   ### Beams in Engineering

"Beam...a long sturdy piece of squared timber or steel used in house building ".
This page refers to a beam is a structural member considered because it is primarily loaded in a transverse direction to its length..
A column is a structural member compressively loaded along its length..
A beam is typically a structural member which has a constant section and has a length dimension which is long compared to its section dimensions. It primarily support loads in a transverse direction to its length. Beams are primarily used in construction of structures. In mechanical engineering shafts, levers, springs, gear teeth are designed using beam theory..
This section only relates to beams subject to transverse loads, it does not include notes on beams withstanding axial or torsional loads

#### Beam Theory

Beam theory, also known as beam mechanics or the theory of beams, is a fundamental concept in structural engineering and mechanics that describes the behavior of beams under various loading conditions. Beams are structural elements that can support loads by resisting bending, shear, and axial forces. Beam theory is based on several assumptions, including: Bernoulli's Hypothesis, Small Deformation and Linear Elastic Material

#### Shear & Bending Moment Diagrams

Shear and bending moment (shear-bending moment) diagrams are graphical representations used in structural analysis to visualize the internal forces and moments along the length of a beam. These diagrams are essential for understanding the distribution of shear forces and bending moments and aid in designing beams and determining their structural integrity. The shear diagram shows the variation of shear force along the length of the beam, while the bending moment diagram illustrates the distribution of bending moments along the beam's span. Both diagrams are typically plotted with distance along the beam's length on the x-axis and shear force or bending moment on the y-axis.

#### Singularity Functions

Singularity functions, also known as unit impulse functions or Dirac delta functions, are mathematical functions that are often used in engineering and physics to represent concentrated loads or moments acting on a structure. Singularity functions are used to simplify the analysis of structures by converting concentrated loads or moments into mathematical expressions that can be integrated and differentiated.The most common singularity function is the Dirac delta function (δ(x)), which is not a true function in the traditional sense but rather a generalized function or distribution. The Dirac delta function is typically used to represent a point load or a point moment acting on a structure.

#### Strain Energy

Strain energy is a concept in physics and mechanics that refers to the energy stored in a material due to deformation or strain. When a material is subjected to external forces or loads, it undergoes deformation, and this deformation requires energy. The energy stored in the material as a result of this deformation is known as strain energy. Strain energy is closely related to the elastic behavior of materials. When a material is elastically deformed, it can return to its original shape once the external forces are removed. This elastic deformation is characterized by the strain, which is the relative change in shape or size of the material. The strain energy is directly proportional to the amount of strain and the stiffness of the material.

#### Curved Beams

In curved beams, the distribution of stresses varies along the cross-section due to the curvature of the beam. The stresses in curved beams are influenced by bending moments, shear forces, and the geometric properties of the beam. When analyzing the stresses in curved beams, engineers often consider two types of stresses: longitudinal stresses and circumferential stresses. Analysing the stresses in curved beams requires considering the combined effects of bending, shear, and the curved geometry. This can be done using beam theory, such as the Euler-Bernoulli or by employing more advanced analysis methods like finite element analysis.

#### Continous Beams

A continuous beam refers to a structural beam that extends over multiple supports, with no interruptions in its span. Unlike a single-span beam that has two distinct supports, a continuous beam rests on three or more supports, creating a continuous load path throughout its length. Designing a continuous beam requires a more complex analysis compared to single-span beams. Engineers utilise structural analysis methods such as the slope-deflection method, moment distribution method, or finite element analysis to determine the reactions, internal forces, and deflections along the beam. The goal is to ensure that the beam can safely carry the anticipated loads while satisfying strength and deflection criteria.

#### Single Span Beams

A single-span beam refers to a beam that is supported at two points and has a clear span between those supports. It is one of the simplest and most common types of beam configurations used in structural engineering and construction. Designing a single-span beam involves analyzing the loads, determining the appropriate beam size and material, and assessing the beam's strength and deflection limits. Engineers typically use beam theory and structural analysis techniques to calculate the bending moments, shear forces, and deflections along the beam. This information is crucial for selecting an appropriate beam section and ensuring that the beam can safely support the anticipated loads. Common types of single-span beams include simply supported beams, cantilever beams, and overhanging beams. Each type has specific support and loading conditions, and the design considerations may vary accordingly.

#### Energy Methods

Energy methods, also known as energy principles or variational methods, are powerful techniques used in structural analysis and mechanics to determine the response of structures or systems. These methods utilise the concept of energy to analyse the behavior of structures under various loading and boundary conditions. There are several energy methods commonly used in structural analysis, including the following:
- Strain Energy Method
- Castigliano's Theorem
- Virtual Work Method
- Principle of Minimum Potential Energy

#### Shear Stresses & Deflections

In beams, shear stresses result from the shearing forces acting perpendicular to the longitudinal axis of the beam. The magnitude of shear stress varies along the cross-section of the beam. At any given point within the cross-section, the shear stress is directly proportional to the shear force and inversely proportional to the area over which the shear force is distributed. In beams, deflections primarily result from bending moments and shear forces. The magnitude of deflection depends on factors such as the beam's geometry, material properties, loading conditions, and support conditions.

#### Online Beam Calculators

Roymech online beam calculator provides a simple interface to determine the bending moment, shear force, and deflection for a given beam. You can specify beam properties, supports, and applied loads to obtain the desired results.