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Related Concept Videos

Beams01:30

Beams

1.9K
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...
1.9K
Deflection of a Beam01:19

Deflection of a Beam

756
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...
756
Prismatic Beams: Problem Solving01:15

Prismatic Beams: Problem Solving

481
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...
481
Principal Stresses in a Beam01:11

Principal Stresses in a Beam

758
In prismatic beams subject to arbitrary transverse loading, It is essential to analyze the interaction between shear forces and bending moments in order to understand stress distribution and ensure structural integrity. The highest normal or bending stress occurs at the outer fibers of the beam, decreasing linearly to zero at the neutral axis. In contrast, shear stress peaks at the neutral axis and diminishes toward the outer surfaces.
Analyzing principal stresses is crucial, especially in...
758
Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

432
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...
432
Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

450
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.
The first moment-area theorem determines the slope at any point on the beam. This theorem indicates that the change in slope between two points on a beam...
450

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Related Experiment Video

Updated: Feb 14, 2026

Single-Digit Nanometer Electron-Beam Lithography with an Aberration-Corrected Scanning Transmission Electron Microscope
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Aberration laser beams with autofocusing properties.

Svetlana N Khonina, Andrey V Ustinov, Alexey P Porfirev

    Applied Optics
    |February 23, 2018
    PubMed
    Summary

    Researchers developed aberration laser beams (ALBs) with flexible autofocusing properties and controllable symmetry. These novel beams offer more versatility than traditional optical aberrations for advanced laser applications.

    Area of Science:

    • Optics and Photonics
    • Laser Physics
    • Diffractive Optics

    Background:

    • Traditional optical aberrations, like Zernike polynomials, have limitations in controlling light beam properties.
    • Novel laser beam generation techniques are crucial for advancing optical technologies.

    Purpose of the Study:

    • To theoretically, numerically, and experimentally investigate a new class of laser beams: aberration laser beams (ALBs).
    • To explore the unique autofocusing and symmetry properties of ALBs.
    • To demonstrate the flexibility of ALBs compared to existing optical beam types.

    Main Methods:

    • Utilized diffractive optical elements with tailored phase transmission functions.
    • Investigated radial power dependence (q) and angular dependence (sin(mϕ) or cos(mϕ)).

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  • Conducted theoretical analysis, numerical simulations, and experimental validation.

    • Main Results:

    • Demonstrated that radial nonuniformity in ALBs leads to inherent autofocusing capabilities.
    • Showcased that ALBs possess more flexible autofocusing than circular Airy beams.
    • Confirmed that angular dependence creates diffraction patterns with mth-order symmetry.
    • Observed that ALBs exhibit arbitrary integer orders of symmetry during propagation.
    • Found that a focusing lens forms focal plane fields with 2mth-order symmetry (even m) or mth-order symmetry (odd m).

    Conclusions:

    • Aberration laser beams (ALBs) represent a generalized class of optical beams with tunable autofocusing and symmetry.
    • ALBs offer enhanced control over beam propagation and focal plane characteristics.
    • The findings pave the way for new applications in laser manipulation and optical systems.