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

非対称荷重下で支持された梁を解析するには、レベル ポイントを特定するために、既知の勾配を持つ参照接線が必要です。
参照接線の傾きは、両端間の接線シフトを決定することによって計算されます。
任意の点でのビームの傾きは、最初のモーメント面積定理を使用してから、その傾きの方程式を書き込むことで決定できます。
ある点から参照接線までの垂直距離は、サポートに対するその点の接線方向の偏差とも呼ばれ、2 次モーメント面積定理を使用して決定できます。
この偏差は特定のセグメントに等しく、点から参照接線までの垂直距離を示します。ある点での変形は、水平線からの垂直距離を示します。
このたわみは特定のセグメントのサイズに等しいため、2 つのセグメント間の差として定義できます。
類似の三角形を調べることにより、これらのセグメントの1つを決定できます。たわみと接線方向の偏差の符号規則を考慮すると、ある点でのたわみが注目されます。

非対称荷重を受ける梁

非対称荷重を受ける支持梁を解析することは、梁がさまざまな力の分布にどのように反応するかを理解するために、構造工学において不可欠です。この解析には、たわみを計算し、梁の傾斜がゼロになる点を特定することが含まれます。これは、構造の安定性と機能を確保するために重要です。

第一モーメント面積定理は、梁上の任意の点での傾きを決定します。この定理は、梁上の 2 点間の傾きの変化が、その区間にわたるモーメント図の下の領域に対応することを示しています。偏差の測定に役立つ基準接線は、梁の両端間の接線方向のシフトから計算された既知の傾きによって識別されます。

次に、第 2 モーメント面積定理により、接線偏差として知られる、この基準接線からの任意の点の垂直偏差が計算されます。この測定は、荷重下での梁の曲げ挙動を理解し、設計を考慮する上で重要な最大たわみ点を特定するために不可欠です。

最後に、第 1 の定理を使用して目的の点での傾きを決定した後、第 2 の定理は、この点が基準線から垂直方向にどれだけ逸脱しているかを測定します。この偏差は最大許容たわみを定義し、構造上の破損や過度の変形を防止することで梁が安全性と運用基準を確実に満たすようにします。このような解析を通じて、動作負荷下でも完全性を維持し、安全要件を満たす梁を設計できます。

非対称荷重下で支持された梁を解析するには、レベル ポイントを特定するために、既知の勾配を持つ参照接線が必要です。
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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

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Beam Deflection and Level Points

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