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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

485
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...
485
Principal Stresses in a Beam01:11

Principal Stresses in a Beam

764
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...
764
Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

437
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...
437
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 15, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Beam shape coefficient calculation for a Gaussian beam: localized approximation, quadrature and angular spectrum

Juncheng Qiu, Jianqi Shen

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    |January 13, 2018
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    Summary
    This summary is machine-generated.

    This study compares three beam shape coefficient calculation methods: quadrature, angular spectrum decomposition (ASD), and localized approximation. It optimizes calculations and analyzes differences in Gaussian beam reconstruction, explaining pseudodistribution in the localized model.

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    Area of Science:

    • Optics and Photonics
    • Computational Physics

    Background:

    • Accurate calculation of beam shape coefficients is crucial for modeling optical beams.
    • Existing methods may face computational challenges like overflow and slow processing.

    Purpose of the Study:

    • To investigate and compare three distinct methods for calculating beam shape coefficients.
    • To optimize computational efficiency and analyze the fidelity of Gaussian beam reconstruction.

    Main Methods:

    • Employed normalized associated Legendre functions to prevent numerical overflow.
    • Reduced two-dimensional integrations in quadrature and ASD methods to one-dimensional for faster computation.
    • Compared Gaussian beams reconstructed using quadrature, ASD, and localized approximation methods.

    Main Results:

    • Demonstrated optimized one-dimensional integration for quadrature and ASD methods.
    • Highlighted differences in Gaussian beam remodeling effects across the studied methods.
    • Provided an explanation for the occurrence of pseudodistribution within the localized model.

    Conclusions:

    • The choice of method impacts Gaussian beam reconstruction fidelity.
    • Optimized integration techniques enhance computational speed.
    • Understanding pseudodistribution is key for localized model accuracy.