Name: beam deflection and level points
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 17 s
Channel: JoVE

El análisis de una viga apoyada bajo cargas asimétricas requiere una tangente de referencia con una pendiente conocida para identificar el punto de nivel.
La pendiente de la tangente de referencia se calcula determinando el desplazamiento tangencial entre los extremos.
La pendiente del haz en cualquier punto se puede determinar usando el teorema del área del primer momento y luego escribiendo una ecuación para esa pendiente.
La distancia vertical desde un punto hasta la tangente de referencia, también conocida como la desviación tangencial de ese punto con respecto al apoyo, se puede determinar utilizando el teorema del área del segundo momento.
Esta desviación es igual a un determinado segmento y significa la distancia vertical desde el punto hasta la tangente de referencia. La deflexión en un punto indica su distancia vertical desde una línea horizontal.
Dado que esta deflexión es igual a un segmento específico en tamaño, se puede definir como la diferencia entre dos segmentos.
Al estudiar triángulos similares, se puede determinar uno de estos segmentos. Teniendo en cuenta las convenciones de signos para deflexiones y desviaciones tangenciales, se observa la deflexión en un punto.

vigas con cargas asimétricas

Analizar una viga apoyada bajo cargas asimétricas es esencial en ingeniería estructural para comprender cómo responden las vigas a diversas distribuciones de fuerzas. Este análisis implica calcular la deflexión e identificar los puntos donde la pendiente de la viga es cero, que son cruciales para garantizar la estabilidad estructural y la funcionalidad.

El primer teorema momento-área determina la pendiente en cualquier punto de la viga. Este teorema indica que el cambio de pendiente entre dos puntos de una viga corresponde al área bajo el diagrama de momentos en ese intervalo. La tangente de referencia, que ayuda a medir las desviaciones, se identifica por su pendiente conocida, calculada a partir de los desplazamientos tangenciales entre los extremos de la viga.

El segundo teorema momento-área calcula la desviación vertical de cualquier punto desde esta tangente de referencia, conocida como desviación tangencial. Esta medida es vital para comprender el comportamiento de flexión de la viga bajo carga e identificar los puntos de deflexión máxima críticos para las consideraciones de diseño.

Finalmente, después de determinar la pendiente en un punto deseado usando el primer teorema, el segundo teorema mide cuánto se desvía verticalmente este punto de la línea de referencia. Esta desviación define la deflexión máxima permitida, asegurando que la viga cumpla con los estándares operativos y de seguridad al evitar fallas estructurales o deformaciones excesivas. A través de tales análisis, se pueden diseñar vigas que mantengan la integridad bajo cargas operativas y cumplan con los requisitos de seguridad.

Vigas con cargas asimétricas

El análisis de una viga apoyada bajo cargas asimétricas requiere una tangente de referencia con una pendiente conocida para identificar el punto de nivel.
La pendiente de la ...

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

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 that interval. A reference tangent is then chosen. This line has a known slope and is used to measure how the beam bends away from it.

The second moment-area theorem gives the vertical deviation from the reference tangent. This deviation is called the tangential deviation. It helps identify the level point, where the slope of the beam is zero, and it also shows the maximum deflection point.

Finding slope and tangential deviation is essential for checking structural stability and function. These values help engineers confirm that the beam stays within the maximum allowable deflection. That reduces the risk of failure or excessive deformation and supports safe design under operational loads.

Beam Deflection and Level Points

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 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 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 moment-area theorems are used to find beam slope, tangential deviation, and level points for a supported beam under unsymmetrical loading.

Beam Deflection and Level Points | 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 and Level Points

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