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

A flecha de uma viga em uma estrutura de telhado pode ser determinada usando o método de integração, desde que uma única função analítica possa representar o momento fletor.
No entanto, se o carregamento da viga exigir múltiplas funções para representar o momento fletor, constantes e equações adicionais seriam necessárias, levando a cálculos demorados.
Essa complexidade pode ser simplificada usando funções de singularidade. Considere uma viga prismática apoiada nas extremidades carregando uma carga excêntrica.
A função de força de cisalhamento desta viga pode ser representada usando uma função de singularidade apropriada. A função de momento de flexão pode ser derivada integrando esta função de força de cisalhamento.
A inclinação e a deflexão da viga podem ser obtidas integrando esta função de momento. As constantes nas equações podem ser determinadas a partir das condições de contorno.
O uso de funções de singularidade elimina a necessidade de constantes e equações adicionais, simplificando a computação. Como resultado, as funções de singularidade fornecem uma maneira eficiente de calcular a inclinação e a deflexão de uma viga sob cargas complexas.

deflexão de uma viga

Determinar com precisão a deflexão e a inclinação de uma viga sob diversas condições de cargas é crucial na engenharia estrutural para garantir a segurança e a integridade estrutural. As funções de singularidade oferecem uma abordagem simplificada para a análise de vigas, especialmente quando múltiplas funções de carregamento complicam a equação do momento fletor.

As funções de singularidade, descritas em uma lição anterior, são ferramentas matemáticas poderosas que representam descontinuidades dentro de uma função comumente encontrada em cenários de carregamento estrutural. Essas funções ajudam a expressar as equações de forças de cisalhamento e momento fletor de forma compacta, mesmo sob cargas complexas ou múltiplas.

<img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="height: 32px;">
<img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="height: 32px;">

Para uma viga prismática, normalmente uniforme ao longo do seu comprimento e apoiada em ambas as extremidades, uma carga excêntrica apresenta desafios específicos. A força de cisalhamento em qualquer ponto dessa viga pode ser modelada usando funções de singularidade. Estas funções lidam facilmente com as descontinuidades introduzidas por cargas aplicadas em pontos específicos ou em determinados intervalos, como uma carga representada como uma função degrau no diagrama de força de cisalhamento.


O momento fletor, derivado da integração da função de força de cisalhamento, é crucial para avaliar o desempenho da viga. Esta etapa afeta a distribuição de tensões da viga e a deflexão geral. A deflexão da viga é determinada integrando a função momento fletor duas vezes e aplicando as condições de contorno da viga para resolver as constantes de integração. O uso de funções de singularidade para modelar forças de cisalhamento e momentos fletores elimina a necessidade de múltiplas constantes adicionais e equações complexas, simplificando os cálculos e melhorando a eficiência computacional. Este método permite uma avaliação simples de diferentes condições de carregamento na deflexão da viga e na distribuição de tensões, o que é essencial para o projeto e manutenção seguro de sistemas estruturais.

Deflexão de uma Viga

A deflexão de uma viga em uma estrutura de telhado pode ser determinada usando o método de integração, desde que uma única função analítica possa representar o ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Deflection of Beams

Deflexão de Vigas

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

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mechanical engineering

Beam Slope and Deflection by Singularity

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