Name: beam deflection and boundary conditions
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 23 s
Channel: JoVE

A curvatura da curva plana em um ponto da curva pode ser expressa usando uma expressão envolvendo a primeira e a segunda derivadas da curva.
A inclinação é insignificante para a curva elástica da viga, portanto, a equação governante para a curva elástica é expressa como uma equação diferencial linear de segunda ordem, considerando a rigidez à flexão.
Se a rigidez à flexão varia ao longo da viga, ela é expressa em função de x. No entanto, para uma viga prismática, a rigidez à flexão permanece constante.
A integração desta equação fornece o ângulo formado pela tangente à curva elástica em um ponto com a horizontal.
Este pequeno ângulo, quando integrado, nos dá a deflexão do feixe em qualquer ponto. As condições de contorno dos suportes de viga determinam as constantes nas equações.
Apoiado, saliente e cantilever são os três tipos de vigas que são considerados principalmente. Para vigas apoiadas e salientes, a deflexão nos pontos de apoio é zero.
Tanto a deflexão quanto a inclinação no ponto de apoio são zero para vigas em balanço. Essas condições permitem o cálculo das constantes.

equação da curva elástica

Equation of the Elastic Curve

O conceito de curvatura de curvas planas, crucial na engenharia estrutural, define o quão acentuadamente uma viga se curva sob o efeito de uma carga. Esta curvatura é determinada usando a primeira e a segunda derivadas da curva.

Considere uma viga cantilever com uma carga pontual em sua extremidade livre (por exemplo, um trampolim). Ao analisar a deflexão da viga com pequenas inclinações, o formato da curva elástica da viga se torna um fator crucial. A equação governante para esta análise envolve o momento fletor e a rigidez à flexão da viga, que é um produto do módulo de elasticidade e do momento de inércia da seção transversal da viga.

<img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style=" height: 32px;">

Para vigas prismáticas, onde a seção transversal permanece constante, a análise é simplificada, tornando a rigidez à flexão constante ao longo do comprimento da viga. A integração da equação governante permite o cálculo do ângulo formado pela tangente à curva em qualquer ponto, o que, após integração subsequente, produz a deflexão da viga naquele ponto.

<img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style=" height: 28px;">

As condições de contorno nos apoios da viga são vitais para completar estes cálculos. Suportadas, suspensas e cantilever são tipos comuns de vigas, cada uma com condições de contorno distintas. Por exemplo, a deflexão e a inclinação no ponto de apoio de uma viga cantilever são zero, o que é essencial para calcular as constantes das equações de deflexão.

Prever com precisão a deflexão da viga é crucial para garantir a segurança estrutural e a funcionalidade. Uma deflexão excessiva pode causar falhas estruturais ou problemas de manutenção, ressaltando a importância de se compreender o comportamento da viga sob carga.

Equação da Curva Elástica

A curvatura da curva plana em um ponto da curva pode ser expressa usando uma expressão envolvendo a primeira e a segunda derivadas da curva.
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

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 moment with flexural rigidity. Flexural rigidity is the product of the modulus of elasticity and the moment of inertia of the beam’s cross-section. For a prismatic beam, the cross-section stays constant, so the flexural rigidity is constant along the beam.

Integrating the governing equation gives the angle of the tangent to the curve at any point. A second integration gives the deflection at that point. Boundary conditions at the beam supports are needed to find the constants in these equations.

Supported, overhanging, and cantilever beams each have different boundary conditions. For a cantilever beam, the deflection and slope at the support point are zero. Predicting deflection is important because too much bending can lead to structural failure or serviceability problems.

Beam Deflection and Boundary Conditions

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 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

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 beam deflection, curvature, and boundary conditions work together in structural engineering, using cantilever and other beam types.

Beam Deflection and Boundary Conditions | 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 Boundary Conditions

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