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Related Concept Videos

Deflection of a Beam01:19

Deflection of a Beam

504
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...
504
Singularity Functions for Bending Moment01:18

Singularity Functions for Bending Moment

398
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...
398
Singularity Functions for Shear01:26

Singularity Functions for Shear

313
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...
313
Beams with Unsymmetric Loadings01:17

Beams with Unsymmetric Loadings

281
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...
281
Shear on the Horizontal Face of a Beam Element01:16

Shear on the Horizontal Face of a Beam Element

403
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...
403
Beams with Symmetric Loadings01:15

Beams with Symmetric Loadings

314
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...
314

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Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads
07:58

Automatic Laser-based Geometry Capture for Finite Element Analysis of Weld Beads

Published on: July 25, 2025

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Accelerating triangle-like singular beam.

Shaohui Yan, Manman Li, Yuan Zhou

    Journal of the Optical Society of America. A, Optics, Image Science, and Vision
    |December 28, 2020
    PubMed
    Summary
    This summary is machine-generated.

    Researchers created a novel singular beam with a unique triangular ring pattern that accelerates parabolically. This beam

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    Area of Science:

    • Optics and Photonics
    • Beam Physics

    Background:

    • Singular beams offer unique light manipulation capabilities.
    • Controlling beam trajectories and intensity patterns is crucial for advanced applications.

    Purpose of the Study:

    • To demonstrate a novel singular beam with a parabolic trajectory.
    • To engineer a beam with a unique triangular-ring intensity pattern.

    Main Methods:

    • Designing a phase mask using six connected circular arcs to form a rounded triangle.
    • Analytically computing individual phase mask components.
    • Numerically simulating beam propagation and energy flow.

    Main Results:

    • Successfully generated a singular beam accelerating along a parabolic path.
    • Observed a distinct intensity pattern with a dark core and a main ring resembling a rounded equilateral triangle.
    • Identified discrete field continuity analogous to topological charge in vortex beams.
    • Demonstrated energy flow circulation around the dark center along the triangular ring.

    Conclusions:

    • The novel rounded-triangle phase mask enables unique beam characteristics.
    • The beam's trajectory and intensity profile suggest potential for particle manipulation.
    • The discrete field continuity offers a new parameter for beam characterization.