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

A análise de uma viga apoiada sob cargas assimétricas requer uma tangente de referência com uma inclinação conhecida para identificar o ponto de nível.
A inclinação da tangente de referência é calculada determinando o deslocamento tangencial entre as extremidades.
A inclinação da viga em qualquer ponto pode ser determinada usando o teorema da área do primeiro momento e, em seguida, escrevendo uma equação para essa inclinação.
A distância vertical de um ponto à tangente de referência, também conhecida como desvio tangencial desse ponto em relação ao apoio, pode ser determinada usando o teorema da área do segundo momento.
Esse desvio é igual a um determinado segmento e significa a distância vertical do ponto à tangente de referência. A deflexão em um ponto indica sua distância vertical de uma linha horizontal.
Como essa deflexão é igual a um segmento específico em tamanho, ela pode ser definida como a diferença entre dois segmentos.
Ao estudar triângulos semelhantes, um desses segmentos pode ser determinado. Considerando as convenções de sinal para deflexões e desvios tangenciais, a deflexão em um ponto é anotada.

vigas com cargas assimétricas

A análise de uma viga apoiada sob carregamentos assimétricos é essencial na engenharia estrutural para compreender como as vigas respondem a distribuições de forças variadas. Esta análise envolve o cálculo da flecha e a identificação de pontos onde a inclinação da viga é zero, cruciais para garantir a estabilidade e funcionalidade estrutural.


O primeiro teorema momento-área determina a inclinação em qualquer ponto da viga. Este teorema indica que a mudança na inclinação entre dois pontos de uma viga corresponde à área sob o diagrama de momentos nesse intervalo. A tangente de referência, que auxilia na medição dos desvios, é identificada pela sua inclinação conhecida, calculada a partir dos deslocamentos tangenciais entre as extremidades da viga.


O segundo teorema momento-área calcula então o desvio vertical de qualquer ponto desta tangente de referência, conhecido como desvio tangencial. Esta medida é vital para compreender o comportamento de flexão da viga sob carga e identificar pontos de deflexão máxima críticos para considerações de projeto.


Finalmente, depois de determinar a inclinação num ponto desejado utilizando o primeiro teorema, o segundo teorema mede até que ponto este ponto se desvia verticalmente da linha de referência. Este desvio define a deflexão máxima permitida, garantindo que a viga atenda às normas de segurança e operacionais, evitando falhas estruturais ou deformações excessivas. Através dessas análises é possível projetar vigas que mantenham a integridade sob cargas operacionais e atendam aos requisitos de segurança.

Vigas com Cargas Assimétricas

A análise de uma viga apoiada sob cargas assimétricas requer uma tangente de referência com uma inclinação conhecida para identificar o ponto de nível.
A inclinação do ...

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

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

O JoVE Core Mechanical Engineering oferece vídeos animados concisos e de fácil compreensão que abrangem tópicos-chave como equilíbrio, forças, análise de treliças e mecânica estrutural. Explore suas aplicações no mundo real e observe esses conceitos em ação por meio de experimentos conduzidos por cientistas.

mechanical engineering

Beam Deflection and Level Points

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