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Related Concept Videos

Deflection of a Beam01:19

Deflection of a Beam

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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...
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Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

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Analyzing a supported beam under unsymmetrical loadings is essential in structural engineering to understand how beams respond to varied force distributions. This analysis involves calculating the deflection and identifying points where the slope of the beam is zero, which are crucial for ensuring structural stability and functionality.
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Singularity Functions for Bending Moment01:18

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Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...
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Prismatic Beams: Problem Solving01:15

Prismatic Beams: Problem Solving

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In the design of a supported timber beam subjected to a distributed load, both the beam's physical dimensions and the timber's characteristics, such as its grade and species, are critical. These factors determine the allowable stress values, which are crucial for calculating the necessary beam depth to ensure structural integrity and safety.
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Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates01:21

Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates

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Understanding the motion of particles is a fundamental aspect of classical mechanics, and the choice of the coordinate system plays a pivotal role in unraveling the complexities of their dynamics.
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Deformations in a Symmetric Member in Bending01:18

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When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
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Related Experiment Video

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Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
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Complete confined bases for beam propagation in Cartesian coordinates.

Rodrigo Gutiérrez-Cuevas, Miguel A Alonso

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |October 17, 2017
    PubMed
    Summary

    New spatially confined bases are proposed for beam propagation. These novel Gaussian polynomial bases offer improved fitting and analytical modeling for confined fields.

    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Beam propagation problems often utilize basis functions for analysis.
    • Standard bases like Hermite-Gaussian have limitations in spatial confinement.
    • Efficiently representing confined initial fields is crucial for accuracy.

    Purpose of the Study:

    • To propose novel complete bases for beam propagation.
    • To introduce bases with the distinct property of spatial confinement.
    • To enable optimal fitting of confined initial fields and analytical propagation modeling.

    Main Methods:

    • Construction of new basis elements using polynomials of Gaussians.
    • Contrast with standard Gaussian-based functions (e.g., Hermite-Gaussian).
    • Analytical modeling of paraxial propagation for the proposed basis elements.

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    Main Results:

    • The proposed bases exhibit spatial confinement at the initial plane.
    • Basis elements possess similar spatial extents, facilitating field fitting.
    • An optimal scaling parameter, independent of truncation order, is identified.

    Conclusions:

    • The novel Gaussian polynomial bases are effective for beam propagation.
    • Spatial confinement property enhances the fitting of confined initial fields.
    • Analytical tractability of paraxial propagation is achieved.