Name: beam deflection and level points
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 17 s
Channel: JoVE

비대칭 하중 하에서 지지되는 빔을 분석하려면 표고 점을 식별하기 위해 알려진 경사가 있는 참조 접선이 필요합니다.
참조 접선의 기울기는 끝 사이의 접선 이동을 결정하여 계산됩니다.
임의의 지점에서 빔의 기울기는 첫 번째 모멘트 면적 정리를 사용한 다음 해당 기울기에 대한 방정식을 작성하여 결정할 수 있습니다.
한 점에서 참조 접선까지의 수직 거리(지지부에 대한 해당 점의 접선 편차라고도 함)는 두 번째 모멘트 면적 정리를 사용하여 결정할 수 있습니다.
이 편차는 특정 세그먼트와 같으며 점에서 참조 접선까지의 수직 거리를 나타냅니다. 한 점에서의 처짐은 수평선으로부터의 수직 거리를 나타냅니다.
이 편향은 크기가 특정 세그먼트와 같기 때문에 두 세그먼트 간의 차이로 정의할 수 있습니다.
유사한 삼각형을 연구하여 이러한 세그먼트 중 하나를 결정할 수 있습니다. 처짐과 접선 편차에 대한 기호 규칙을 고려할 때, 한 점에서의 처짐이 표시됩니다.

비대칭 하중을 받는 빔

비대칭 하중 하에서 지지 빔을 분석하는 것은 구조 엔지니어링에서 빔이 다양한 힘 분포에 어떻게 반응하는지 이해하는 데 필수적입니다. 이 분석에는 처짐을 계산하고 빔의 기울기가 0인 지점을 식별하는 작업이 포함되며, 이는 구조적 안정성과 기능성을 보장하는 데 중요합니다.

첫 번째 모멘트 면적 정리는 빔의 임의 지점에서의 기울기를 결정합니다. 이 정리는 빔의 두 점 사이의 기울기 변화가 해당 간격에 대한 모멘트 다이어그램 아래의 영역에 해당함을 나타냅니다. 편차를 측정하는 데 도움이 되는 기준 접선은 빔 끝 사이의 접선 이동에서 계산된 알려진 기울기로 식별됩니다.

그런 다음 두 번째 모멘트 면적 정리는 접선 편차라고 알려진 이 기준 접선에서 임의 점의 수직 편차를 계산합니다. 이 측정은 하중을 받는 빔의 굽힘 동작을 이해하고 설계 고려 사항에 중요한 최대 처짐 지점을 식별하는 데 필수적입니다.

마지막으로 첫 번째 정리를 이용하여 원하는 점에서의 기울기를 결정한 후 두 번째 정리는 이 점이 기준선에서 수직으로 얼마나 벗어나는지 측정합니다. 이 편차는 최대 허용 편향을 정의하여 구조적 결함이나 과도한 변형을 방지하여 빔이 안전 및 작동 표준을 충족하도록 보장합니다. 이러한 분석을 통해 작동 부하 하에서 무결성을 유지하고 안전 요구 사항을 충족하는 빔을 설계할 수 있습니다.

비대칭 하중 하에서 지지되는 빔을 분석하려면 표고 점을 식별하기 위해 알려진 경사가 있는 참조 접선이 필요합니다.
의 경사 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 that interval. A reference tangent is then chosen. This line has a known slope and is used to measure how the beam bends away from it.

The second moment-area theorem gives the vertical deviation from the reference tangent. This deviation is called the tangential deviation. It helps identify the level point, where the slope of the beam is zero, and it also shows the maximum deflection point.

Finding slope and tangential deviation is essential for checking structural stability and function. These values help engineers confirm that the beam stays within the maximum allowable deflection. That reduces the risk of failure or excessive deformation and supports safe design under operational loads.

Beam Deflection and Level Points

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

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 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 moment-area theorems are used to find beam slope, tangential deviation, and level points for a supported beam under unsymmetrical loading.

Beam Deflection and Level Points | 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 and Level Points

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