Name: moment-area method for cantilever deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 15 s
Channel: JoVE

El método de momento-área se puede implementar en una viga en voladizo bajo una carga concentrada y un momento para identificar la pendiente y la deflexión.
El proceso comienza con el dibujo del diagrama de cuerpo libre de la viga y el cálculo de las reacciones en el extremo fijo.
A continuación, se dibuja el diagrama de momento flector para determinar el punto en el que el momento es igual a cero.
A continuación, se dibuja el diagrama M/EI para identificar las áreas correspondientes a los segmentos AO y OB, asignando signos positivos o negativos en función de su ubicación en relación con el eje x.
El primer teorema de momento-área se aplica para calcular el ángulo entre las tangentes, y el segundo teorema de momento-área se usa para determinar la desviación tangencial, que es igual al primer momento alrededor de un eje vertical a través de B del área total entre A y B.
Finalmente, se determina la deflexión en B, que es igual a la desviación tangencial. A continuación, se esboza la forma desviada de la viga para su comprensión visual.

estabilidad de estructuras

Beams with Symmetric Loadings

En ingeniería mecánica, la estabilidad de los sistemas sometidos a diversas fuerzas es fundamental para diseñar estructuras duraderas y eficientes. Una forma fundamental de explorar estos conceptos es analizando sistemas como dos varillas conectadas en un punto de pivote, O, con un resorte de torsión de constante elástica k en el punto de pivote. Este sistema es similar en apariencia a un gato de tijera que se usa para cambiar neumáticos en un automóvil. En este caso, los brazos del varillaje (equivalentes a las varillas en este sistema) son totalmente verticales, correspondiendo al caso en el que el vehículo ha sido elevado lo máximo posible.

Estas varillas se colocan inicialmente verticalmente y la acción del resorte evita que giren. Si dos cargas externas F y F’, de igual magnitud y dirección opuesta, se aplican al sistema de tal manera que comparten la misma línea de acción a lo largo de las varillas, entonces el sistema permanece en equilibrio. Esta alineación asegura que no haya ningún momento o par neto que actúe para desplazar al sistema de su estado de equilibrio.

Sin embargo, si el punto de pivote, O, se mueve ligeramente hacia los lados, este movimiento da como resultado una pequeña desviación angular de las varillas con respecto a la vertical. Esto hace que los extremos de cada varilla giren con respecto a sus puntos de pivote, introduciendo pares en el sistema. El primer par resulta del apoyo de reacción en el punto O. Esta fuerza, que ahora actúa en ángulo, intenta desplazar aún más la varilla de su orientación vertical inicial, alejándola del equilibrio. El segundo par, resultante de la resistencia del resorte de torsión, ejerce una fuerza de restauración para devolver la varilla a su posición vertical original. El sistema alcanza un punto de carga crítica Fcr cuando estos dos momentos son iguales. Si la carga aplicada excede la carga crítica, el sistema se vuelve inestable; si es menor que la carga crítica, el sistema permanece estable.

Este principio es universalmente aplicable en sistemas estructurales y se utiliza ampliamente en diseños de ingeniería mecánica, incluida la estabilización de edificios, vehículos y maquinaria.

Estabilidad de Estructuras

El método de momento-área se puede implementar en una viga en voladizo bajo una carga concentrada y un momento para identificar la pendiente y la deflexión.
El proceso ...

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

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, the M/EI diagram is constructed. In this diagram, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam cross-section. The areas under the curve are identified between the zero-moment points. Each area is given a positive or negative sign based on its position relative to the x-axis.

The first moment-area theorem is then used to find the slope at a point on the beam. It does this by using the area under the M/EI diagram between two points. The second moment-area theorem is used to find deflection. It equals the first moment of the area under the M/EI diagram about a vertical axis through the endpoint.

The final step is to sketch the deflected shape of the beam. This gives a clear picture of how the beam bends under the applied load and moment. The method provides precise insight into the structural behavior of cantilever beams.

Moment-Area Method for Cantilever Deflection

moment-area-method-for-cantilever-deflection

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...

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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...

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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...

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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...

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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

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

Use the moment-area method to analyze a cantilever beam with a load and end moment. Learn how free-body diagrams, M/EI diagrams, slope, and deflection are found.

Moment-Area Method for Cantilever Deflection | JoVE K12

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mechanical engineering

Moment-Area Method for Cantilever Deflection

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