Stress state of hinged supported thin-wall elastic structures

The paper studies the stress-strain state of flat elastic isotropic thin-walled shell structures in the framework of the S. P. Timoshenko shear model with pivotally supported edges. The stress-strain state of shell structures is described by a system of five second-order nonlinear partial differential equations under given static boundary conditions with respect to generalized displacements. The system of equations under study is linear in terms of tangential displacements, rotation angles, and nonlinear in terms of normal displacement. To find a solution to the system that satisfies the given static boundary conditions, integral representations for generalized displacements containing arbitrary holomorphic functions are used. Finding holomorphic functions is one of the main and difficult points in the proposed study. The integral representations constructed in this way allow us to reduce the original problem to a single nonlinear operator equation with respect to the deflection, the solvability of which is established using the principle of compressed maps.


Introduction
Currently, there are a large number of works devoted to the strength of thin-walled shell structures, taking into account the geometric and/or physical nonlinearity [1][2][3][4][5][6][7][8][9]. This is due to the widespread use of elastic thin-walled shell structures in aviation, space technology, shipbuilding, mechanical engineering and construction. In very rare cases, nonlinear problems are solved in a closed form. For this reason, a wide range of approximate methods with the use of computers is used to solve them. When solving problems numerically, the problem of convergence of the numerical solution to the exact (real) solution of the problem always comes to the fore. The solution to this problem, as is known, is based on a rigorous mathematical study of the stress-strain state of thin-walled shell structures. At present, this problem is sufficiently fully studied in the framework of the simplest Kirchhoff-Love model [10][11][12][13][14][15][16][17]. Questions related to the qualitative study of the stress-strain state within the framework of more general models of the theory of thin-walled shell structures that do not rely on the Kirchhoff-Love hypotheses were included in the well-known list of unsolved problems of the mathematical theory of shells by I.I. Vorovich [10] and remained open until recently. To date, there are a number of works [18][19][20][21][22][23][24] devoted to the study of the stress-strain state in the framework of the Timoshenko shear model. The research in [18][19][20][21][22][23][24] is based on integral representations for generalized displacements containing arbitrary holomorphic functions that are found in such a way that the generalized displacements satisfy the given boundary conditions. Two approaches are used to build them. The first approach is based on the application of explicit representations of solutions to Riemann -Hilbert problems for holomorphic functions in the unit circle. Therefore, a flat region homeomorphic to the median surface of the shell is either assumed to be a unit circle from the very beginning [18][19][20], or conformally mapped to a unit circle [22], [24]. In the second approach, the theory of onedimensional singular integral equations is used to determine holomorphic functions [21], [23]. In this paper, the conformal mapping method is used to study a nonlinear problem for arbitrary flat shells under different boundary conditions.

Problem A
Find a solution to the system (1) satisfying the boundary conditions (2)-(4).
We will study the boundary value problem A in a generalized setting. We consider the following conditions to be satisfied: а)  is a simply connected domain with the boundary ГС 1 ; b) external forces R j (j = 1̅ ,̅ 3̅ ), L k (k = 1,2)  Lp(), the components of external forces N 1 , P 1  C(Г); here and further everywhere: p  2, 0    1.
Let's consider a system of the first two equations in (1), in which the deflection w3 is temporarily assumed to be fixed. The general solution of the system (1) has the form [18]: We find the function 2(z) from the boundary condition w2=0 on Г. We obtain the Riemann-Hilbert problem with the boundary condition for a holomorphic function 2(z) in the domain : Let us denote by the function z =  () the conformal mapping of the single circle K ̅ : ||  1 onto the domain ̅ . Since the condition: a) is satisfied for the region , it follows from [26, p. 25] that the function  () belongs to the space C 1 (K ̅ ). In the boundary condition (7), we make the change t → (t), 2((t)) → 2(t), leaving the same designations for the new variables. Thus, in the single circle K we arrive at the Riemann -Hilbert problem for a holomorphic function 2(z) with the boundary condition: where c0 an arbitrary real constant. We find the holomorphic function 1(z) using the boundary condition (3). The expressions of the functions w1, w2 from (6) are introduced into (3). Taking into account the ratios: (10) the boundary condition (3) can be represented as: , (5). Through Sd[1] + (t) they denote the limit of the function Sd[1](z) at z → t  Г from inside the region . (t) the boundary value of the holomorphic function : Thus, for the function (z) in an arbitrary domain  we have the Riemann-Hilbert problem with the boundary condition (11). We will reduce this problem to the problem in a single circle. Using the conformal mapping of the domain ̅ onto the single circle K ̅ , we obtain: Therefore, leaving the previous designations for the new variables, taking into account (10) and (14), we arrive at the Riemann-Hilbert problem for a holomorphic function (z)'(z) in the single circle K ̅ with the boundary condition: .
Let us study the problem (15). The index of the problem (15) is -1. Then, following [27, p. 253], the solution of the problem (15) has the following form: in this case, they satisfy the condition of this problem solvability: Let's transform the representation (18). To do this, you need to find Sd[Φ1] + (t), Φ'2(z), Φ1(z). The representation for Sd[Φ1] + (t) is found in [22]. For the function Φ'2(z) (z  K ̅ ) from (9), using the representation d[g] from (6) and taking into account that ((t+z)/(t-z)) = 2t/(t-z) 2 we have the following: Let us calculate all the integrals on the right-hand side of the representation (19). Taking into account the operator T representation (6) and changing the variable t → φ(t), we obtain: Further, taking into account the representation for the function φ0(τ;z) = (φ(τ) -φ(z))/(τ -z) and the Cauchy formula, we find the following integrals easily: Then the first integral in the formula (19), taking into account (20) and the representation Tf from (6), is transformed to the form: Applying the formula (4.9) [26, p. 29], we have: Then, taking into account (21), we obtain: Let us transform the second integral in representation (19). First, for this we use (20), then we apply the formula (4.9) to the integral over the domain T [10, p. 29], and then -the last formula in (21), and calculate the integral: It's obvious that: Let us calculate the third integral in representation (19). Applying the formulas (20) and (21), we obtain: Let Kε the area obtained from the area K by throwing out the circle centered at the point z : |ζ -z| ≤ ε with the boundary ∂Kε = ∂K ∪ γε̅ . Then we have: Using the formula (4.7) [26, p. 28], we calculate the following integral: Here the first term is equal to zero, since the function Φ1(τ)φ'(τ)τ/(1 -z ̅ τ) is the boundary value of the function Φ1(ζ)φ'(ζ)ζ/(1 -z ̅ ζ), holomorphic by ζ in K ̅ . We use the replacement for the second term: i Therefore, the integral over the region K: Similarly, using the formula (4.7) [26, p. 28] in the case of the domain Kε, we find the integral: Thus, for the third integral in representation (19), we have: Now we calculate the fourth integral in the representation (19). Taking into account (20), (21), the formula (4.7) [26, p. 28] and Cauchy's formula, we obtain: Substituting the found expressions of the integrals into the formula (19), we obtain the following representation for Φ2'(z): We substitute the expressions Φ2'(z) from (22) and Φ(z) from (23) into the formula (18). After cumbersome transformations, we get the following representation for Φ1(z): where the functions gj ≡ gj(w3) are defined in (5), the operator H0[f;g] -in (26), c1 an arbitrary real constant.
Besides, the solvability condition must be satisfied: , Further, proceeding as in [22], we reduce this equation to the following form: In this case, the solvability condition (29) will be satisfied identically. Thus, we obtain an unambiguous representation for the functions Ψ1, Ψ2 through the deflection w3 of the form (30).
Problem A will be reduced to a single operator equation for the function w3. For this, in the third equation of system (1), the functions w1, w2, Ψ1, Ψ2 and their first-order derivatives are replaced by expressions from (26), (30). Then we arrive at a nonlinear second-order partial differential equation with respect to w3 of the form: , 0 Thus, finding the solution to the problem A was reduced to the equation (31) solution for the function w3 with the condition w3=0 on the boundary Г. The problem is the equivalent equation: w3 + G*w3 = 0, G*w3 = (I + K) -1 Gw3, -the harmonic Green's function for the domain Ω, the operator G1w3 is defined in the formula (31). The operator G*w3 is nonlinear bounded operator in Wp (2) (Ω). Moreover, for any two values of w3 j  Wp (2) (Ω), j = 1.2, belonging to the ball

Results and discussion
Let us assume that the external forces acting on the shell and the radius r of the sphere are such that the following conditions are satisfied: q* < 1, Then the equation (32) in the ball has a unique solution w3  Wp (2) (Ω), 2 < p < 2/(1 -β) according to [28]. Substituting the found solution into the formulas (26), (30), we find the functions w1, w2, Ψ1, Ψ2  Wp (2) (Ω). Moreover, the functions w2, Ψ1, Ψ2 are determined uniquely, and the function w1 up to a constant term c0. The condition (27) is not only sufficient, but also a necessary condition for the solvability of the problem A.

Conclusion
Let conditions а), b) from the problem A, and the inequality (33) be satisfied. Then, to solve the problem A, it is necessary and sufficient that the condition (27) be satisfied. In the case of its fulfillment, the problem has a generalized solution a = (w1, w2, w3, Ψ1, Ψ2)  Wp (2) (Ω),