Bending and Stretching of an Elastic U-Beam

The normal stresses (aka longitudinal stresses) in an elastic beam with U-shaped cross section loaded by bending moments and by a normal force are computed by the generalized Navier's formula.
This Demonstration shows the central ellipse of inertia and the principal inertia axes of the cross section, displaying the axis of flexure (in red) and the neutral axis (in blue), and plotting the stress diagram.

SNAPSHOTS

  • [Snapshot]
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DETAILS

The generalized Navier's solution for bending and stretching depends on the geometry of the cross section of the beam, on the position of the applied normal force and on its intensity. The point at the intersection of the axis of the normal force with the plane of the cross section is called the center of pressure. The line joining it with the barycenter of the cross section is called the axis of loading. Pure bending corresponds to the limit of a null normal force applied at an infinite distance; pure tension or compression to a normal force applied on the barycenter (aka centroid).
The solution in terms of normal stresses is an affine field, characterized by a neutral axis (where the field is null). The neutral axis and the center of pressure are related by an anti-polarity property depending on the ellipse of inertia: the neutral axis is parallel to the tangent to the ellipse where it intersects with the axis of loading; its distance from the barycenter is inversely proportional to that of the same point from the center of pressure.
Details are given in textbooks on the theory of elasticity and structural mechanics, including elements of the theory of elastic beams [1].
The symbols of the input parameters are the following:
= width or length of the flange,
= height or length of the web,
= thickness of the web,
= thickness of the flange,
= normal force,
= coordinates of the center of pressure (red point).
The parameters of the plot allow one to displace it along the direction of the neutral axis and to set its amplitude. The computed geometric characteristics of the section, displayed under the controls, are:
= surface area,
= static moments (aka first moments of area),
= coordinates of the barycenter,
= inertial moments (aka second moments of area),
= angle between the first principal axis of inertia and the axis,
= principal inertial moments,
= radii of gyration.
The results displayed under the image are:
= angle between the axis of bending and the axis,
= angle between the neutral axis and the axis,
and = intersections of the neutral axis with the principal coordinate axes (blue points),
= stress at point on the perimeter according to the numbering displayed in the figure (shown by a segment in the plot).
Reference
[1] M. Brocato, Cours de mécanique des structures. Volume 1. Poutres élastiques, Paris: Presses des Ponts, forthcoming.
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