PSI - Issue 78

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

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where σ 11 , m is the peak longitudinal tensile stress at mid-span, f m is the characteristic bending strength of the grade, σ i j , h i is the peak stress around hole h i in direction i j , and f i j is the strength in that direction. Note that f m di ff ers from f 11 : the first comes from bending tests used for grading, while f 11 is the clear-wood tensile strength along the grain [4]. The target is the drop in bending capacity of a perforated beam versus a sound beam, obtained through linear analyses only. To do so the authors treat the stress check at each hole as an equivalent bending check and introduce an e ff ective bending strength f m , i j that accounts for the hole. The next algebraic steps lead from the local stress ratios to that equivalent strength and, finally, to a simple reduction factor that the classifier can use.

k strength , i j k stress , i j = f m , i j    Equivalent bending verification

σ 11 , m σ 11 , m ≤ f m f m    Algebraic manipulation f i j

→ σ 11 , m ≤ f m

(2)

σ i j , h i ≤ f i j    Stress verification at the h i hole

→ σ i j , h i

where k strength , i j , k stress , i j and f m , i j , are defined as follows

f i j f m

(3)

k strength , i j =

σ i j , h i σ 11 , m

k stress , i j =

(4)

k strength , i j k stress , i j

f m , i j = f m

(5)

Namely, k strength , i j in Eq. 3 expresses the ratio between the strength along the i j direction ( f i j ) and the bending strength ( f m ), while k stress , i j in Eq. 4 expresses a ratio among maximum stresses around the hole in the i j direction ( σ i j , h i ) and the maximum longitudinal tensile stress at the midspan ( σ 11 , m ). Eq. 5 represents the bending capacity of the beam equivalent to the hole verification along the i j direction. Eq. 2 reduces to a bending check if k strength , i j = k stress , i j = 1. Therefore, Eq. 2 specializes in four distinct equations in the plane case. Specifically, it reduces to a bending verification if k strength , i j = k stress , i j = 1 and in a verification around the generic i -th hole in the 11, 22, and 12 directions, knowing the strength values of the respective directions, parallel to the grain, perpendicular to the grain, and tangential, respectively. Therefore, the bending capacity of the beam with hole ( f m , hi ) can be considered as the minimum between the four, as in

    f m , f m f m f m

k stress , 11 k strength , 11 k stress , 12 k strength , 12 k stress , 22 k strength , 22

,

f m , hi = min

(6)

,