Interaction of elastic plate weakened by multiple holes with a set of patches

Reinforcement of a plate weakened by multiple holes with several patches for different types of plate-patch attachment

“…. , n + r, while the condition (4.18) provides that the resultant force applied to the patch S k is equal to zero [8], [10]. It can be shown that the system (4.13)-(4.16) has a unique solution.…”

“…While the precise investigation of the convergence of the solution of the resulting system of linear algebraic equations to the solution of the system of singular integral equations (4.13)-(4.16) is outside of the scope of this paper, the numerical results obtained by using this method and also comparison with the numerical results obtained by the method of mechanical quadratures in [10] suggests that even relatively small number of terms N (N ≤ 20 in most cases) is sufficient to obtain a satisfactory convergence, thus, reducing the problem to the solution of a relatively small system of linear algebraic equations. For instance, consider a plate with two holes L 1 and L 2 in the shape of the squares with rounded corners given by the equations τ 1,2 (θ) = 0.45(e iθ + e −3iθ /9) ∓ 1 which is reinforced by two patches in the shape of the squares with rounded corners with the boundaries Γ 1 and Γ 2 given by the equations ξ 1,2 (θ) = 0.7(e iθ + e −3iθ /14) ∓ 1 (fig.…”

“…Similar results have been observed in other considered examples. The comparison of the results with the method of mechanical quadratures [10] is presented on the fig. 4.…”

“…The patch repair of a single hole in a thin plate reinforced by a single patch is considered for two different types of the plate-patch attachment in [8], [9]. To the author's knowledge, a patch repair of multiple holes in a plate by multiple patches of arbitrary shapes has not been a subject of many studies with an exception of [10].…”

“…One of the main goals of this paper is to propose an alternative technique for a numerical solution of the derived system of singular integral equations. The method of mechanical quadratures applied for the solution of similar systems in [8], [9], [10] is difficult to implement for a large number of contours due to decreasing precision. The idea behind the method proposed here is in the approximation of the unknown functions by trigonometric polynomials.…”