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

지붕 구조에서 보의 처짐은 단일 해석 함수가 굽힘 모멘트를 나타낼 수 있는 경우 적분법을 사용하여 결정할 수 있습니다.
그러나 빔의 하중에 굽힘 모멘트를 나타내기 위해 여러 기능이 필요한 경우 추가 상수와 방정식이 필요하여 계산이 오래 걸립니다.
이러한 복잡성은 특이점 함수를 사용하여 단순화할 수 있습니다. 편심 하중을 전달하는 끝에서 지지되는 프리즘 빔을 고려하십시오.
이 빔의 전단력 함수는 적절한 특이점 함수를 사용하여 나타낼 수 있습니다. 굽힘 모멘트 함수는 이 전단력 함수를 적분하여 도출할 수 있습니다.
빔의 기울기와 편향은 이 모멘트 함수를 통합하여 얻을 수 있습니다. 방정식의 상수는 경계 조건에서 결정할 수 있습니다.
특이점 함수를 사용하면 추가 상수와 방정식이 필요하지 않아 계산이 단순화됩니다. 결과적으로, 특이점 함수는 복잡한 하중 하에서 빔의 기울기와 처짐을 계산하는 효율적인 방법을 제공합니다.

빔의 처짐

구조 엔지니어링의 다양한 하중 조건에서 빔 처짐과 경사를 정확하게 결정하는 것은 안전과 구조적 무결성을 보장하는 데 중요합니다. 특이점 기능은 특히 여러 하중 기능으로 인해 굽힘 모멘트 방정식이 복잡해지는 경우 빔 분석에 대한 간소화된 접근 방식을 제공합니다.

이전 단원에서 설명한 특이점 함수는 구조 하중 시나리오에서 흔히 발생하는 함수 내의 불연속성을 나타내는 강력한 수학적 도구입니다. 이러한 기능은 복잡하거나 다중 하중 하에서도 전단력과 굽힘 모멘트 방정식을 간결하게 표현하는 데 도움이 됩니다.

<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;">

일반적으로 길이에 따라 균일하고 양쪽 끝에서 지지되는 프리즘형 빔의 경우 편심 하중이 특정 문제를 야기합니다. 이러한 빔의 임의 지점에서의 전단력은 특이점 함수를 사용하여 모델링할 수 있습니다. 이러한 함수는 전단력 다이어그램에서 계단 함수로 표시되는 하중과 같이 특정 지점이나 특정 간격에 걸쳐 적용되는 하중으로 인해 발생하는 불연속성을 쉽게 처리합니다.

전단력 함수를 통합하여 파생된 굽힘 모멘트는 빔 성능을 평가하는 데 중요합니다. 이 단계는 빔의 응력 분포와 전반적인 처짐에 영향을 미칩니다. 빔의 편향은 굽힘 모멘트 함수를 두 번 통합하고 빔의 경계 조건을 적용하여 적분 상수를 해결함으로써 결정됩니다. 특이점 함수를 사용하여 전단력과 굽힘 모멘트를 모델링하면 여러 개의 추가 상수와 복잡한 방정식이 필요하지 않아 계산이 단순화되고 계산 효율성이 향상됩니다. 이 방법을 사용하면 구조 시스템의 안전한 설계 및 유지 관리에 필수적인 빔 편향 및 응력 분포에 대한 다양한 하중 조건을 쉽게 평가할 수 있습니다.

지붕 구조에서 빔의 처짐은 단일 분석 기능이 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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 Slope and Deflection by Singularity

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