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Related Concept Videos

Stress: General Loading Conditions01:15

Stress: General Loading Conditions

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To grasp the intricacy of real-world conditions where multiple loads are applied simultaneously to a structure, one might visualize a section passing through a specific point within a body, aligned parallel to the xy plane. This section is subjected to various forces, including original loads, normal forces, and shearing forces.
The shearing force, possessing potential directionality within the plane of the section, is simplified into two component forces running parallel to the x and y axes....
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Eccentric Axial Loading in a Plane of Symmetry01:16

Eccentric Axial Loading in a Plane of Symmetry

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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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Transformation of Plane Stress01:18

Transformation of Plane Stress

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Studying stress transformation is essential in understanding how stress components within a material, like a cube under plane stress, change with rotation. This change is analyzed by considering a prismatic element within the cube. As the element rotates, the stress components acting on it—both normal and shearing stresses—change in magnitude and orientation. This change is quantified using trigonometric functions of the rotation angle, relating the forces acting on the rotated element's...
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Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

676
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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Applications of Stress01:04

Applications of Stress

746
Consider a structure made of a boom and a rod designed to support a load. These two components are connected by a pin and stabilized by brackets and pins. The boom and the rod are detached from their supports to assess the different stresses imposed on this structure, and a free-body diagram is drawn. Then, all the forces applied, including the load acting on the structure, are identified. The reaction forces exerted on both the boom and the rod are computed using the equilibrium equations.
The...
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Transformation of Plane Strain01:12

Transformation of Plane Strain

607
When analyzing elongated structures like bars subjected to uniformly distributed loads, it is essential to understand the transformation of plane strain when coordinate axes are rotated. This transformation helps to assess how material deformation characteristics vary with orientation, which is crucial in materials science and structural engineering.
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Liftings and stresses for planar periodic frameworks.

Ciprian Borcea1, Ileana Streinu2

  • 1Department of Mathematics, Rider University, Lawrenceville, NJ 08648, USA.

Discrete & Computational Geometry
|March 15, 2016
PubMed
Summary

This study proves a periodic Maxwell

Area of Science:

  • Mathematics
  • Materials Science
  • Crystallography

Background:

  • Maxwell's theorem connects stressed planar frameworks to polyhedral surfaces.
  • Periodic structures in materials science and crystallography exhibit unique properties.
  • Understanding rigidity and deformation is crucial for designing advanced materials.

Purpose of the Study:

  • To establish a periodic version of Maxwell's theorem for planar frameworks.
  • To investigate deformation and rigidity properties of periodic pseudo-triangulations.
  • To apply these findings to auxetic structures and ultrarigid periodic frameworks.

Main Methods:

  • Formulation and proof of a periodic analog of Maxwell's theorem.
  • Utilizing a lifting theorem to analyze periodic pseudo-triangulations.
Keywords:
Maxwell’s theoremauxeticsexpansive motionliftingsperiodic frameworkperiodic pseudo-triangulationperiodic stressultrarigidity

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  • Generalizing known properties from finite frameworks to periodic systems.

    • Main Results:

    • A novel periodic analog of Maxwell's theorem is established.
    • Deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations are proven.
    • These properties generalize findings for finite frameworks.

    Conclusions:

    • The study provides a theoretical framework for analyzing periodic structures.
    • The results have direct implications for mathematical crystallography and materials science.
    • New insights into auxetic and ultrarigid periodic frameworks are offered.