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Related Concept Videos

Beams01:30

Beams

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Beams are integral components of structural engineering and construction, designed to support loads applied at various points along their length. These long, straight members can be classified based on geometry, cross-section, support type, and equilibrium condition.
Based on geometry, beams can be straight, tapered, or curved. Straight beams are the most common type and have a constant cross-section throughout their length. Tapered beams, on the other hand, have a varying cross-section along...
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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.
The design begins with analyzing the beam as a free body to identify moments and force balances, thereby determining support reactions. Next, the...
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Shear on the Horizontal Face of a Beam Element01:16

Shear on the Horizontal Face of a Beam Element

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To understand shear on the flat side of a prismatic beam element, consider the vertical and horizontal shearing forces, and the normal forces, acting on the element. The element's upper (U) and lower (L) sections, which are divided by the beam's neutral axis, are examined. The equilibrium of these forces is determined by applying the equilibrium equation, which helps identify the horizontal shearing force. This force is directly related to the bending moments and the cross-section's...
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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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Distribution of Stresses in a Narrow Rectangular Beam01:11

Distribution of Stresses in a Narrow Rectangular Beam

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In studying beam stress distribution, examining an elemental section is essential. To determine the average shearing stress on this face, the calculated shear is divided by the surface area. Importantly, shearing stresses on the beam's transverse and horizontal planes mirror each other, indicating a consistent stress distribution along the upper region of the beam. Notably, shearing stresses are absent at the beam's upper and lower surfaces due to the absence of applied forces in these...
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Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

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The moment-area method is an analytical tool used in structural engineering to determine the slope and deflection of beams under various loads. Consider a cantilever with a concentrated load and moment at the free end. The first step is constructing a free-body diagram to calculate the reactions at the fixed end. Next, the bending moment diagram is plotted to visualize how the bending moment varies along the beam's length, focusing on points where the bending moment equals zero.
The M/EI...
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Author Spotlight: Introduction to Active Probe Atomic Force Microscopy with Quattro-Parallel Cantilever Arrays
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Fractional Airy beams.

S N Khonina, A V Ustinov

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |November 2, 2017
    PubMed
    Summary
    This summary is machine-generated.

    Scientists introduce fractional Airy beams (FrAiB), a novel generalization of Airy beams. These beams exhibit non-diffracting and autofocusing properties, offering potential in laser manipulation and patterning.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Airy beams are crucial in various applications due to their unique properties.
    • Existing generalizations of Airy functions modify differential equations or integral representations.
    • The search for novel Airy beam variants is driven by their multifunctionality.

    Purpose of the Study:

    • To propose and investigate a new class of Airy beams: fractional Airy beams (FrAiB).
    • To explore a wider range of power-law dependencies, including fractional values, in the integral representation.
    • To analyze the properties and potential applications of these novel beams.

    Main Methods:

    • Theoretical and numerical analysis of FrAiBs and their symmetrized variants.
    • Numerical investigation of non-diffracting and autofocusing properties.
    • Utilizing the fractional Fourier transform to describe beam transformations in paraxial optical systems.

    Main Results:

    • FrAiBs are introduced based on a generalized integral representation, extending Olver functions.
    • The study confirms the non-diffracting and autofocusing characteristics of FrAiBs.
    • Fractional Fourier transform analysis provides insights into their behavior in optical systems.

    Conclusions:

    • Fractional Airy beams represent a significant advancement in beam engineering.
    • These novel beams demonstrate promising properties for advanced optical applications.
    • FrAiBs are expected to be valuable for laser manipulation and lensless laser patterning techniques.