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

橋の上を移動する列車を考えてみましょう。ここでは、通常、中央で最大のたわみが発生しない支持ビームに非対称の荷重が加えられます。
ビームの最大変形は、ビームの接線が水平であるカーブ上の点Oを特定することによって計算されます。
点Xにおける接線の傾きは、サポート間の接線方向の偏差を計算し、それをサポートの距離で割ることによって決定されます。
点 O の傾きは 0 であるため、O と X の間の傾きは X の負の傾きと等しくなります。
最初のモーメント-面積定理は、サポート X の負の傾きに等しい M/EI ダイアグラムの下の領域を測定することにより、点 O を特定するために使用されます。
最大たわみは、点Oを中心としたサポートXの接線方向の偏差に等しくなります。この値は、X と O の間の領域の X を通る垂直軸に関連する最初のモーメントを計算することによって取得できます。

最大たわみ

橋の上を移動する列車など、非対称な荷重を受ける梁を解析する場合、最大応力とたわみ点を正確に決定することが重要です。このプロセスには、梁の最大たわみを特定することが含まれますが、荷重の分布が不均一であるため、梁の最大たわみが常に中間点で発生するとは限りません。

最大のたわみは、たわみ曲線の接線が水平になる、点 O として知られる特定の点で発生します。点 O を見つけるには、梁に沿った任意の点 X における接線の傾きを調べます。点 X での傾きは、支持間の接線方向の偏差を考慮し、それを距離で割ることによって計算できます。 この傾きは点 O でゼロとなり、最大のたわみ位置を示します。

第一モーメント面積定理は、点 O を特定する際に重要な役割を果たします。この定理によれば、梁に沿った任意の 2 点間の曲げモーメント線図の下の面積は、これらの点間の傾きの変化に対応します。点 O は、支持 X の負の傾きまでのこの面積を計算することで特定できます。

点 O が決定されると、点 O を中心とした支持 X の接線方向の偏差を分析することによって最大たわみが計算されます。このアプローチは、非対称荷重下の梁の構造挙動を評価する系統的な方法を提供し、鉄道橋などの構造物の安全性と安定性を確保します。

橋の上を移動する列車を考えてみましょう。ここでは、非対称の荷重が支持されたビームに加えられますが、通常は最大たわみが発生しません...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Deflection of Beams

梁のたわみ

KEY TERMS AND CONCEPTS

重要用語と概念

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

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mechanical engineering

Locating the Beam’s Highest Deflection

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