Curved Beam

(in strength of materials and the theory of elasticity), a body whose geometric shape is formed by the motion in space of a plane figure (called the cross section of the curved beam); its center of gravity always follows a certain curve (the axis), and the plane of the figure is normal to the curve. A distinction is made between curved beams with constant cross section (for example, the link of a chain composed of oval or circular rings) and with variable cross section (for example, the hook of a crane) and between plane beams (with a plane axis) and three-dimensional beams (with a three-dimensional axis). A special variety of curved beam is the naturally twisted curved beam, whose plane cross-sectional figure moves along its axis and simultaneously rotates around a tangent to the axis (for example, the blade of an aircraft propeller or fan).

The design of a plane curved beam (Figure 1) with a symmetrical cross section (the axis of symmetry lies in the plane of curvature) taking into account the effect of a load lying in the plane of symmetry consists in the determination of stresses normal to the cross section according to the formula

where F is the area of the cross section, N is the longitudinal force, M is the bending moment in the cross section defined with respect to the axis Z₀ passing through the center of gravity of the cross section (C), y is the distance from the fiber being examined to the neutral axis z, p is the radius of curvature of the fiber being examined, and S_z = Fy₀ is the static moment of the cross-sectional area with respect to the axis z. The displacement Y₀ of the neutral axis relative to the center of curvature of the curved beam is always directed toward the center of curvature of the curved beam and is usually determined from special tables. For a circular cross section, Y₀ ≈ d²/16R; for a rectangular cross section, Y ≈ h²/12R (R is the radius of curvature of the axis of the curved beam; d and h are the diameter and height of the cross section of the beam, respectively). Normal stresses in a curved beam have their maximum values (in absolute magnitudes) near the concave edge of a beam and vary in the cross section according to a hyperbolic law. For small curvatures (R > 5h) the determination of normal stresses can be made in the same way as for a straight beam.

Figure 1. Distribution of normal stresses in the cross section of a curved beam upon pure bending