Stress state of a laminated beam under normal load

An analytical method for determining the bending stresses of a beam of a layered structure with an arbitrary number of layers, including the last layer of semi-infinite thickness, is proposed. The method consists in taking the indicators of the stress-strain state of the layers using the Eri stress functions of the linear theory of elasticity. They are taken in the form of sums of products of exponential and trigonometric functions and such products multiplied by the coordinate of the axis normal to the beam. Based on the boundary conditions, conditions of compatibility and continuity of deformation of the connected layers, a system of linear algebraic equations is formulated. The solutions of the systems are found to the values of the coefficients of the stress functions. The known coefficients of the Eri functions allow determining the stress-strain state of the layers of the beam. The cases of supporting the beam by parts of its surface and its holding by the ends are considered. The following is established. Compressive deformations are small relative to deflection deformations provided that the length of the beam significantly exceeds its thickness. The calculated deflection of the lower surface of the beam is close to parabolic in nature. The dimensions of the supporting surfaces exceeding 1% of its length significantly affect its deflection - they reduce it. The analytically determined average deflection of the beam is 0.7% less than the deflection calculated by the methods of material resistance. Beam deflections also depend on the properties of the material of the beam layers, so the increase in Poisson's ratio from a minimum, equal to zero, to a practically maximum, taken equal to 0,45, leads to a decrease in the deflections of the surfaces of the layers. The decrease does not exceed 10%. The proposed method is based on the provisions of the linear theory of elasticity. The transformations are carried out without simplifications. The results obtained do not contradict the results obtained by the methods of material resistance, within the limits of the linear formulation have a sufficiently high level of reliability. They can be used to calculate the stress-strain state of beams of a layered structure of rectangular cross-section, including beams on an elastic base.