The effect of transverse shear strain on laminated curved beams

Typescript (photocopy).A theory is developed to take in to account the effects of transverse shear strain on the shear stress and circumferential normal stress distributions in homogeneous isotropic or nonhomogeneous laminated curved beams. The theory is based on Kozik's linear exact displacement variations which explicitly show the necessary correction terms which must be added to the classical Kirchhoff displacement variations to take into account the effects of the transverse shear and radial normal strains. An approximate method for determining the shear stress and circumferential normal stress distributions including the effects of transverse shear strain only is presented. The solution scheme is achieved by discretizing the beam into an arbitrary number of layers of equal thickness. A system of differential equations is developed by writing local equilibrium equations at the interfaces between the laminae. The shear stresses at each interface are solved for explicitly and these values are then used to determine the circumferential normal stress distribution. An approximate method for determining shear stress distributions in curved beams based on the classical Kirchhoff assumption of nondeformable normals is presented, since no simple mechanics of materials formulation is available. This solution scheme is also accomplished by discretizing the beam into an arbitrary number of layers or equal thickness. Example problems are presented to demonstrate the application of the theory. A group of homogeneous isotropic beams and a group of three-layered nonhomogeneous beams are solved. The results show that for isotropic circular curved beams with rectangular cross sections, the Kirchhoff assumption of nondeformable normals is quite good for a wide range of curvatures. The warping effect of transverse shear strain increases as the curvature of the beam becomes sharper. In the three-layered nonhomogeneous beam problems, the effect of the transverse shear clearly depends on the degree of curvature and the difference between the elastic moduli of the beam layers. The Kirchhoff assumption of nondeformable normals is clearly inadequate in cases where the beam curvature is sharp and the elastic moduli of the various layers are significantly different