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Loss in moment capacity of tree stems induced by decay

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We model varying decay in tree cross-sections by considering bending theory to estimate moment capacity loss (MCL) for the sections. We compare MCL with experiments on selected oak trees.

Abstract

Tree failures can damage property and injure people, sometimes with fatal consequences. Arborists assess the likelihood of failure by examining many factors, including strength loss in the stem or branch due to decay. Current methods for assessing strength loss due to decay are limited by not accounting for offset areas of decay and assuming that the neutral axis of the cross-section corresponds to the centroidal axis. This paper considers that strength loss of a tree can be related to moment capacity loss (MCL) of the decayed tree cross-section, because tree failures are assumed to occur when induced moments exceed the moment capacity of the tree cross-section. An estimation of MCL is theoretically derived to account for offset areas of decay and for differences in properties of wood under compressive and tensile stresses. Field measurements are used to validate the theoretical approach, and predictions of loss in moment capacity are plotted for a range of scenarios of decayed stems or branches. Results show that the location and size of decay in the cross-section and relative to the direction of sway are important to determine MCL. The effect of wood properties on MCL was most evident for concentric decay and decreased as the location of decay moved to the periphery of the stem. The effect of the ratio of tensile to compressive moduli of elasticity on calculations of MCL was negligible. Practitioners are cautioned against using certain existing methods because the degree to which they over- or underestimate the likelihood of failure depended on the amount and location of decay in the cross-section.

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Acknowledgments

This study was funded by the TREE Fund’s Mark S. McClure Biomechanics Fellowship. Nevin Gomez, Sherry Hu, Alex Julius, Dan Pepin, Alex Sherman, and Joseph Scharf (University of Massachusetts-Amherst) helped collect data.

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Correspondence to Cihan Ciftci.

Additional information

Communicated by T. Fourcaud.

Appendices

Appendix A: Acronyms and abbreviations

c :

Perpendicular distance from the neutral axis to a point whose tension or compression stress will be calculated

E C :

Wood compressive Young’s modulus parallel to grain

E T :

Wood tensile Young’s modulus parallel to grain

F C :

Resultant compressive force at the compression side in a stem cross-section

F T :

Resultant tensile force at the tension side in a stem cross-section

H :

Height (see Fig. 4) between the puling force and the strain meter of the experiments described in “Experiments on MCL

I :

Moment of inertia of a cross-sectional area computed about the neutral axis

k d :

The slope of the curve in Fig. 4 after notching

k ud :

The slope of the curve in Fig. 4 before notching

M :

Bending moment

M d :

Calculated bending moment [Eq. (8)] for the experiments (after notching) described in “Experiments on MCL

M ud :

Calculated bending moment [Eq. (8)] for the experiments (before notching) described in “Experiments on MCL

n :

Modular ratio of E T to E C

P d :

Pulling force measures from the experiments (after notching) described in “Experiments on MCL

P ud :

Pulling force measures from the experiments (before notching) described in “Experiments on MCL

r :

Radius of the circular decay defined in Fig. 1

r 0 :

Distance between the centers of the tree section and decayed area defined in Fig. 1

R :

Radius of the circular tree cross-section defined in Fig. 1

S :

Section modulus of tree cross-sections

ε c :

Strain at the outer face of the compression side in a stem cross-section

ε d :

Strain meter measures from the experiments (after notching) described in “Experiments on MCL

ε t :

Strain at the outer face of the tension side in a stem cross-section

ε ud :

Strain meter measures from the experiments (before notching) described in “Experiments on MCL

σ :

Bending stress

σ c-yield :

Yield compressive stress at the outer face of the compression side in a stem cross-section

σ t :

Tensile stress at the outer face of the tension side in a stem cross-section

θ :

Angle (see Fig. 4) between the pulling force and the tree stem for the experiments described in “Experiments on MCL

Appendix B: MCL for areas of decay confined within the cross-section

If n = 1, MCL of areas of decay confined within the cross-section can be determined from the ratio of the section modulus of the decay to that of the tree cross-section, assuming each is circular. To obtain the section moduli, first, the moments of inertia (I) for the tree cross-section and decay area are:

$$I = \frac{\pi }{4}R^{4}$$
(18)
$$I_{\text{d}} = \frac{\pi }{4}r^{4}$$
(19)

in which R and r are the radii of cross-section and the decayed area, respectively, and the subscript d indicates the area of decay. The areas (A) of these sections are:

$$A\, = \,\pi R^{2}$$
(20)
$$A_{\text{d}} = \pi r^{2}$$
(21)

To find the location of the neutral axis (ΔX) of the tree cross-section with decay:

$$\Delta X = \frac{{A_{\text{d}} r_{0} }}{{A - A_{\text{d}} }}$$
(22)

where r 0 refers to the distance between the centers of the tree section and decayed area. The moment of inertia of the hollow section (the tree cross-section with decay) (I H) can be determined:

$$I_{\text{H}} = I - I_{\text{d}} + A\left( {\Delta X} \right)^{2} - A_{\text{d}} \left( {\Delta X + r_{0} } \right)^{2}$$
(23)

Then, the distance (c) between the circumference of tree cross-section and the neutral axis is:

$$c = R + \Delta X$$
(24)

Finally, loss of section modulus can be calculated as:

$$s_{\text{loss}} \, = \,\frac{{{\raise0.7ex\hbox{$I$} \!\mathord{\left/ {\vphantom {I R}}\right.\kern-0pt} \!\lower0.7ex\hbox{$R$}} - {\raise0.7ex\hbox{${{}^{I}H}$} \!\mathord{\left/ {\vphantom {{{}^{I}H} c}}\right.\kern-0pt} \!\lower0.7ex\hbox{$c$}}}}{{{\raise0.7ex\hbox{$I$} \!\mathord{\left/ {\vphantom {I R}}\right.\kern-0pt} \!\lower0.7ex\hbox{$R$}}}}$$
(25)

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Ciftci, C., Kane, B., Brena, S.F. et al. Loss in moment capacity of tree stems induced by decay. Trees 28, 517–529 (2014). https://doi.org/10.1007/s00468-013-0968-8

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