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

빔이 분산 하중을 전달할 때 빔의 모든 지점에서 전단력과 굽힘 모멘트는 차동 형태로 표현될 수 있습니다.
3차 선형 미분 방정식은 표현식을 추가로 미분하고 굴곡 강성이 일정하다고 가정하여 생성됩니다.
다시 한 번 미분하면 4차 선형 미분 방정식이 생성됩니다. 이 방정식은 기본적으로 분산 하중을 지지할 때 빔이 취하는 형상 또는 탄성 곡선을 제어합니다.
4차 선형 미분 방정식에 굴곡 강성 상수를 곱하고 4번 적분하면 곡선에 대한 표현식을 얻을 수 있습니다.
빔의 곡선을 정의하는 방정식의 상수는 지지대의 기울기 또는 처짐과 같은 끝 조건과 캔틸레버 빔의 자유 끝에서 전단력과 굽힘 모멘트가 null인지 또는 지지된 빔의 양쪽 끝에서 굽힘 모멘트만 0인지 여부에 따라 결정됩니다.
이 기술은 분산 하중이 있는 캔틸레버 또는 지지된 빔의 곡선을 정확하게 계산합니다. 그러나 돌출된 빔의 지지 반응은 전단 불규칙성을 유발하여 전체 길이 곡선 정의를 위해 고유한 기능이 필요합니다.

하중 분포의 탄성 곡선

분산 하중 하에서 빔의 구조적 거동은 이러한 조건에서 빔이 어떻게 구부러지고 반응하는지 예측하는 데 중점을 두는 엔지니어링 분석에 매우 중요합니다. 다양한 유형의 빔(예: 캔틸레버, 지지 또는 돌출부)은 분산 하중 조건에서 다르게 동작합니다.

모든 빔에 대해 분산 하중에 대한 빔의 반응 분석은 빔의 하중과 그에 따른 전단력 및 굽힘 모멘트 간의 관계를 이해하는 것에서 시작됩니다. 처음에 이 관계는 미분방정식으로 표현됩니다.

<img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="height: 32px;">

일정한 빔 유연성을 가정하여 이 표현을 더욱 차별화하면 빔의 편향 곡선 또는 하중 하에서 어떻게 구부러지는지를 설명하는 더 복잡한 미분 방정식이 생성됩니다.

<img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="height: 40px;">

이 복잡한 방정식은 이 곡선의 실제 모양을 결정하기 위해 여러 번 통합됩니다. 각 통합 단계에는 빔이 지지되거나 끝에서 연결되는 방식과 같은 빔의 경계 조건에 의해 정의되어야 하는 상수가 도입됩니다.

<img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="height: 32px;">

경계 조건은 빔이 어떻게 지지되는지에 따라 달라집니다. 예를 들어, 한쪽 끝은 고정되고 다른 쪽 끝은 자유로운 캔틸레버 빔은 각 끝의 편향 및 힘과 관련하여 서로 다른 제약 조건을 갖습니다. 반대로, 지지된 빔은 주로 지지점에 초점을 맞춘 조건을 갖습니다. 특히 어려운 점은 빔의 일부가 지지대 너머로 확장되는 돌출된 빔을 분석하는 것입니다. 이러한 세그먼트는 고유한 힘과 굽힘 모멘트를 경험하므로 전체 길이에 따른 빔의 동작을 정확하게 설명하려면 별도의 계산이 필요합니다. 이러한 상세한 이해는 구조물의 안전과 기능을 보장합니다.

빔이 분산 하중을 전달할 때 빔의 모든 지점에서 전단력과 굽힘 모멘트는 차동 형태로 표현될 수 있습니다.
세 번째 순서 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Beam Deflection Under Distributed Load

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