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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...

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...

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...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Distribution of Stresses in a Narrow Rectangular Beam

Distribution of Stresses in a Narrow Rectangular Beam

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 areas.

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Related Experiment Video

Updated: Jun 9, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

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Published on: August 12, 2013

Gaussian beams from variable groove depth grating couplers in planar waveguides.

K A Bates, L Li, R L Roncone

|September 8, 2010

Summary

Researchers fabricated a variable groove depth grating coupler for waveguides. This device can shape light beams, demonstrating a method to produce a Gaussian beam profile using ion-beam etching.

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Area of Science:

Photonics
Optical Engineering
Waveguide Technology

Background:

Grating couplers are essential for efficiently coupling light into and out of planar waveguides.
Controlling the output beam profile from waveguide devices is crucial for integrated optics applications.
Existing grating couplers often produce fixed output profiles, limiting their versatility.

Purpose of the Study:

To analyze, fabricate, and characterize variable groove depth (VGD) planar waveguide grating couplers.
To develop a formula for designing VGD gratings to achieve arbitrary output beam profiles.
To demonstrate the fabrication of a VGD grating coupler capable of producing a Gaussian beam profile.

Main Methods:

Derivation of a theoretical formula relating grating groove depth variation to output beam profile.
Fabrication of a VGD grating coupler on a waveguide using ion-beam etching.
Utilizing a photoresist grating as a mask for precise ion-beam etching.
Characterization of the outcoupled beam's near-field irradiance profile.

Main Results:

Successful derivation of a formula for VGD grating design.
Fabrication of a VGD grating coupler with a scanning slit apparatus.
Experimental measurement showing the outcoupled beam's near-field irradiance closely approximates a Gaussian profile.
Demonstration of precise control over the output beam shape.

Conclusions:

Variable groove depth grating couplers offer a method for controlling the output beam profile from waveguides.
The developed fabrication technique allows for the precise creation of complex grating structures.
This technology has potential applications in integrated optics and optical beam shaping.