Name: locating the beam’s highest deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 13 s
Channel: JoVE

다리 위를 달리는 기차를 생각해 보십시오. 여기서 비대칭 하중은 최대 처짐이 일반적으로 중간에서 발생하지 않는 지지된 빔에 적용됩니다.
빔의 최대 처짐은 곡선에서 빔의 접선이 수평인 점 O를 식별하여 계산됩니다.
점 X에서 접선의 기울기는 지지부 사이의 접선 편차를 계산하고 거리로 나누어 결정됩니다.
점 O에서의 기울기가 0이므로 O와 X 사이의 기울기는 X에서의 음의 기울기와 같습니다.
첫 번째 모멘트 면적 정리는 지지 X에서 음의 기울기와 동일한 M/EI 다이어그램 아래의 면적을 측정하여 점 O를 찾는 데 사용됩니다.
최대 처짐은 점 O에 대한 지지 X의 접선 편차와 같습니다. 이 값은 X와 O 사이의 영역에서 X를 통과하는 수직 축과 관련된 첫 번째 모멘트를 계산하여 얻을 수 있습니다.

최대 처짐

교량 위를 이동하는 열차와 같이 비대칭 하중을 받는 빔을 해석할 때 최대 응력 및 처짐 지점을 정확하게 결정하는 것이 중요합니다. 이 과정에는 하중의 고르지 않은 분포로 인해 중간점에서 항상 발생하지 않을 수 있는 빔의 최대 처짐을 식별하는 작업이 포함됩니다.

최대 처짐은 처짐 곡선의 접선이 수평인 점 O로 알려진 특정 지점에서 발생합니다. 점 O를 찾기 위해 빔을 따라 주어진 점 X에서 접선의 기울기를 검사합니다. 점 X의 기울기는 지지점 사이의 접선 편차를 고려하고 이를 거리로 나누어 계산할 수 있습니다. 이 기울기는 점 O에서 0이며 최대 처짐 위치를 나타냅니다.

첫 번째 모멘트-영역 정리는 점 O를 찾는 데 중요한 역할을 합니다. 이 정리에 따르면 빔을 따라 있는 두 점 사이의 굽힘 모멘트 다이어그램 아래 영역은 이 점 사이의 기울기 변화에 해당합니다. 점 O는 지지점 X에서 음의 기울기까지 이 면적을 계산하여 식별할 수 있습니다.

점 O가 결정되면 점 O에 대한 지지점 X의 접선 편차를 분석하여 최대 처짐이 계산됩니다. 이 접근법은 비대칭 하중 하에서 보의 구조적 거동을 평가하는 체계적인 방법을 제공하여 철도와 같은 구조물의 안전성과 안정성을 보장합니다. 교량.

다리 위를 달리는 기차를 생각해 보십시오. 여기서, 비대칭 하중은 최대 처짐이 일반적으로 발생하지 않는 지지된 빔에 적용됩니다.

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 is examined.

The slope at point X can be found from the tangential deviation between the supports divided by the distance between them. When this slope becomes zero, point O has been reached. The First Moment-Area Theorem helps with this step. This theorem says the area under the bending moment diagram between two points equals the change in slope between those points.

Point O is identified by using that area up to the negative slope at support X. After point O is found, the maximum deflection is calculated by studying the tangential deviation of support X about point O. This method gives a clear way to analyze beams under unsymmetrical loading and helps support the safety and stability of structures like railway bridges.

Locating the Beam’s Highest 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

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

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 unsymmetrical loads shift the point of maximum deflection in a supported beam and how the First Moment-Area Theorem helps locate it.

Locating the Beam’s Highest Deflection | 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

Locating the Beam’s Highest Deflection

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