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Related Concept Videos

Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

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In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
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Deformations in a Symmetric Member in Bending01:18

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When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
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Pure bending is a fundamental concept in structural mechanics, essential for understanding how materials deform under symmetrical loads without direct forces. Pure bending occurs when prismatic members, such as beams, are subjected to equal and opposite moments that induce bending. The phenomenon is crucial as it allows for predicting stress distributions without the influence of axial or shear forces.
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The living membranes are flexible due to their fluid mosaic nature; however, their bending into different shapes is an active process regulated by specific lipids and proteins. The membrane bending can be transient as seen in vesicles or stable for a long time as in microvilli. Cells regulate the size, location, and duration of the membrane curvature.
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The mechanics of deformation in curved members, such as beams or arches, under bending moments, involve complex responses. When such a member, symmetric about the y-axis and shaped like a segment of a circle centered at point C, is subjected to equal and opposite forces, its curvature and surface lengths change significantly. This alteration results in the shift of the curvature's center from C to C', indicating a tighter curve.
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Unsymmetric Bending01:18

Unsymmetric Bending

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Unsymmetrical bending occurs when the bending moment applied to a structural member does not align with its principal axis. This misalignment leads to complex stress distributions and deflection patterns that differ from those in symmetrical bending, and are essential for designing structures to withstand different loading conditions. In unsymmetrical bending, the neutral axis—where stress is zero—does not necessarily align with the geometric axes of the cross-section. The...
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How periodic surfaces bend.

Hussein Nassar1

  • 1Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.

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|October 7, 2024
PubMed
Summary
This summary is machine-generated.

Researchers explored deformation modes in periodic surfaces, finding membrane and bending modes are often inversely related. The total number of these modes is limited to three, impacting surface mechanics and origami structures.

Keywords:
Poisson’s coefficientcurved creasesisometric deformationsorigami tessellationsrigidity theory

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Area of Science:

  • Materials Science
  • Mechanical Engineering
  • Computational Geometry

Background:

  • Periodic surfaces possess invariance under a 2D lattice of translations.
  • Deformation modes are categorized as effective membrane modes (lattice stretching) or effective bending modes (lattice bending).

Purpose of the Study:

  • To investigate the relationship between effective membrane and bending modes in periodic surfaces.
  • To establish constraints on the total number of deformation modes for such surfaces.

Main Methods:

  • Theoretical analysis of deformation modes for periodic piecewise smooth simply connected surfaces.
  • Mathematical formulation to demonstrate orthogonality between membrane and bending modes.
  • Illustrative examples derived from curved-crease origami tessellations.

Main Results:

  • Effective membrane and bending modes exhibit a form of orthogonality for periodic surfaces.
  • An increase in membrane modes corresponds to a decrease in bending modes, and vice versa.
  • The combined total number of effective membrane and bending modes is constrained, never exceeding three.

Conclusions:

  • The interplay between membrane and bending modes in periodic surfaces is fundamentally constrained.
  • This constraint limits the total degrees of freedom for deformation, influencing structural behavior.
  • Findings are relevant to the design and analysis of origami/kirigami-inspired structures.