Name: beam deflection under distributed load
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 16 s
Channel: JoVE

Quando uma viga carrega uma carga distribuída, a força de cisalhamento e o momento de flexão em qualquer ponto da viga podem ser expressos de forma diferencial.
Uma equação diferencial linear de terceira ordem é criada diferenciando ainda mais a expressão e assumindo que a rigidez à flexão é constante.
Diferenciar mais uma vez resulta em uma equação diferencial linear de quarta ordem. Esta equação governa essencialmente a forma ou curva elástica que a viga tomará ao suportar uma carga distribuída.
Multiplicando a equação diferencial linear de quarta ordem pela constante de rigidez à flexão e integrando-a quatro vezes, pode-se obter uma expressão para a curva.
As constantes na equação que define a curva da viga são ditadas pelas condições finais, como inclinação ou deflexão dos apoios e se a força de cisalhamento e o momento fletor são nulos na extremidade livre de uma viga em balanço ou apenas o momento fletor é zero em ambas as extremidades de uma viga apoiada.
Esta técnica calcula com precisão as curvas para vigas em balanço ou apoiadas com cargas distribuídas. No entanto, as reações de apoio das vigas salientes causam irregularidades de cisalhamento, exigindo funções distintas para a definição da curva de comprimento total.

curva elástica da distribuição de carga

O comportamento estrutural das vigas sob cargas distribuídas é fundamental para a análise de engenharia, que se concentra na previsão de como as vigas se dobram e reagem sob tais condições. Diferentes tipos de vigas (por exemplo, cantilever, apoiadas ou suspensas) comportam-se de maneira diferente sob condições de carga distribuída.

Para todas as vigas, a análise da reação da viga às cargas distribuídas começa pela compreensão da relação entre a carga de uma viga e as forças de cisalhamento e momentos fletores resultantes. Inicialmente, essa relação é expressa como uma equação diferencial.

<img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="height: 32px;">

Uma diferenciação adicional desta expressão, assumindo flexibilidade constante da viga, leva a uma equação diferencial mais complexa que descreve a curva de deflexão da viga, ou como ela se dobrará sob a carga.

<img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="height: 40px;">

Esta equação complexa é integrada várias vezes para determinar a forma real desta curva. Cada etapa de integração introduz uma constante que deve ser definida pelas condições de contorno da viga, como a forma como a viga é apoiada ou conectada em suas extremidades.

<img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="height: 32px;">
As condições de contorno variam com base em como uma viga é suportada. Por exemplo, uma viga cantilever, fixada numa extremidade e livre na outra, terá diferentes restrições relativamente à deflexão e à força em cada extremidade. Por outro lado, uma viga apoiada terá condições focadas principalmente nos pontos de apoio. Particularmente desafiadora é a análise de vigas pendentes, onde partes da viga se estendem além dos seus apoios. Esses segmentos sofrem forças e momentos fletores únicos, exigindo cálculos distintos para descrever com precisão o comportamento da viga ao longo de todo o seu comprimento. Esse entendimento detalhado garante que as estruturas sejam seguras e funcionais.

Curva Elástica da Distribuição de Carga

Quando uma viga carrega uma carga distribuída, a força de cisalhamento e o momento de flexão em qualquer ponto da viga podem ser expressos de forma diferencial.
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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

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 differential equation for the deflection curve, which is the shape the beam takes as it bends.

Finding the actual curve requires integrating the equation several times. Each integration adds a constant that must be determined from boundary conditions. Boundary conditions depend on how the beam is supported or connected at its ends.

Different beam types need different boundary conditions. A cantilever beam is fixed at one end and free at the other, so its deflection and force limits are set differently at each end. A supported beam is handled mainly at the support points.

Overhanging beams are more complex because part of the beam extends beyond the supports. Those outer sections experience unique forces and bending moments. They need separate calculations so the behavior along the full beam length can be described accurately.

Beam Deflection Under Distributed Load

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 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 distributed loads create shear force, bending moment, and beam deflection. See how boundary conditions change for cantilever, supported, and overhanging beams.

Beam Deflection Under Distributed Load | JoVE K12

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mechanical engineering

Beam Deflection Under Distributed Load

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