000 min and fitted data using Eq. (3). Schniewind and Barrett (1972) conducted tensile creep tests in the radial-longitudinal and tangential-longitudinal planes of Douglas-fir at 10% MC and used the form of Eq. (1) as model function to fit tests. Gressel (1984) fitted existing experimental data from 10 year creep tests using Eq. (1). Hoyle et al. (1985) implemented Eq. (1) based on experimental results from bending Douglas-fir beams. Bond (1993) used Eq. (1) to fit tension and compression creep test of pine and yellow poplar.

2.3 Optimization and fitting

For fitting the compliance functions, an unbiased statistical evaluation of weighted least square method was used. The algorithm was devised by Bažant and associates to handle the spurious effects of non-uniform data distribution. This has consistently plagued time-related data sets for concrete responses due to a greater amount of short term data (Bažant and Baweja 2000; Rasoolinejad et al. 2019). It is clear that for the current database, the majority of data points concentrate in early time periods. Thus, the data for each curve was divided by intervals of equal size in log scale, logΔ(ti)=log(ti+1)−log(ti)=1, i=1, 2, 3 … n. The number of intervals, n, was defined such that each interval has a similar number of data points, mi. The points are weighted based on Eq. (4). This method was also adopted in Pathirage et al. (2019). The weights, w¯, for each interval were calculated as follows,

where the wi is the ith interval weight.

Once weights are obtained, the weighted standard deviation, s, may be computed as follows,

The error (or residual) e=Yij−yij is regarded as the error of predictions where Yij is the experimental results and yij is the numerical prediction of compliance as a function of time. N is the total number of data points. P is the number of parameters (e. g., Three for power law fitting). The multiplier N/(N−P), used to avoid a bias, is close to one because the value for N is much larger than P. This multiplier is necessary primarily to prevent the variance of regression errors of the database, which has a finite number N data points, from being smaller than the variance of a theoretical database where N→∞. Additionally, only a set of P data points can be fitted exactly without error (Bažant and Li 2008). The weighted mean value for experimental data is y¯, s is defined as the overall weighted standard error of numerical predictions per curve, and s¯ is the overall weighted deviation of all the data.

where

With s, y¯, and s¯, the unbiased coefficient of variation, ωn=s/y¯, and the coefficient of determination, r2=1−s2/s¯2, can be computed for each curve to identify the goodness of fit. The nonlinear regression fitting results are presented in Table 2 in detail for Eq. (1). After an analysis of the results, some abnormal values of parameters were removed, for instance when the value for c was much smaller than the inverse of the elastic modulus.

The overall unbiased coefficient of variation ω was found based on the individual curve unbiased coefficient of variation ωn, such that ω=(ω12+ω22+ω32+⋯+ωN2)/N. Here n represents the individual curve and N=39, the total number of curves in the database.