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Deflection of a Beam

Deflection of a Beam

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...

Deformation of a Beam under Transverse Loading

Deformation of a Beam under Transverse Loading

Understanding beam deflection, particularly for indeterminate beams with overhanging segments and multiple concentrated loads, is crucial for ensuring structural integrity and functionality. The process begins with constructing an accurate free-body diagram, which helps identify the forces and moments acting on the beam. This diagram is vital for visualizing how bending moments vary along the beam's length, influencing its curvature.
The insights from the bending moment diagram extend to...

Beams with Symmetric Loadings

Beams with Symmetric Loadings

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...

Beams with Unsymmetric Loadings

Beams with Unsymmetric Loadings

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...

Design of Prismatic Beams for Bending

Design of Prismatic Beams for Bending

The design of prismatic beams, structural elements with a uniform cross-section, focuses on ensuring safety and structural integrity under load. The design process begins by determining the allowable stress, either from material properties tables, or by dividing the material's ultimate strength by a safety factor. This safety factor is essential for accommodating uncertainties, and varies depending on the material—timber, steel, or concrete—with each having unique strength and stress...

Shear on the Horizontal Face of a Beam Element

Shear on the Horizontal Face of a Beam Element

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 first...

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Related Experiment Video

Updated: Jun 7, 2026

Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating

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Published on: October 11, 2016

Asymmetric beam deflection by doubly grooved binary gratings.

J Saarinen, E Noponen, J Turunen

|November 6, 2010

Summary

Nonsymmetric binary gratings with wavelength-scale features can break inversion symmetry in diffraction patterns. This enables efficient deflection of light into specific orders, with up to 80% directed into the minus first order.

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Last Updated: Jun 7, 2026

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Area of Science:

Optics and Photonics
Diffraction Gratings
Nanophotonics

Background:

Diffraction patterns of binary gratings typically exhibit inversion symmetry.
Breaking this symmetry is crucial for advanced optical applications.

Purpose of the Study:

To demonstrate theoretically and experimentally the breaking of inversion symmetry in binary grating diffraction patterns.
To achieve efficient deflection of normally incident radiation using nonsymmetric grating profiles.

Main Methods:

Theoretical analysis of binary gratings with nonsymmetric profiles.
Experimental validation using gratings with wavelength-scale transverse features.
Investigation of gratings with two unequal grooves within the period.

Main Results:

Efficient breaking of inversion symmetry in diffraction patterns was achieved.
Nonsymmetric binary gratings deflected up to 80% of normally incident radiation.
Deflection into the minus first diffraction order at a 45° angle was observed.

Conclusions:

Nonsymmetric binary gratings with specific profiles are effective for controlling light direction.
This method offers a pathway to enhanced performance in diffractive optical elements.
The findings have implications for optical devices requiring precise light manipulation.