PSI - Issue 68

Gül Demirer et al. / Procedia Structural Integrity 68 (2025) 190–196 G. Demirer and A. Kayran / Procedia Structural Integrity 00 (2024) 000–000

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3

The fiber orientation varies along the x-direction and the reference path must be shifted in the y-direction to cover the plate. The fiber orientation with a constant curvature can be expressed as follows:

[ cosT 1 − cosT 0 ] w / 2 .

(2)

cos φ = cosT 0 + | x |

(b)

(c)

(a)

Fig. 1. (a) Boundary conditions, (b) reference path with a constant curvature, (c) centerline and course boundaries for a constant D (top-sided tow drop). ( T 1 < T 0 )

The lower and upper portion of the centerline are formulized separately. It can be seen in Fig. 1b that the upper portion is a part of a circle with radius of ρ and has its center at O ( x 0 , y 0 ) where x 0 = − ρ cosT 0 and y 0 = ρ sinT 0 . Similarly, the lower half of the centerline forms part of a circle with the same radius ρ but its center is located at O ′ ( x 0 ′ , y 0 ′ )where x 0 ′ = ρ cosT 0 and y 0 ′ = − ρ sinT 0 . Alternatively, the equation can also be obtained by considering the OAB triangle in Fig. 1b and applying the Pythagorean theorem. Finally, the resulting equations for the centerline are as follows: y centerline = ρ sinT 0 − ρ 2 − ( ρ cosT 0 + x ) 2 for 0 ≤ x ≤ w / 2 (3a) y centerline = − ρ sinT 0 + ρ 2 − ( ρ cosT 0 − x ) 2 for − w / 2 ≤ x ≤ 0 . (3b) In the current study, the vertical distance D, between the upper and lower course boundaries is taken as constant to avoid formation of large gaps and overlaps while o ff setting the reference course. This is achieved by keeping the lower boundary fixed and adjusting the upper one. A geometrical procedure, as outlined by Fayazbakhsh (2013), is followed to locate the gaps and overlaps for a top-sided tow drop scenario. The formulation of the lower half of the centerline and the bottom boundary as well as the derivation of the vertical distance, D , are explained with the help of Fig. 1c, which shows a case where T 1 < T 0 . The centerline equation is already obtained in Eq. 3. It can be seen in Fig. 1c that the bottom boundary resembles the centerline with a radius of ( ρ − W course / 2) rather than ρ , where W course is the maximum course width, defined as the product of tow width and number of tows. Therefore the lower half of the bottom boundary is expressed as y bottom = − ρ sinT 0 + ( ρ − W course 2 ) 2 − ( ρ cosT 0 − x ) 2 for − w / 2 ≤ x ≤ 0 . (4) 2.2. Identification of gap and overlap locations