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Stability of structures

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In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...
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Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...
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In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
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Three-Dimensional Analysis of Strain01:29

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Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal...
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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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Mohr's Circle for Plane Strain01:18

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Mohr's circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain ∈ and the ordinate is shear strain γ. Similarly to Mohr’s circle for plane stress, two points X and Y are plotted. Their coordinates are (∈x, -γXY) and (∈Y, γXY), respectively.
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Flexural Rigidity Measurements of Biopolymers Using Gliding Assays
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Rigidity percolation in a random tensegrity via analytic graph theory.

William Stephenson1,2, Vishal Sudhakar1, James McInerney2

  • 1School of Physics, Georgia Institute of Technology, Atlanta, GA 30332.

Proceedings of the National Academy of Sciences of the United States of America
|November 21, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a model system for tensegrity structures, revealing how adding flexible elements to rigid backbones creates a mechanical critical point and collective avalanche behavior. This impacts various nonlinear mechanical systems.

Keywords:
percolationphase transitionrigiditytensegrity

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Area of Science:

  • Engineering
  • Physics
  • Materials Science

Background:

  • Functional structures combine rigid and flexible elements, known as tensegrities.
  • Tensegrity elements, like cables, support only tension and exhibit nonlinear properties.
  • Marginally rigid systems are key for flexible deformation and load support.

Purpose of the Study:

  • To model and analyze the mechanical properties of tensegrity systems with randomly added elements.
  • To investigate the impact of tensegrity elements on mechanical critical points and transitions to rigidity.
  • To explore collective behaviors like avalanche effects in tensegrity networks.

Main Methods:

  • A model system with tensegrity elements added randomly to a regular backbone was developed.
  • Analytical solutions were derived using directed graph theory.
  • The study analyzed the transition points and collective behaviors.

Main Results:

  • A mechanical critical point generalizing Maxwell's was identified.
  • The addition of tensegrity elements fundamentally alters transition points to rigidity.
  • Collective avalanche behavior was observed, where adding one element eliminates multiple floppy modes.

Conclusions:

  • Tensegrity networks exhibit unique mechanical transitions and collective behaviors.
  • These findings have implications for biopolymer networks, soft robots, jammed packings, and origami.
  • The study provides insights into nonlinear mechanical systems through a generalized critical point analysis.