Bending

In engineering mechanics, bending refers to a mechanical change in the geometry of slender components (beams or arches) or thin components (shells or plates).

Typical of bending are changes in the curvature of the centre line or surface compared to the curvature that the component had in the unstressed state, due to static and dynamic stresses. Such curvatures lead to bending moments and thus to bending stresses.

By dimension reduction of the original 3D problem, the description of the geometry change is approximated:

• in the case of beams or arcs by a 1D theory
• in the case of shells or plates by a 2D theory.

By determining the bending deformation (the centerline, i.e. the bending line, in the case of a 1D theory, or the mid-surface in the case of a 2D theory), the deformation and stress state at each point of the component can be calculated using the kinematic laws of the respective bending theories.

Bending Theories

Depending on whether the bends are small, moderate or large compared to the dimensions of the cross-section (for beams and arches) or the thickness (for plates or shells), different 1D or 2D bending theories can be used to obtain a physically and mathematically sufficient approximation of the original 3D problem:

• The best known 1D bending theory is Bernoulli's bending beam theory. It is valid when the deflections of the original straight centerline are small compared to the cross-sectional dimensions.
• For the plate theory according to Kirchhoff to be valid, the deflection of the originally flat central surface must be small compared to the plate thickness.
• The plate theory according to von Kármán is valid if the deflection is of the same order of magnitude as the plate thickness, i.e. if the deflection is moderate.

Bending in beam theory

bending stiffness is defined as:

with

• the modulus of elasticity
• the area moment of inertia
• the bending moment
• of curvature κ .

Straight and oblique bending

• Straight Bend: Bending of a beam or an arch curved in one plane only in the direction of one of the principal axes of inertia of the cross-section.

• Skew bending: Bending of a beam or an arch curved in one plane only in a direction deviating from the principal axes of inertia.

The bending line of a beam for which a linear theory is applicable can be determined for composite stresses using the superposition of standard bending cases. There are corresponding tables for standard bending cases.