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Beams

Beams

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...

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...

Prismatic Beams: Problem Solving

Prismatic Beams: Problem Solving

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 designer...

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...

Elastic Curve from the Load Distribution

Elastic Curve from the Load Distribution

The structural behavior of beams under distributed loads is critical for engineering analysis, which focuses on predicting how beams bend and react under such conditions. Different types of beams (e.g., cantilever, supported, or overhanging) behave differently under distributed load conditions.
For all beams, the analysis of the beam's reaction to distributed loads begins by understanding the relationship between a beam's load and the resulting shear forces and bending moments. Initially, this...

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Related Experiment Video

Updated: Jul 8, 2026

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

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

Published on: August 12, 2013

Circular beams.

Miguel A Bandres¹, Julio C Gutiérrez-Vega

¹California Institute of Technology, Pasadena, California 91125, USA.

|January 17, 2008

Summary

A new general solution for optical beams, termed circular beams (CiB), is introduced. This framework unifies various known beam types and analyzes their behavior in optical systems.

Area of Science:

Optics and Photonics
Mathematical Physics
Wave Propagation

Background:

The paraxial wave equation is fundamental for describing light propagation in many optical systems.
Existing solutions, like Laguerre-Gauss and Bessel-Gauss beams, represent specific cases of optical beam behavior.
A unified framework for describing diverse optical beams is lacking.

Purpose of the Study:

To present a general beam solution to the paraxial wave equation in circular cylindrical coordinates.
To introduce and define the circular beam (CiB) as a comprehensive optical beam model.
To analyze the propagation characteristics and square integrability of CiBs in complex optical systems.

Main Methods:

Derivation of a general beam solution using Whittaker functions or confluent hypergeometric functions.

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Last Updated: Jul 8, 2026

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Mathematical analysis of the complex amplitude and propagation parameters of the circular beam.

Investigation of beam propagation through ABCD optical systems and conditions for square integrability.

Main Results:

The complex amplitude of the circular beam is described by Whittaker or confluent hypergeometric functions.
The circular beam is characterized by three generally complex parameters.
The study details the propagation of circular beams through complex ABCD optical systems and their square integrability.

Conclusions:

The circular beam (CiB) provides a unified and general framework for describing various optical beam types.
Special cases of the CiB include standard, elegant, and generalized Laguerre-Gauss beams, Bessel-Gauss beams, and optical vortex beams.
This generalized solution enhances the understanding of beam propagation and optical system design.