We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas. The obtained formulas remain the same for other types of homogeneous boundary conditions, for example, when the horizontal sides of the rectangle are firmly clamped, have stiffening ribs that work in tension–compression and/or bending, etc. Obviously, in this case, the Papkovich–Fadle eigenfunctions will change, as well as the corresponding biorthogonal functions and normalizing factors. To solve a specific boundary value problem, it is enough to find the Lagrange coefficients, which are determined from simple formulas, as Fourier integrals of boundary functions specified at the ends of the rectangle, and then substitute them into the necessary formulas. Examples of solving two problems (even-symmetric deformation relative to the central coordinate axes) are given: (a) The normal stresses are known at the rectangle ends, and the tangential ones are zero; and (b) the longitudinal displacements conditioned by the action of some normal stresses are known at the rectangle ends (the tangential stresses are zero). These solutions are compared with the known solutions in trigonometric Fourier series. The basis of the exact solutions obtained is the theory of expansions in Papkovich–Fadle eigenfunctions based on the Borel transform in the class of quasi-entire functions of exponential type (developed by the authors in their previous studies).

我们推导了公式，该公式描述了弹性理论中边值问题的精确解，其中矩形的两个相对（水平）侧是自由的，而另一侧指定了应力（所有相对于中心轴对称的情况）两侧（矩形末端）。还给出了半条带的公式。解以Papkovich-Fadle本征函数表示为级数，其系数由简单公式确定。对于其他类型的齐次边界条件，例如，当矩形的水平边被牢固地夹紧，具有在拉伸，压缩和/或弯曲等情况下起作用的加劲肋时，所获得的公式将保持不变。显然，在这种情况下， Papkovich–Fadle本征函数将发生变化，以及相应的双正交函数和归一化因子。要解决特定的边值问题，只需将由简单公式确定的拉格朗日系数作为在矩形末端指定的边界函数的傅立叶积分，然后将其代入必要的公式即可。给出了解决两个问题的示例（相对于中心坐标轴的对称变形）：（a）在矩形端已知法向应力，并且切向应力为零；（b）在矩形的端部（切向应力为零）已知受某些法向应力作用调节的纵向位移。将这些解决方案与三角傅里叶级数中的已知解决方案进行比较。