Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

La deflexión de una viga en una estructura de cubierta se puede determinar utilizando el método de integración, siempre que una sola función analítica pueda representar el momento flector.
Sin embargo, si la carga de la viga requiere múltiples funciones para representar el momento flector, serían necesarias constantes y ecuaciones adicionales, lo que llevaría a cálculos largos.
Esta complejidad se puede simplificar utilizando funciones de singularidad. Considere una viga prismática apoyada en los extremos que soporta una carga excéntrica.
La función de fuerza cortante de esta viga se puede representar utilizando una función de singularidad apropiada. La función de momento flector se puede derivar integrando esta función de fuerza cortante.
La pendiente y la deflexión de la viga se pueden obtener integrando esta función de momento. Las constantes de las ecuaciones se pueden determinar a partir de las condiciones de contorno.
El uso de funciones de singularidad elimina la necesidad de constantes y ecuaciones adicionales, lo que simplifica el cálculo. Como resultado, las funciones de singularidad proporcionan una forma eficiente de calcular la pendiente y la deflexión de una viga bajo cargas complejas.

deflexión de una viga

<div>
 Determinar con precisión la deflexión y la pendiente de la viga bajo diversas condiciones de carga en ingeniería estructural es crucial para garantizar la seguridad y la integridad estructural. Las funciones de singularidad ofrecen un enfoque simplificado para analizar vigas, especialmente cuando múltiples funciones de carga complican la ecuación del momento flector.
 
 Las funciones de singularidad, descritas en una lección anterior, son poderosas herramientas matemáticas que representan discontinuidades dentro de una función que se encuentra comúnmente en escenarios de carga estructural. Estas funciones ayudan a expresar las ecuaciones de fuerza cortante y momento flector de forma compacta, incluso bajo cargas complejas o múltiples.
 
 <img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="text-align: center; height: 32px;">
 <img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="text-align: center; height: 32px;">
 Para una viga prismática, típicamente uniforme a lo largo de su longitud y apoyada en ambos extremos, una carga excéntrica presenta desafíos específicos. La fuerza cortante en cualquier punto de dicha viga se puede modelar utilizando funciones de singularidad. Estas funciones manejan fácilmente las discontinuidades introducidas por cargas aplicadas en puntos específicos o durante ciertos intervalos, como una carga representada como una función escalonada en el diagrama de fuerza cortante.
 
 El momento flector, derivado de la integración de la función de fuerza cortante, es fundamental para evaluar el desempeño de la viga. Este paso afecta la distribución de tensiones de la viga y la deflexión general. La deflexión de la viga se determina integrando la función del momento flector dos veces y aplicando las condiciones de contorno de la viga para resolver las constantes de integración. El uso de funciones de singularidad para modelar fuerzas cortantes y momentos flectores elimina la necesidad de múltiples constantes adicionales y ecuaciones complejas, lo que simplifica los cálculos y mejora la eficiencia computacional. Este método permite la fácil evaluación de diferentes condiciones de carga en la deflexión de la viga y la distribución de tensiones, esenciales para el diseño y mantenimiento seguros de sistemas estructurales.
 
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Deflexión de una viga

La deflexión de una viga en una estructura de cubierta puede determinarse utilizando el método de integración, siempre que una sola función analítica pueda representar el ...

Education

Educación

JoVE Core

Mechanical Engineering

Ingeniería Mecánica

Unit 1

Unidad 1

Deflection of Beams

Deflexión de vigas

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Deflection of a Beam

beam slope and deflection by singularity

Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases.

Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

deformation of a beam under transverse loading

deformation-of-a-beam-under-transverse-loading

Beam Deflection from Loads and Supports

Beam deflection depends on the loads on a beam and the support conditions at its ends. For indeterminate beams, especially those with overhanging parts and multiple concentrated loads, careful analysis is needed to predict how the beam will bend. The first step is to draw an accurate free-body diagram. A free-body diagram shows the forces and moments acting on the beam. It helps reveal how the bending moment changes along the beam length. The bending moment diagram also shows how the beam...

equation of the elastic curve

Beam Deflection and Boundary Conditions

Beam deflection and boundary conditions help describe how a beam bends under load. In structural engineering, the curvature of a plane curve shows how sharply the beam bends. That curvature is found from the curve’s first and second derivatives. A cantilever beam with a point load at its free end is a common example. A diving board works like this type of beam. When slopes are small, the beam’s elastic curve becomes the main shape used in the analysis. The governing equation connects bending...

elastic curve from the load distribution

Beam Deflection Under Distributed Load

Beam deflection under distributed load is a key part of beam analysis in engineering. It helps predict how a beam bends and reacts when a load is spread along its length. Engineers use this relationship to judge whether a structure will stay safe and functional. The analysis starts with the link between load, shear force, and bending moment. These quantities can be written in differential form. When beam flexibility is treated as constant, further differentiation leads to a higher-order...

deflection of a beam

Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases. Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the...

method of superposition

Beam Deflection Under Combined Loads

Beam deflection under combined loads can be found with the method of superposition. This structural engineering method is used to analyze how multiple loads affect a beam. It works by finding the slope and deflection from each load separately, then adding those results together. The method applies only when the beam stays within its elastic limit. In that range, the material deforms in a linearly elastic way. That makes it valid to combine the effects of separate loads without changing the...

moment-area theorems

Beam Deflection Using Moment-Area Rules

The moment-area theorems help describe beam bending in structural engineering. They are often used for building floor supports and other beams under load. The theorems use the elastic curve, which is the shape a beam takes as it deforms. The first theorem links slope change to area on an M/EI diagram. Here, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia. When M/EI is plotted along the beam length, the area under the curve gives the total rotation...

beams with symmetric loadings

Moment-Area Method for Cantilever Deflection

The moment-area method is used in structural engineering to find the slope and deflection of a beam under load. It is especially useful for a cantilever beam with a concentrated load and a moment at the free end. The analysis begins with a free-body diagram. This diagram helps calculate the reactions at the fixed end. After that, the bending moment diagram is drawn to show how the bending moment changes along the beam. Points where the bending moment is zero are important in this step. Next,...

beams with unsymmetric loadings

Beam Deflection and Level Points

Beam deflection and level points can be found for a supported beam under unsymmetrical loadings using moment-area theorems. This type of analysis helps show how a beam bends when forces are spread unevenly along it. It is an important part of structural engineering because it links load patterns to beam response. The first moment-area theorem is used to find the slope at any point on the beam. It states that the change in slope between two points equals the area under the moment diagram over...

maximum deflection

Locating the Beam’s Highest Deflection

Maximum deflection in a supported beam depends on where the load is placed. With an unsymmetrical load, such as a train moving across a bridge, the beam may bend most at a point that is not the midpoint. Finding that point is important for judging stress and deflection in the structure. The point of maximum deflection is called point O. At point O, the tangent to the deflection curve is horizontal. That means the slope is zero there. To locate point O, the slope at any point X along the beam...

key terms and concepts

Learn how singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Beam Slope and Deflection by Singularity

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