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Singularity Functions for Shear

Singularity Functions for Shear

In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the shear...

Deflection of a Beam

Deflection of a Beam

Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...

Thin-Walled Hollow Shafts

Thin-Walled Hollow Shafts

In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of...

Shearing Stress

Shearing Stress

Shearing stress, denoted by the Greek letter tau (τ), is stress caused by forces acting transversely on an object. These forces create internal ones within the entity in the plane where the external forces are applied. The resultant of these internal forces is the shear in the section.
The average shearing stress can be calculated by dividing the shear by the area of the cross-section.

Shearing Stresses in a Beam: Problem Solving

Shearing Stresses in a Beam: Problem Solving

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...

Shear and Bending Moment Diagram: Problem Solving

Shear and Bending Moment Diagram: Problem Solving

When analyzing a beam supporting concentrated loads and a distributed load, drawing the shear and bending moment diagrams is essential. These diagrams help understand the internal forces and moments acting on the beam, which is crucial for designing safe and efficient structures. Follow these steps to create the shear and bending moment diagrams:
Draw a Free-Body Diagram: Start by drawing a free-body diagram of the entire beam, including the concentrated loads, distributed load, and reaction...

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Related Experiment Video

Updated: May 24, 2026

4D Printed Bifurcated Stents with Kirigami-Inspired Structures

4D Printed Bifurcated Stents with Kirigami-Inspired Structures

Published on: July 25, 2019

Fold bifurcations in shearing radiative collapse.

K S Govinder¹, S D Maharaj²

¹Astrophysics Research Centre, Discipline of Mathematics, School of Agriculture and Science, University of KwaZulu-Natal, Private Bag X54001, 4000, Durban, South Africa. govinder@ukzn.ac.za.

Scientific Reports

|May 22, 2026

Summary

Researchers studied radiating relativistic stars with shear, finding distinct behaviors and potential collapse or superdense star formation due to shear. This analysis helps understand stellar evolution and gravitational fields.

Keywords:

Asymptotic behaviour Fold bifurcations Radiating stars Shearing spacetimes

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Visualization of Failure and the Associated Grain-Scale Mechanical Behavior of Granular Soils under Shear using Synchrotron X-Ray Micro-Tomography

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Published on: September 29, 2019

Area of Science:

Relativistic astrophysics
Gravitational physics
Stellar evolution

Background:

Understanding the behavior of radiating relativistic stars is crucial in astrophysics.
Previous models often simplified by assuming shear-free conditions.

Purpose of the Study:

To investigate a simple model of a radiating relativistic star with non-zero shear.
To analyze the impact of shear on the star's temporal evolution and asymptotic behavior.

Main Methods:

Studied a simplified gravitational field model for radiating relativistic stars.
Reduced junction conditions to a nonlinear first-order Riccati differential equation.
Employed phase plane analysis for examining the differential equation.

Main Results:

Identified exact solutions and analyzed the differential equation using phase plane methods.
Determined the temporal evolution and asymptotic behavior, including shear, charge, and cosmological constant effects.
Observed distinct behaviors compared to shear-free models, including fold bifurcations leading to collapse or superdense star formation.

Conclusions:

Non-zero shear significantly alters the behavior of radiating relativistic stars.
The presence of shear can lead to catastrophic collapse or the formation of superdense cold stars.
Areal distance, charge, and cosmological constant play interconnected roles in the asymptotic analysis.