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Related Concept Videos

Principal Stresses: Problem Solving01:15

Principal Stresses: Problem Solving

663
When analyzing two planes intersecting at right angles under the influence of shearing, tensile, and compressive stresses, it is essential to identify principal planes, maximum shearing stress, and principal stresses. To find the principal planes, apply a formula that equates them to twice the shearing stress divided by the difference between tensile and compressive stresses.
663
Yield Criteria for Ductile Materials under Plane Stress01:25

Yield Criteria for Ductile Materials under Plane Stress

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In designing structural elements and machine parts using ductile materials, it is crucial to ensure that these components withstand applied stresses without yielding. Yielding is initially determined through a tensile test, which evaluates the material's response to uniaxial stress. However, tensile stress is insufficient when components face biaxial or plane stress conditions This condition requires advanced criteria to predict failure.
The Maximum Shearing Stress Criterion, also known as...
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Elastic Strain Energy for Shearing Stresses01:20

Elastic Strain Energy for Shearing Stresses

592
As discussed in previous lessons, strain energy in a material is the energy stored when it is elastically deformed, a concept crucial in materials science and mechanical engineering. This energy results from the internal work done against the cohesive forces within the material. When a material undergoes shearing stress and corresponding shearing strain, the strain energy density, which is the energy stored per unit volume, is calculated. Within the elastic limit, where the stress is...
592
Stresses under Combined Loadings01:23

Stresses under Combined Loadings

524
When analyzing a bent tube with a circular cross-section subjected to multiple forces, it is crucial to determine the stress distribution in order to maintain structural integrity under varied load conditions.
The process begins by slicing the tube at critical points and analyzing the internal forces and stress components at these sections, focusing on the centroid. Normal stresses, generated by axial forces and bending moments, are either compressive or tensile and vary across the section from...
524
Principal Stresses01:24

Principal Stresses

1.0K
The graphical depiction of normal and shearing stress equations is represented by a circle, demonstrating the interplay between these stresses under different angular conditions. The center of this circle C, located on the vertical axis, represents the average normal stress, while its radius shows the range of stress variations. At points A and B, where the circle intersects the horizontal axis, the maximum and minimum normal stresses are observed, occurring without shearing stress. These...
1.0K
Shearing Stresses in a Beam: Problem Solving01:14

Shearing Stresses in a Beam: Problem Solving

771
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...
771

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Related Experiment Video

Updated: Mar 21, 2026

The Assembly and Application of 'Shear Rings': A Novel Endothelial Model for Orbital, Unidirectional and Periodic Fluid Flow and Shear Stress
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Optimal bounds with semidefinite programming: An application to stress-driven shear flows.

G Fantuzzi1, A Wynn1

  • 1Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom.

Physical Review. E
|May 14, 2016
PubMed
Summary
This summary is machine-generated.

We developed a new numerical method using convex optimization and semidefinite programming (SDP) to solve fluid flow problems. This technique provides significantly improved bounds on dissipation coefficients for surface-driven flows.

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Area of Science:

  • Fluid dynamics
  • Numerical analysis
  • Optimization

Background:

  • Infinite-dimensional variational problems are common in fluid dynamics.
  • Traditional methods often rely on Euler-Lagrange equations, which can be challenging for certain flow conditions like imposed boundary fluxes.

Purpose of the Study:

  • To introduce a novel numerical technique for solving variational problems in fluid flows.
  • To develop a method that overcomes limitations of existing schemes, particularly for surface-driven flows and boundary flux conditions.

Main Methods:

  • Utilized convex optimization and series expansions to formulate a finite-dimensional semidefinite program (SDP).
  • The SDP formulation converges to the solution of the original infinite-dimensional variational problem.
  • Applied the SDP technique to compute upper bounds on dissipation coefficients for surface-driven flows.

Main Results:

  • Achieved rigorous and near-optimal upper bounds on dissipation coefficients, improving previous analytical bounds by over 10 times.
  • Demonstrated that these bounds become independent of domain aspect ratio at vanishing viscosity.
  • Confirmed similarities in dissipation properties between stress-driven flows and body-force driven flows in a narrow surface layer.

Conclusions:

  • SDP relaxations offer an efficient and powerful method for investigating energy stability in laminar surface-driven flows.
  • The new technique successfully handles complex boundary conditions, such as imposed fluxes.
  • This approach provides significant improvements in bounding dissipation coefficients for fluid flows.