Name: beam deflection under distributed load
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 16 s
Channel: JoVE

Cuando una viga soporta una carga distribuida, la fuerza cortante y el momento flector en cualquier punto de la viga se pueden expresar en forma diferencial.
Se crea una ecuación diferencial lineal de tercer orden diferenciando aún más la expresión y asumiendo que la rigidez a la flexión es constante.
Derivando una vez más, se obtiene una ecuación diferencial lineal de cuarto orden. Esta ecuación gobierna esencialmente la forma o curva elástica que tomará la viga al soportar una carga distribuida.
Multiplicando la ecuación diferencial lineal de cuarto orden por la constante de rigidez a flexión e integrándola cuatro veces, se puede obtener una expresión para la curva.
Las constantes en la ecuación que define la curva de la viga están dictadas por las condiciones finales, como la inclinación o la deflexión de los apoyos y si la fuerza cortante y el momento flector son nulos en el extremo libre de una viga en voladizo o solo el momento flector es cero en ambos extremos de una viga apoyada.
Esta técnica calcula con precisión las curvas para vigas en voladizo o apoyadas con cargas distribuidas. Sin embargo, las reacciones de apoyo de las vigas en voladizo causan irregularidades de cortante, lo que requiere funciones distintas para la definición de la curva de longitud completa.

curva elástica a partir de la distribución de carga

<div>
 El comportamiento estructural de las vigas bajo cargas distribuidas es fundamental para el análisis de ingeniería, que se centra en predecir cómo las vigas se doblan y reaccionan en tales condiciones. Los diferentes tipos de vigas (por ejemplo, en voladizo, apoyadas o en voladizo) se comportan de manera diferente bajo condiciones de carga distribuida.
 
 Para todas las vigas, el análisis de la reacción de la viga a las cargas distribuidas comienza con la comprensión de la relación entre la carga de una viga y las fuerzas cortantes y los momentos flexionantes resultantes. Inicialmente, esta relación se expresa como una ecuación diferencial.
 
 <img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="text-align: center; height: 32px;">
 Una diferenciación adicional de esta expresión, suponiendo una flexibilidad de la viga constante, conduce a una ecuación diferencial más compleja que describe la curva de deflexión de la viga o cómo se doblará bajo la carga.
 
 <img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="text-align: center; height: 40px;">
 Esta compleja ecuación se integra varias veces para determinar la forma real de esta curva. Cada paso de integración introduce una constante que debe estar definida por las condiciones de contorno de la viga, como por ejemplo cómo se soporta o conecta la viga en sus extremos.
 
 <img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="text-align: center; height: 32px;">
 Las condiciones de contorno varían según cómo se soporta una viga. Por ejemplo, una viga en voladizo, fija en un extremo y libre en el otro, tendrá diferentes restricciones en cuanto a deflexión y fuerza en cada extremo. Por el contrario, una viga apoyada tendrá condiciones enfocadas principalmente en los puntos de apoyo. Particularmente desafiante es el análisis de vigas salientes, donde partes de la viga se extienden más allá de sus soportes. Estos segmentos experimentan fuerzas y momentos de flexión únicos, lo que requiere cálculos distintos para describir con precisión el comportamiento de la viga en toda su longitud. Esta comprensión detallada garantiza que las estructuras sean seguras y funcionales.
 
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Curva elástica a partir de la distribución de carga

Cuando una viga soporta una carga distribuida, la fuerza cortante y el momento flector en cualquier punto de la viga se pueden expresar en forma diferencial.
Un ...

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

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 differential equation for the deflection curve, which is the shape the beam takes as it bends.

Finding the actual curve requires integrating the equation several times. Each integration adds a constant that must be determined from boundary conditions. Boundary conditions depend on how the beam is supported or connected at its ends.

Different beam types need different boundary conditions. A cantilever beam is fixed at one end and free at the other, so its deflection and force limits are set differently at each end. A supported beam is handled mainly at the support points.

Overhanging beams are more complex because part of the beam extends beyond the supports. Those outer sections experience unique forces and bending moments. They need separate calculations so the behavior along the full beam length can be described accurately.

Beam Deflection Under Distributed Load

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 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 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...

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 distributed loads create shear force, bending moment, and beam deflection. See how boundary conditions change for cantilever, supported, and overhanging beams.

Beam Deflection Under Distributed Load | 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 Deflection Under Distributed Load

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