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Eccentric Axial Loading in a Plane of Symmetry01:16

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Eccentric axial loading occurs when an axial load is applied away from the centroidal axis of a structural member. This scenario is common in engineering, where structural elements may not be directly aligned due to various design or functional requirements.
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Principal Stresses in a Beam01:11

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In prismatic beams subject to arbitrary transverse loading, It is essential to analyze the interaction between shear forces and bending moments in order to understand stress distribution and ensure structural integrity. The highest normal or bending stress occurs at the outer fibers of the beam, decreasing linearly to zero at the neutral axis. In contrast, shear stress peaks at the neutral axis and diminishes toward the outer surfaces.
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Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Behavior of Concrete Under Compressive Load01:23

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Concrete exhibits specific behaviors under different compressive loads. Understanding this is crucial for understanding its structural integrity. When concrete undergoes uniaxial compression, it tends to develop cracks that run parallel to the direction of the force. These parallel cracks stem from localized tensile stresses that occur perpendicular to the compression direction. Additionally, angled cracks may appear due to the formation of shear planes.
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A cantilever beam with a rectangular cross-section under distributed and point loads experiences shearing stresses. The analysis begins by identifying the loads acting on the beam. Then, the reactions at the beam's fixed end are calculated using equilibrium equations. The vertical reaction is a combination of the distributed and point loads, while the moment reaction is the sum of their moments. The shear force distribution along the beam, resulting from these loads, is established by creating...
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When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.
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Related Experiment Video

Updated: May 2, 2026

Finite Element Modeling for the Simulation of the Quasi-Static Compression of Corrugated Tapered Tubes
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Soft beams: when capillarity induces axial compression.

S Neukirch1, A Antkowiak1, J-J Marigo2

  • 1CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France and UPMC Université Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 4, 2014
PubMed
Summary

This study explores elastic beam and liquid drop interactions, revealing how surface energy modifies Hooke's law. The research details forces at the triple point and confirms the Young-Dupré relation for wetting angles.

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Area of Science:

  • * Physics of soft matter
  • * Materials science
  • * Fluid dynamics

Background:

  • * Understanding the behavior of elastic materials interacting with liquid interfaces is crucial in various scientific and engineering fields.
  • * Existing models often simplify the complex interplay between bending, extension, and surface energy effects in such systems.

Purpose of the Study:

  • * To investigate the coupled effects of bending and extension in an elastic beam interacting with a liquid drop.
  • * To derive a modified constitutive relation for the elastic beam that incorporates surface energy contributions.
  • * To analyze the force balance and wetting phenomena at the triple point (solid-liquid-vapor).

Main Methods:

  • * A variational approach was employed to derive the equilibrium equations and the constitutive relation for the elastic beam.
  • * The study mathematically models the interaction, considering both mechanical deformation and surface energy effects.
  • * Analysis focused on the conditions at the triple point and the implications for wetting angles.

Main Results:

  • * A modified Hooke's law was derived, including a surface energy term alongside the Young's modulus.
  • * At the triple point, the external force applied to the beam is found to be parallel to the liquid-vapor interface.
  • * Under specific conditions (interface energies independent of extension), beam extension is continuous, and the wetting angle follows the Young-Dupré relation.

Conclusions:

  • * Surface energy significantly influences the mechanical response of elastic beams in contact with liquid drops.
  • * The derived constitutive relation provides a more accurate description of beam behavior under these conditions.
  • * The findings confirm classical wetting theories under simplified interface energy assumptions, offering insights into microscale fluid-structure interactions.