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Related Concept Videos

Euler's Formula for Pin-Ended Columns01:21

Euler's Formula for Pin-Ended Columns

653
In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load, envision...
653
Euler's Formula to Columns with Other End Conditions01:15

Euler's Formula to Columns with Other End Conditions

947
Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.
947
Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

457
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.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...
457
Euler's Formula to Columns: Problem Solving01:23

Euler's Formula to Columns: Problem Solving

908
Euler's formula is used in structural engineering to determine the buckling load of columns under various conditions. However, when dealing with systems that incorporate both rigid elements and elastic components, such as springs, the analysis requires a finer approach to determine the critical load. The problem described involves two rigid bars connected at a pivot point with a spring attached and a vertical load applied at one end.
The system comprises two vertical rigid bars, AB and BC, of...
908
General Case of Eccentric Axial Loading01:12

General Case of Eccentric Axial Loading

446
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 symmetrical bending, which are essential for designing structures to withstand different loading conditions.
Consider a member subjected to equal and opposite forces that are applied along a line that does not coincide with the member's neutral axis. In unsymmetrical...
446
Equation of the Elastic Curve01:23

Equation of the Elastic Curve

952
The concept of curvature in plane curves, crucial in structural engineering, defines how sharply a beam bends under load. This curvature is determined using the curve's first and second derivatives.
Consider a cantilever beam with a point load at its free end (for instance, a diving board). When analyzing beam deflection with small slopes, the shape of the beam's elastic curve becomes key. The governing equation for this analysis involves the bending moment and the beam's flexural rigidity,...
952

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Updated: Jan 7, 2026

Finite Element Modeling for the Simulation of the Quasi-Static Compression of Corrugated Tapered Tubes
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Euler Buckling on Curved Surfaces.

Shiheng Zhao1, Pierre A Haas1

  • 1Center for Systems Biology Dresden, Max Planck Institute of Molecular Cell Biology and Genetics, Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany; , Pfotenhauerstraße 108, 01307 Dresden, Germany; and , Pfotenhauerstraße 108, 01307 Dresden, Germany.

Physical Review Letters
|January 2, 2026
PubMed
Summary
This summary is machine-generated.

The buckling of elastic lines on curved surfaces fundamentally changes, with the critical force for the lowest mode becoming zero. New bifurcation structures emerge, altering how these lines buckle and snap under compression.

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Area of Science:

  • Mechanics of Materials
  • Nonlinear Dynamics
  • Elasticity Theory

Background:

  • Euler buckling describes elastic line instability under compression.
  • Classical buckling occurs in straight lines, with a positive critical force.
  • Instability in curved surfaces remains less understood.

Purpose of the Study:

  • To investigate how elastic lines buckle on general curved surfaces.
  • To analyze the fundamental changes in classical Euler buckling under curvature.
  • To discover new bifurcation phenomena and critical force behaviors.

Main Methods:

  • Weakly nonlinear analysis of asymptotically short elastic lines.
  • Numerical simulations for long elastic lines under compression.
  • Investigating bifurcation structures and mode splitting.

Main Results:

  • The critical force for the lowest buckling mode on curved surfaces is zero (F_{*}=0).
  • Classical Euler buckling modes split into pairs, revealing a new bifurcation structure.
  • Long elastic lines exhibit an additional bifurcation and discontinuous snapping at high compression.

Conclusions:

  • Buckling instabilities in curved surfaces differ fundamentally from classical Euler buckling.
  • The findings provide a foundation for understanding buckling in curved geometries, relevant to biological development.
  • New theoretical frameworks are needed for elastic instabilities on curved manifolds.