Name: moment-area method for cantilever deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 15 s
Channel: JoVE

모멘트 영역 방법은 집중된 하중과 모멘트 하에서 캔틸레버 빔에 구현하여 기울기와 처짐을 식별할 수 있습니다.
이 프로세스는 빔의 자유물체 다이어그램을 그리고 고정 단부에서의 반응을 계산하는 것으로 시작됩니다.
다음으로, 모멘트가 0과 같은 지점을 결정하기 위해 굽힘 모멘트 다이어그램이 그려집니다.
그런 다음 M/EI 다이어그램을 그려 세그먼트(AO 및 OB)에 해당하는 영역을 식별하고 x축에 상대적인 위치를 기반으로 양수 또는 음수 부호를 할당합니다.
첫 번째 모멘트 영역 정리는 접선 사이의 각도를 계산하기 위해 적용되고, 두 번째 모멘트 영역 정리는 A와 B 사이의 총 영역에서 B를 통과하는 수직 축에 대한 첫 번째 모멘트와 동일한 접선 편차를 결정하는 데 사용됩니다.
마지막으로, B에서의 처짐이 결정되며, 이는 접선 편차와 같습니다. 그런 다음 시각적 이해를 위해 빔의 편향된 모양을 스케치합니다.

대칭 하중을 받는 빔

Beams with Symmetric Loadings

모멘트 영역 방법은 다양한 하중 하에서 빔의 경사와 처짐을 결정하기 위해 구조 공학에서 사용되는 분석 도구입니다. 자유단에 집중된 하중과 모멘트가 있는 캔틸레버를 생각해 보세요. 첫 번째 단계는 고정 끝에서의 반응을 계산하기 위해 자유 물체 다이어그램을 구성하는 것입니다. 다음으로, 굽힘 모멘트가 0인 지점에 초점을 맞춰 굽힘 모멘트가 빔의 길이에 따라 어떻게 변하는지 시각화하기 위해 굽힘 모멘트 다이어그램이 그려집니다.

그런 다음 M/EI 다이어그램이 그려집니다. 여기서 M은 굽힘 모멘트, E는 탄성 계수, I는 빔 단면의 관성 모멘트입니다. 이 다이어그램은 굽힘 모멘트가 0인 점 사이의 세그먼트에 대한 곡선 아래 영역을 식별하고 x축을 기준으로 한 위치를 기준으로 해당 영역에 양수 또는 음수 부호를 할당합니다.

첫 번째 모멘트-면적 정리는 두 지점 사이의 M/EI 다이어그램 아래 면적을 통합하여 빔의 임의 지점에서 기울기를 계산하는 데 적용됩니다. 두 번째 모멘트-영역 정리는 빔의 편향을 찾는 데 사용되며 이를 끝점을 통과하는 수직 축에 대한 M/EI 다이어그램 아래 영역의 첫 번째 모멘트와 동일시합니다.

마지막으로, 빔의 편향된 형상이 스케치되어 분석 결과를 시각적으로 표현하고 부과된 하중 조건에서 빔의 굽힘을 보여줍니다. 이 방법은 캔틸레버 빔의 구조적 성능에 대한 정확한 통찰력을 제공합니다.

모멘트 영역 방법은 집중된 하중과 모멘트 하에서 캔틸레버 빔에 구현하여 기울기와 처짐을 식별할 수 있습니다.
그 과정은 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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, the M/EI diagram is constructed. In this diagram, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam cross-section. The areas under the curve are identified between the zero-moment points. Each area is given a positive or negative sign based on its position relative to the x-axis.

The first moment-area theorem is then used to find the slope at a point on the beam. It does this by using the area under the M/EI diagram between two points. The second moment-area theorem is used to find deflection. It equals the first moment of the area under the M/EI diagram about a vertical axis through the endpoint.

The final step is to sketch the deflected shape of the beam. This gives a clear picture of how the beam bends under the applied load and moment. The method provides precise insight into the structural behavior of cantilever beams.

Moment-Area Method for Cantilever Deflection

moment-area-method-for-cantilever-deflection

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

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

Use the moment-area method to analyze a cantilever beam with a load and end moment. Learn how free-body diagrams, M/EI diagrams, slope, and deflection are found.

Moment-Area Method for Cantilever Deflection | JoVE K12

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Moment-Area Method for Cantilever Deflection

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