Name: beam deflection under combined loads
Uploaded: 2025-05-22T09:06:18.000Z
Duration: 1 min 20 s
Channel: JoVE

重ね合わせの方法は、構造工学で複数の荷重を受ける梁の勾配とたわみを計算するために使用されます。
ビームにさまざまな集中荷重または分散荷重がかかる場合、多くの場合、各荷重によって生じる勾配とたわみを個別に計算する方が実用的です。
次に、重ね合わせの原理が適用され、各荷重に対応する勾配またはたわみの値を合計して、全体的な効果を決定します。
分散荷重と集中荷重を受ける支持梁について考えてみましょう。
まず、集中荷重による傾きとたわみを決定し、続いて分散荷重によるたわみを弾性曲線方程式を使用して決定します。
傾き方程式は、たわみ方程式を微分することによって得られます。次に、分布荷重と集中荷重の両方によって生じる勾配と変形が組み合わされ、ビームの全体的な変形と傾斜が得られます。
重ね合わせの方法は、複雑な荷重シナリオを簡素化し、安全で効果的な構造設計に不可欠なビームのたわみと傾きを正確に計算します。

重ね合わせの原理

重ね合わせの原理は、構造工学において重要な技術であり、梁にかかる複数の荷重の影響を分析するために使用されます。このアプローチでは、梁にかかる各荷重のたわみと傾斜を個別に計算し、これらの影響を合計して全体的な影響を決定します。これは、梁材料がその弾性限界内に留まり、変形が線形弾性であることが保証される場合にのみ適用されます。

重ね合わせの原理を適用する場合、集中または分散にかかわらず、各タイプの荷重が独立して考慮されます。集中荷重の場合、たわみと傾斜は標準的な梁の理論式から導出されます。分布荷重の場合、たわみは通常、弾性曲線方程式を使用して計算されます。この方程式を積分して傾きを導出し、さらにたわみを取得します。

<img alt="Equation 1" src="/files/ftp_upload/15670/15670_Equation_1v2.svg" style="height: 42px;">

個々の荷重によるたわみと傾きの合計が、梁に対する全体的な影響を与えます。たとえば、梁に均一分布荷重と中心点荷重がかかる場合、全体のたわみは各荷重によるたわみの合計になります。この方法により、複数の荷重下での梁の挙動を正確に予測する複雑な荷重シナリオが簡素化されます。重ね合わせの原理は、問題をより単純な要素に分解することで、安全かつ効果的な構造システムの設計に役立ち、安全基準と性能要件への準拠を保証します。

重ね合わせの方法は、構造工学で複数の荷重を受ける梁の勾配とたわみを計算するために使用されます。
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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

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 result.

Each load is treated on its own. A concentrated load, or point load, uses standard beam theory formulas to find slope and deflection. A distributed load is handled with the elastic curve equation, which is integrated to find slope and then integrated again to find deflection.

The total beam response is the sum of the individual responses. For example, if a beam carries a uniform distributed load and a central point load, the overall deflection is the sum of the deflections from both loads. This approach simplifies complex loading problems and helps predict beam behavior for safe and effective structural design.

Beam Deflection Under Combined Loads

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 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

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 the method of superposition finds beam slope and deflection under combined point and distributed loads using elastic beam theory.

Beam Deflection Under Combined Loads | 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 Under Combined Loads

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