PSI - Issue 78

Angelo Aloisio et al. / Procedia Structural Integrity 78 (2026) 25–32

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where subscripts 11, 22, and 12 identify the directions parallel to the grain, perpendicular to the grain, and tangential. The ultimate goal is to estimate the reduction capacity factor ( k red ) compared to the reference configuration without holes, as reported below

Wf m , hi Wf m

f m , hi f m

R m , hi R m

k red =

(7)

=

=

where R m , hi is the bending resistance in the perforated beam and R m the bending resistance in the reference configu ration without holes. As reported in the same equation, this expression can be written in terms of the ratio between the bending strengths, assuming the same bending modulus of resistance ( W ). The section modulus W governs elastic bending sti ff ness. Lab tests on beams drilled with small holes [22] show that the global sti ff ness, and thus W , hardly changes when the holes are no larger than other natural defects. The authors can therefore treat W as constant when they compare a sound beam with a perforated one. Equation (7) turns the elastic stresses into a capacity-reduction factor that the authors express as a percentage. Recall that the ratio k strength scales each stress by the matching strength value, so results along and across the grain are on the same footing. Because wood is much weaker perpendicular to the grain, a modest cross-grain stress can still control failure even if it is smaller than the parallel stress. Finally, notice that k red depends only on relative stress levels inside the beam, not on the actual load applied. Any convenient load in the four-point-bending FE model will therefore give the same reduction factor.

4. Dataset generation from the stochastic FE model

The Monte-Carlo workflow contained three key stages: (i) create random layouts of holes, (ii) build a 2-D Abaqus model and run an elastic bending analysis, (iii) post-process the results to obtain the capacity-loss factor k red viaEq. 7. The authors considered the three beam sizes most common on Norwegian sites—50 × 100, 50 × 150 and 50 × 200 mm. For every simulation the variables in Table 1 were sampled uniformly: number of hole clusters N g ; holes per cluster N h ; hole diameter d ; and the Cartesian coordinates of each cluster centre and its holes. Beam span was fixed to L = 19 H (depth H ).

Table 1. Parameters varied in the Monte-Carlo runs and their bounds. Description Symbol

Lower bound Upper bound

Total number of hole clusters

N g

1 1 1

5

Hole diameter Holes per cluster

d

30

5

N h

Cluster centre ( x C , y C ) [mm]

[ x C , y C ] [ L / 19, 10]

[18 L / 19, H − 10] [50mm, 2 π ]

Hole position in polar coords [ ρ,θ ]

[0, 0]

Hole positions are generated in polar form ( ρ,θ ) around a randomly chosen reference point ( x C , y C ); the point is merely a seed, not the true centroid of the set. A geometric filter then removes cases with overlapping holes or holes outside the beam envelope. The geometry is passed to Abaqus through a Python script that meshes the beam, applies four-point bending and records the stress field automatically. Orthotropic properties follow EN 338 values for C24 spruce (Table 2). For each cluster the maximum stresses are extracted, divided by the corresponding clear-wood strengths (Table 3) and inserted into Eq. 7. Reference stresses for an intact beam of each depth are pre-computed once, so the dimension less ratio k stress andhence k red are obtained rapidly for every Monte-Carlo sample.

5. Classification model and key findings

The dataset, geometric descriptors of every hole pattern plus the capacity, loss factor k red from Eq. 7—was used to train two families of classifiers: