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Probabilistic models for the modulus of elasticity and shear in serial and parallel acting timber elements

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Abstract

Elastic properties, like modulus of elasticity and shear, are basic design parameters. Besides calculation of deformations in serviceability limit state design, these properties are also applied to ultimate limit state design, considering stability analysis of columns and girders and the calculation of internal forces in statically indeterminate structures. However, analytical probabilistic models explicitly considering the stochastics of elastic properties in timber design and product modelling are scarce or even missing. Focus was put on elastic properties of structural timber and timber products, composed of face-by-face rigidly and continuously bonded timber members. Considering timber members as rigid composites of sub-elements in series and timber products as rigid composites of timber members as serial system of parallel sub-elements, probabilistic models for parallel and serial acting elements are obtained assuming linear elastic theory and lognormal distribution. Motivated by the material structure, a two-level hierarchical model was modelled explicitly considering within- and between-element variation and correlation. Parameter settings were defined for three timber qualities. The authors (1) investigated structural timber and timber products with and without finger joints and quantified the homogenization in dependence on input parameters, i.e. the coefficient of variation and the number of serial and/or parallel interacting elements and sub-elements, (2) compared the outcome with current regulations in European standards, e.g. EN 338:2009 and EN 14080:2013 and JCSS:2006, and (3) gave recommendations for revisions. Although not explicitly demonstrated, the probabilistic models have additional potential in calculating the stiffness of joints consisting of multiple fasteners, as well as of joint stiffness in flexible composites, e.g. mechanically jointed beams where the stiffness parameters are later used in shear analogy or γ-method.

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Abbreviations

CLT:

Cross-laminated timber

CV[X]:

Coefficient of variation of random variable X

CoVar[X]:

Covariance of random variable X

d :

Depth

d WZ :

Distance between weak zones

E c :

Elastic modulus in compression

E m :

Elastic modulus in bending

E t :

Elastic modulus in tension

E[X]:

Expectation of random variable X

E 0 :

Elastic modulus parallel to grain

E 90 :

Elastic modulus perpendicular to grain

f c,0 :

Compression strength parallel to grain

FJ:

Finger joint

f m :

Bending strength

F X (x):

Distribution function of random variable X

GLT:

Glued laminated timber (glulam)

G 090 :

Longitudinal shear modulus

G 9090 :

Transverse (rolling) shear modulus

iid:

Independently and identically distributed

I tor :

Torsion moment of inertia

I z :

Moment of inertia in z-direction

HQ:

High quality

k crit :

Factor for lateral (torsional) buckling

k c,y ; k c,z :

Instability factor in y-/z-direction

k sys :

System factor

l ef :

Effective length

l ref :

Reference length

LQ:

Low quality

LVL:

Laminated veneer lumber

M :

Number of serial acting elements

m :

Number of finger joints per reference length

MG:

Machine graded

MQ:

Medium quality

M y,crit :

Critical bending moment

N :

Number of parallel acting elements

ND:

Normal distribution

RSDM:

Representative statistical distribution model

SLS:

Serviceability limit state

t :

Thickness

ULS:

Ultimate limit state

Var[X]:

Variance of random variable X

VG:

Visual graded

W y :

Section modulus in y-direction

X j |k :

Two-level hierarchical model: jth iid deviation of sub-element j from Y k given element k

X d :

Design value of random variable X

X g :

Property of random variable X for glulam

X k :

Characteristic value of random variable X

X l :

Property of random variable X for lamellas (boards)

X mean :

Mean value of random variable X

X 05 :

5 %-Quantile of random variable X

X 1 :

Property of random variable X for a single element

X N :

Property of random variable X for N parallel acting elements

X M :

Property of random variable X for M serial acting elements

Y k :

Two-level hierarchical model: mean value of the analysed property of the kth element

Z jk :

Two-level hierarchical model: property of sub-element j given element k

λ y ; λ z :

Geometric slenderness in y-/z-direction

λ rel :

Relative slenderness

μ X :

Expectation of random variable X

ρ :

Pearson correlation coefficient

ρ equi :

Equicorrelation coefficient

σ X :

Standard deviation of random variable X

σ c,0 :

Compression stress parallel to grain

σ m; σ m,crit :

(Critical) bending stress

2pLND:

Two-parametric lognormal distribution

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Acknowledgments

The research work was conducted at the Institute of Timber Engineering and Wood Technology at the Graz University of Technology in the course of the dissertation published in Brandner (2013).

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Authors and Affiliations

  1. Institute of Timber Engineering and Wood Technology, Graz University of Technology, Inffeldgasse 24/I, 8010, Graz, Austria

    R. Brandner & G. Schickhofer

  2. Competence Centre holz.bau forschungs gmbh, Graz, Austria

    R. Brandner

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  1. R. Brandner
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  2. G. Schickhofer
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Brandner, R., Schickhofer, G. Probabilistic models for the modulus of elasticity and shear in serial and parallel acting timber elements. Wood Sci Technol 49, 121–146 (2015). https://doi.org/10.1007/s00226-014-0689-1

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