International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Serial correlation of withdrawal properties from axially-loaded self-tapping screws Brandner, Reinhard; Bratulic, Katarina; Ringhofer, Andreas Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Serial Correlation of Withdrawal Properties from Axially-Loaded Self-Tapping Screws Reinhard Brandner Assistant Professor, Institute of Timber Engineering and Wood Technology, Graz University of Technology, Graz, Austria Katarina Bratulic Research Assistant, Competence Centre holz.bau forschungs gmbh, Graz, Austria Andreas Ringhofer Research Associate, Institute of Timber Engineering and Wood Technology, Graz University of Technology, Graz, Austria ABSTRACT: Previous investigations outline the applicability of a two-level hierarchical stochastic material model combined with equicorrelation for the description of timber strength and elasticity, by explicit differentiation in variation within and between timber elements. Consequently, as far as withdrawal of primary axially-loaded self-tapping screws is concerned, the load bearing capacity of screw groups in laminated timber products depends on their positioning relative to the product layup. We analyse the first time the applicability of a two-level hierarchical model on withdrawal strength, stiffness properties and density. By testing a saturated data set, the hypothesis of equicorrelated withdrawal properties could not be rejected. Test setup, examination and accompanied epistemic uncertainties in analysing the stiffness properties are seen as general reason for their relatively high variation and consequently low correlation, whereas the high equicorrelation of withdrawal strength is explained by the homogeneous test material. However, in reality screw groups are influenced by unavoidable flaws which provoke higher variation and lower correlation. In view of previous investigations on timber strengths, an equicorrelation for withdrawal strength in the range of 0.40 to 0.50 (0.60) appears more reasonable.   1. INTRODUCTION Timber is a hierarchical material, structured in five scales (Speck and Rowe 2006). On each scale, properties’ variation, e.g. of strength, depends on variations in dimension, distribution and interaction of flaws which are representative for each scale. Previous investigations (e.g. Isaksson 1999, Köhler 2007, Brandner 2013) show that aleatory (natural) variation within scales can be classified in variation within and between individuals, samples or even between different material batches or proveniences.  We aim on the scale of timber, with dimensions 10 ± 2 m, as basis for laminated engineered timber products. With focus on timber engineering, within the last few decades two important developments have been achieved: one by establishing unidirectional and orthogonal layered linear and laminar engineered timber products, e.g. duo- and trio-beams, glued laminated timber (glulam; GLT), cross laminated timber (CLT) and laminated veneer lumber (LVL). The second development addresses innovations in connection technique, which allow almost utilizing the high load bearing and stiffness capacity of engineered timber products also at joints between structural components; this by recent achievements in bonding technology and in regard to self-tapping screws. Both connection techniques provide high utilization 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 degrees even if the stresses are transferred in grain direction, the main direction of timber, and are additionally useable for reinforcing timber’s weak material properties, e.g. stresses perpendicular to grain and shear.  We concentrate on partly- or fully-threaded self-tapping screws inserted in laminated timber products, e.g. glulam. These screws are optimized for primary axial loading. Joints of up to a few hundreds screws show an enormous load-bearing and stiffness potential with possible failure scenarios: (i) steel failure, (ii) head-pull through, (iii) withdrawal failure, and (iv) block shear failure of a group of screws. Although such joints are often designed to fail by steel fracture, the resistance against withdrawal has to be considered as well. With focus on (iii), the withdrawal strength fax based on tests and the characteristic withdrawal strength fax,k according to EN 1995-1-1 (2006), respectively, are given as  ax,maxaxef Ff d l; 0.5 0.1 0.8ef kax,k0.52 d lf    (1) with Fax,max as the maximum withdrawal resistance, d and lef as nominal diameter and effective penetration length of the screw, respect-tively, π = 3.41… and ρk as the characteristic density of the timber product. Thus, density is the only timber property indicating the withdrawal resistance of screws in timber. Due to the high stiffness combined with limited plastic deformation till the ultimate load, load sharing between common acting screws in a group is limited. Current design procedures take this into account by a reduced chargeable number of screws in calculating the group resistance, e.g. by nef = n0.9 according to EN 1995-1-1 (2006), with nef and n as chargeable and common acting number of screws, respectively, in a joint. In contrast to the use of screws in timber elements, in laminated timber products the withdrawal capacity of a single screw depends on the number and properties of the penetrated elements composing the timber product (Brandner 2013, Ringhofer et al. 2014a). Furthermore, the interaction of screws in a group inserted in such timber products depends additionally on the positioning of the group relative to the layup of the laminated timber product, see Figure 1. However, all these influences are currently not considered in the design process.     Figure 1: Positioning of screws in a butt joint with outer steel plates relative to the layup of a laminated timber product: (a) parallel, (b) perpendicular to lamination’s face   In view of these circumstances, analysing the allocation of the variation of withdrawal properties to variation within and between timber elements appears worthwhile. In fact, the magnitude of stochastic system action or homogenization, as consequence of the reduced variation in cases where more than one element are penetrated by a single screw as well as by the group action, is directly a function of the property’s variation and of the correlation of properties between sub-elements.  For the description of the spatial variation of properties within and between timber elements, recent and earlier results on timber, e.g. from Riberholt and Madsen (1979), Taylor (1988), Williamson (1992), Lam et al. (1994), Källsner et al. (1997), Isaksson (1999), Köhler (2007) and Brandner (2013), demonstrate the applicability of a hierarchical model combined with equicor-relation rather than a serial correlation with Markov property. In probability theory, equicorrelation constitutes the simple stochastic case where all coefficients in the correlation matrix ρ of size M  M of M variables in a set are F FA – A  A – A  A – A  F FA – A  A – A  A – A  (a)(b)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 equal, with ρij = ρequi, for i, j = 1, …, M,  i ≠ j, and ρij and ρequi as pairwise correlation and equicorrelation coefficient, respectively.  We focus further on the simplest case of a two-level hierarchical model and define   kXYZ ikik  , (2) with i = 1, …, M, k = 1, …, N, M as the number of random variables (e.g. number of segments per board), N as the number of elements or realizations per variable (e.g. number of boards), and with Yk as mean value of the kth element, Xi | k as ith independent and identically distributed (iid) deviation of sub-element i from Yk given element k, with expectation μX = E[Xi | k] = 0 and variance Var[Xi | k] = σX2, and covariance CoVar[Yk + Xi | k, Yk + Xl | k] = Var[Yk] = σY2, expectation E[Zik] = E[Yk] = μY, variance Var[Zik] = σX2 + σY2 and with   2equi 2 2   YX Y.  (3) In fact, this hierarchical model, introduced by Källsner and Ditlevsen (1994), Källsner et al. (1997) and Ditlevsen and Källsner (1998, 2005), can be directly inferred from the hierarchical material structure of wood.  The focus is on the correlation between segments (sub-elements) within one element. This information constitutes the basis for a more realistic consideration of the variation in with-drawal properties and serves as input parameter for capacity modelling of common acting groups of screws, by taking into account their positioning relative to the layup of laminated timber products.  2. MATERIAL AND METHODS 2.1. Withdrawal tests Withdrawal of self-tapping screws was tested on 20 boards of Norway spruce (Picea abies), conditioned to u = 12 % moisture content, with final (planed) dimensions w × t × l = 120 × 40 × 3,000 mm³, with w, t, and l as width, thickness and length, respectively. These boards were taken from an ungraded batch of 900, at reference moisture content uref = 12 % with average batch density ρbatch,12,mean = 461 kg/m³ and coefficient of variation CV[ρbatch,12] = 8.3 %. Subsequent grading limited the density of boards to 440 to 500 kg/m³, with ρboard,12,mean = 468 kg/m³ and CV[ρboard,12] = 4.1 %. The unusual low variation consequences also from further reject criteria: compression wood, bark and cracks.  Before testing, each board was divided in 19 to 23 segments with and without knots, in total 421. Due to the fact that withdrawal tests shall only be conducted without penetrating knots, this amount reduced to nine to 14 segments per board, in total 248. Consequently, the distance between the segments’ centres is not constant but their order is known. This approach appears adequate as long as the hypothesis of equi-correlation can be confirmed which corresponds to a serial correlation between segments’ pro-perties within elements which is independent of their in-between distance, also known as lag-distance.   Figure 2: Specimen’s dimensions and screw positioning   The withdrawal tests were conducted according to EN 1382 (1999) by using fully-threaded self-tapping screws (ETA-12-0373 2012) of nominal diameter d = 8 mm. All fasteners were screwed through the segments in radial direction of the timber (thread-fibre angle α = 90°), eliminating any influences of the screw tip, without pre-drilling and with anchoring length lef = 40 mm, see Figure 2. Testing was done by using a push-pull configuration and a pre-load of 150 N. After testing, specimens with 40160808040l ef6090 90 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 w × t × l = 40³ mm³ for determination of the local density and moisture content according to EN 13183-1 (2002) were taken centrically to the screw hole. More details concerning the tests can be found in Bratulic (2012).    Figure 3: Typical load-displacement curve of an axially loaded self-tapping screw: denomination of Fax,max, wf, Kser and Ku   The definitions of Fax,max, Kser and Ku, corresponding to the maximum withdrawal force, the slip modulus and the instantaneous slip modulus for ultimate limit states, respectively, as used in further data processing, are illustrated in Figure 3. The displacement was measured only globally, thus influences from the testing device and test setup are included in Kser and Ku leading to an underestimation. These properties can consequently only considered as “apparent”. However, focusing on the inference of the spatial correlation correct absolute values are not required. By considering the additional displace-ments in steel as approximately deterministic the realizations of Kser and Ku are still usable.  With timber as hygroscopic material, deviations from uref = 12 % conditions correc-tions in the physical properties: 0.5 % for density (EN 384 2010), 3 % for fax (Ringhofer et al. 2014b) and 2 % for Kser and Ku per 1 % Δu. Correction factors for the mechanical properties are motivated by considering that withdrawal primary causes shear stresses and failure in timber.  2.2. Statistical methods Statistical analysis and inference was primary performed in R (R Core Team 2012). After calculation of basic statistics, the focus was on testing the pairwise correlations ρij between segments within elements, with NMNNMMikZZZZZZZZZZ212222111211,  (4) given as       1;ρ;ρ;ρ;ρ1ρ12211kMkkMkkkkMkijZZZZZZZZ.  (5) This was done for all five properties: the density of the tested segments ρseg,12, the local density around the screw hole ρloc,12, and the withdrawal properties fax,12, Kser,12 and Ku,12.  A statistic for testing the hypothesis H0: ρij = ρ = ρequi vs. H1: ρij ≠ ρequi, for  i, j = 1, 2, …, M and i ≠ j, is given as (Lawley 1963)        2 22111MN ij ji j jNT R R A R RR           , (6) with  11Mj iji jR RM   ,  21 iji jR RM M     (7) and        221 22 1M R RAM M R      ,  (8) which follows asymptotically a χ2 distribution  asym. 2 ,~N fT  , with   1 1 22f M M    (9) degrees of freedom. Hereby M variables are assumed to follow a multivariate normal 121086420loadFax[kN]displacement w [mm]0      1      2         3     4    5wfFax,maxtan–1(Kser)tan–1(Ku)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 distribution. Estimates Rij confirm to the Pearson’s sample correlation coefficients between properties of segments at varying lag-distances. The null hypothesis has to be rejected whenever TN exceeds the (1 – α)-quantile of the χ2 distribution. The best estimator for the equi-correlation coefficient ρequi is represented by the weighted averaged correlation coefficient R  gained from the M  M correlation matrix R = Rij (N realizations of M variables). We use Lawley’s test because of three reasons: (i) a likelihood ratio test for equicorrelation is not available in closed form (Lawley 1963), (ii) the asymptotic null distribution of TN is independent from ρequi (Gleser 1968), and (iii) the test procedure can be easily implemented even in standard software packages like Microsoft® Excel.  3. RESULTS AND DISCUSSION 3.1. Basic statistics and outcomes Table 1 shows basic statistics of withdrawal properties and local densities. Nine data sets were excluded; one because of a knot touching the screw and eight because of statistically outliers, classified via box plots according to Tukey (1977). Analysis was performed on the untransformed and logarithmized data set; rea-soning therefore is indicated later in section 3.2.  Table 1: Basic statistics of withdrawal properties (Kser,12 & Ku,12 as apparent values)   ρseg,12 ρloc,12 fax,12 Kser,12 Ku,12  [kg/m³] [MPa] [N/mm] no. [–] 239 mean  456 430 5.61 5,074 3,074 CV [%] 5.0 5.0 7.2 9.0 15.5  In contrast to previous investigations which outline CV[fax] between 10 and 20 % (Ringhofer et al. 2014a), in our data a much lower variation is found. Also contrary to other timber properties the variations in stiffness properties Kser and Ku are higher than in the corresponding strength. Reasoning is seen in the examination procedure and by additional displacements caused by the test setup. Although not provable from this data set it is assumed that a relevant amount of variation in Kser and Ku is epistemic and not aleatoric. Overall, the low variations identified for all five properties underline the homogeneity of the test material. This is in particular related to the use of clear wood samples, a circumstance which necessitates further discussions on the practical applicability of the outcome.  3.2. Tests on equicorrelation  Lawley’s (1963) test presumes multivariate normal variables. Although density is frequently described as normal distributed, the same model can be physically hardly argued for strength and stiffness properties. Kowalski (1972) outlines the vulnerability of the distribution of equicor-relation on nonnormality for all ρequi ≠ 0. JCSS (2006), Brandner (2013) and others confirm a lognormal distribution (2pLND) as repre-sentative for strengths, elastic properties and even for density. In fact, the lognormal distribution can be directly inferred from multi-plicative processes (Gibrat 1930), or reverse, according to Brandner (2013) from subsequent (cascade) fracture processes which are typical for hierarchically organised materials.  With the definition of U ~ 2pLND, V = ln(U) is normal distributed. Consequently, Lawley’s test can be applied after logarithmic transformation of presumed lognormal variables. The coefficients Requi,V can be re-transferred to Requi,U = Requi by (Law and Kelton 2000)       equi,equi,exp Var 1exp Var 1VUVV   .  (10) Setting the significance level α to 5 %, by testing the complete set of 239 segments (nine to 14 per board), for all five variables (ρseg,12, ρloc,12, fax,12, Kser,12, Ku,12) the hypothesis of equi-correlation (see section 2.2) was rejected at a realized significance of p < 0.001 (p < α). Rea-soning is the considerable decrease of the number of realizations for m > 9. As the number of segments per board is not in relation to the timber quality and in particular to the clear wood tested, Lawley’s test and the statistical analysis 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 were repeated on a reduced, saturated data set which contains only the realizations of the first nine segments per board. The basic statistics, average coefficients of variations within boards and the estimated equicorrelations are shown in Table 2.   Table 2: Basic statistics of withdrawal properties (Kser,12 & Ku,12 as apparent values) and estimated equicorrelations (reduced data set)   ρseg,12 ρloc,12 fax,12 Kser,12 Ku,12  [kg/m³] [MPa] [N/mm] no. [–] 180 mean  457 430 5.62 5,055 3,048 CV[Zik] [%] 4.8 4.8 7.2 9.3 15.6 CV[Xi | k] [%] 2.2 1.9 4.1 7.6 13.5 requi,LN  [–] 0.80 p<0.001 0.85 p=0.001 0.71 p=0.559 0.35 p=0.166 0.26 p=0.951 requi [–] 0.80 0.85 0.71 0.34 0.25  The comparison between the basic statistics of Table 1 and 2 indicates the reduced data set as equally representative. Values of CV[Xi | k] already imply the proportions of variation within and between boards and corresponding equicorrelation coefficients.    Figure 4: Serial (pairwise) correlation of ρseg,12  Figures 4 to 8 visualize pairwise corre-lations in dependency of the lag-distance for the complete and the reduced, untransformed data sets. Both, Pearson’s (P) correlation coefficient (together with a 95 % confidence interval; CI) and Spearman’s (SP) rank correlation coefficient are included for comparison. The increasing bandwidths in the confidence interval with increasing lag-distance consequence from the decreasing amount of data pairs.  Figure 5: Serial (pairwise) correlation of ρloc,12   Figure 6: Serial (pairwise) correlation of fax,12  The equicorrelation of density agrees well with results in Brandner and Schickhofer (2014). Consequently, the variation of (clear wood) densities within boards is widely negligible. However, the null hypothesis was rejected with p ≤ 0.001.  In fact, equicorrelation within timber elements, which is motivated by the hierarchical material structure, is especially obvious by observing the regular pattern of knot clusters of a standing coniferous tree in combination with the radial differentiation in juvenile and adult timber 1   2    3   4   5   6  7  8  9 10 11 12 13lag-distance [–]serial correlation [–]rP,95% CIrP,meanrSP,meanrequiall     red.1.00.90.80.70.60.50.40.30.20.10.01   2    3   4   5   6  7  8  9 10 11 12 13lag-distance [–]serial correlation [–]1.00.90.80.70.60.50.40.30.20.10.0rP,95% CIrP,meanrSP,meanrequill     red.1   2    3   4   5   6  7  8  9 10 11 12 13lag-distance [–]serial correlation [–]1 098.7.60.50.40.30.20.10.0rP,95% CIrP,meanrSP,meanrequiall      red.12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 zones. However, in sawn timber this and the formation of different knot zones in radial and longitudinal direction, as consequence of changing environmental conditions and needs of the living tree, together with the cutting process, which is usually oriented parallel to the pith, impose gradually changes in longitudinal properties’ profiles. These circumstances may to some extend explain the decreasing pairwise correlations in densities (Figure 4 and 5) and other timber properties with increasing lag-distance.    Figure 7: Serial (pairwise) correlation of Kser,12   Figure 8: Serial (pairwise) correlation of Ku,12  In contrast to density, the hypothesis of equicorrelation could not be rejected for all tested withdrawal properties, fax,12, Kser,12 and Ku,12. Figures 6 to 8 visualise the corresponding serial pairwise correlations.  Whereas the results for fax,12 appear quite homogenous, the serial pairwise correlations of the stiffness properties are very heterogeneous and in case of Kser,12 even with a distinctive decreasing tendency. Reasons for the hetero-geneous results and the minor equicorrelation of the stiffness properties are again supposed to be related to the examination procedure, whereas requi [Ku,12] < requi [Kser,12] consequences from additional variation caused by variation of the displacement at ultimate load.  Brandner (2013) summarized previous investigations on equicorrelation of timber properties and concluded ρequi[E] = 0.50 to 0.60 and ρequi[f] = 0.40 to 0.50 for elastic moduli and strengths, respectively, with higher values for timber of higher quality and grade. In comparison to these ranges the coefficients for Kser,12 and Ku,12 appear rather low whereas requi[fax,12] = 0.71 appears rather high. By considering that fax,12 bases on testing of already very homogenous clear wood samples the outcome can be judged as reasonable.  However, in timber engineering placing fasteners near or through knots and other growth characteristics, like checks, reaction wood, bark inclusions and resin pockets, cannot be prevented. The additional variation caused by unavoidable timber properties directly influences the variation of withdrawal properties and thus their spatial variation within timber elements.  4. SUMMARY AND CONCLUSIONS We investigated the serial correlation of withdrawal properties and densities based on a data set of Bratulic (2012). Motivated by the hierarchical structure of timber and successful previous investigations, we analysed the applicability of a two-level hierarchical model combined with equicorrelation. Using the test statistic of Lawley (1963) the hypothesis of equicorrelated properties could not be rejected for fax,12, Kser,12 and Ku,12. However, equicorrelation could not be confirmed for the densities of segments and small clear wood samples. In view of the stochastic modelling of the group action of common acting and primarily 1   2    3  4   5  6  7  8  9 10 11 12 13lag- istance [–]serial correlation [–]1.00.90.80.70.60.50.40.30.20.10.0rP,95% CIrP,meanrSP,meanrequiall    red.1   2    3  4   5   6  7  8  9 10 11 12 13lag-distance [–]serial correlation [–]1.00.90.80.70.60.50.40.30.20.10.0rP,95% CIrP,meanrSP,meanrequiall      re .12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 axially loaded self-tapping screws in dependency of the insertion of fasteners relative to the layup of laminated timber products and the failure mode “withdrawal”, for general applications in timber and unavoidable influences of flaws, e.g. knots, checks and reaction wood, ρequi[fax,12] = 0.71 is judged as too high. In view of previous investigations on equicorrelation of timber strength properties (e.g. Brandner 2013) ρequi[fax,12] in the range of 0.40 to 0.50 (0.60) appears more adequate; modelling the influence of equicorrelation on the group action is envisaged.  5. REFERENCES Brandner, R. (2013). “Stochastic System Actions and Effects in Engineered Timber Products and Structures”. ISBN 978-3-85125-263-7. Brandner, R., and Schickhofer, G. (2014). “Spatial correlation of tensile perpendicular to grain properties in Norway spruce timber”. Wood Science and Technology, 48, 337–352. Bratulic, K. (2012). “Alteration of the withdrawal strength of self-tapping screws along the board and over the varying GLT cross section”. Master thesis, TU Graz, pp. 96. Ditlevsen, O., and Källsner, B. (1998). “System effects influencing the bending strength of timber beams”. IFIP, 8th WG 7.5, Krakow, Poland. Ditlevsen, O., and Källsner, B. (2005). “Span-dependent distributions of the bending strength of spruce timber”. Journal of Engineering Mechanics, ASCE, 131(5), 485–499. EN 384 (2010). “Structural timber – Determination of characteristic values of mechanical properties and density”. CEN.  EN 1382 (1999). “Timber structures – Test methods – Withdrawal capacity of timber fasteners”. CEN. EN 13183-1 (2002). “Moisture content of a piece of sawn timber – Part 1: Determination by oven dry method”. CEN. EN 1995-1-1 (2006). “Eurocode 5: Design of timber structures – Part 1-1: General – Common rules and rules for buildings”. CEN. ETA-12-0373 (2012). “Schmid screws RAPID®, STARDRIVE and SP: Self-tapping screws for use in timber constructions”. EOTA, pp. 36. Gibrat, R. (1930). “Une loi des re´partitions e´conomiques: l’effet proportionnell“. Bulletin de la Statistique Ge´ne´rale de la France, 19, 469–513.  Gleser, L. J. (1968). “On testing a set of correlation coefficients for equality: some asymptotic results”. Biometrika, 55(3), 513–517. Isaksson, T. (1999). “Modelling the Variability of Bending Strength in Structural Timber”. Dissertation, Lund, Sweden, ISSN 0349-4969. Joint Committee on Structural Safety JCSS (2006). “Probabilistic Model Code – Part III Resistance Models – Timber”. pp. 16. Källsner, B., and Ditlevsen, O. (1994). “Lengthwise bending strength variation of structural timber”. IUFRO S 5.02 – Timber Engineering, pp. 20. Källsner, B., Ditlevsen, O., and Salmela, K. (1997). “Experimental verification of a weak zone model for timber in bending”. IUFRO S 5.02 – Timber Engineering, pp. 17. Kowalski, C. J. (1972). “On the effects of non-normality on the distribution of the sample product-moment correlation coefficient”. Journal of the Royal Statistical Society, Series C (Applied Statistics), 21(1), 1–12.  Köhler, J. (2007). “Reliability of Timber Structures”. Dissertation, ETH Zürich, IBK No. 301. Lam, F., Wang, Y.-T., and Barrett, J. D. (1994). “Simulation of correlated nonstationary lumber properties”. Journal of Materials in Civil Engineering, 6(1), 34–53. Law, A. M., and Kelton, W. D. (2000). “Simulation Modelling and Analysis”. 3rd Edition, McGraw-Hill, ISBN 978-0-07059-292-6. Lawley, D. N. (1963). “On Testing a Set of Correlation Coefficients for Equality”. The Annals of Mathematical Statistics, 34(1), 149–151. R Core Team (2012). “R: A language and environment for statistical computing”. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.  Riberholt, H., and Madsen, P. H. (1979). “Strength distribution of timber structures – measured variation of the cross sectional strength of structural lumber”. Technical University of Denmark, No. R 114.  Ringhofer, A., Brandner, R., and Schickhofer, G. (2014a). “Development of an optimized screw 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  9 geometry for highstressed steel-to-timber joints“. Bautechnik, 91(1), 31–37.  Ringhofer, A., Grabner, M., Vilaca Silva, C., Branco, J., and Schickhofer, G. (2014b). “The influence of moisture content variation on the withdrawal capacity of self-tapping screws”. Holztechnologie, 55(3), 33–40.  Speck, T., and Rowe, N. P. (2006). “How to become a successful climber – mechanical, anatomical, ultra-structural and biochemical variations during ontogeny in plants with different climbing strategies”. 5th Plant Biomech. Conf., Vol. I, 103–108.  Taylor, S. E. (1988). “Modelling spatial variability of localized lumber properties”. Dissertation, Texas A&M University of Florida, pp. 283. Tukey, J. W. (1977). “Exploratory Data Analysis”. Addison-Wesley, ISBN 0-201-07616-0. Williamson, J. A. (1992). “Statistical models for the effect of length on the strength of lumber”. Dissertation, UBC, pp. 270.  

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