Name: beam deflection and boundary conditions
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 23 s
Channel: JoVE

곡선의 한 지점에서 평면 곡선의 곡률은 곡선의 1차 및 2차 도함수를 포함하는 표현식을 사용하여 표현할 수 있습니다.
경사는 빔의 탄성 곡선에 대해 중요하지 않으므로 탄성 곡선에 대한 지배 방정식은 굴곡 강성을 고려하여 2차 선형 미분 방정식으로 표현됩니다.
굴곡 강성이 빔을 따라 변하는 경우 x의 함수로 표현됩니다. 그러나 프리즘 빔의 경우 굴곡 강성이 일정하게 유지됩니다.
이 방정식의 통합은 수평이 있는 점에서 탄성 곡선에 대한 접선에 의해 형성된 각도를 제공합니다.
이 작은 각도는 통합될 때 모든 지점에서 빔의 편향을 제공합니다. 빔 지지부의 경계 조건은 방정식의 상수를 결정합니다.
지지, 돌출 및 캔틸레버는 주로 고려되는 세 가지 유형의 빔입니다. 지지되고 돌출된 빔의 경우 지지점에서의 처짐은 0입니다.
지지점에서의 처짐과 경사는 모두 캔틸레버 빔에 대해 0입니다. 이러한 조건을 통해 상수를 계산할 수 있습니다.

탄성곡선의 방정식

Equation of the Elastic Curve

구조 공학에서 중요한 평면 곡선의 곡률 개념은 하중을 받을 때 빔이 얼마나 급격하게 구부러지는지를 정의합니다. 이 곡률은 곡선의 1차 및 2차 도함수를 사용하여 결정됩니다.

자유단에 점하중이 있는 캔틸레버 빔(예: 다이빙 보드)을 생각해 보세요. 작은 경사로 빔 편향을 분석할 때는 빔의 탄성 곡선 모양이 중요합니다. 이 분석을 위한 지배 방정식에는 굽힘 모멘트와 빔의 굴곡 강성이 포함됩니다. 이는 탄성 계수와 빔 단면의 관성 모멘트의 곱입니다.

<img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style=" height: 32px;">

단면이 일정하게 유지되는 프리즘형 빔의 경우 분석이 단순화되어 빔 길이에 따라 굽힘 강성이 일정해집니다. 지배 방정식을 통합하면 특정 지점에서 곡선의 접선에 의해 형성된 각도를 계산할 수 있으며, 이를 추가로 통합하면 해당 지점에서 빔의 편향이 발생합니다.

<img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style=" height: 28px;">

빔 지지대의 경계 조건은 이러한 계산을 완료하는 데 매우 중요합니다. 지지형, 돌출형 및 캔틸레버형은 일반적인 유형의 빔이며 각각 고유한 경계 조건을 갖습니다. 예를 들어, 캔틸레버 빔 지지점의 처짐과 경사는 0이며, 이는 처짐 방정식의 상수를 계산하는 데 필수적입니다.

구조적 안전성과 기능성을 보장하려면 빔 편향을 정확하게 예측하는 것이 중요합니다. 과도한 처짐은 구조적 결함이나 서비스 가능성 문제를 일으킬 수 있으므로 하중 하에서 빔 동작을 이해하는 것이 중요합니다.

곡선의 한 지점에서 평면 곡선의 곡률은 곡선의 1차 및 2차 도함수를 포함하는 표현식을 사용하여 표현할 수 있습니다.
그만큼 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Deflection of Beams

보의 처짐

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 moment with flexural rigidity. Flexural rigidity is the product of the modulus of elasticity and the moment of inertia of the beam’s cross-section. For a prismatic beam, the cross-section stays constant, so the flexural rigidity is constant along the beam.

Integrating the governing equation gives the angle of the tangent to the curve at any point. A second integration gives the deflection at that point. Boundary conditions at the beam supports are needed to find the constants in these equations.

Supported, overhanging, and cantilever beams each have different boundary conditions. For a cantilever beam, the deflection and slope at the support point are zero. Predicting deflection is important because too much bending can lead to structural failure or serviceability problems.

Beam Deflection and Boundary Conditions

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 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

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 beam deflection, curvature, and boundary conditions work together in structural engineering, using cantilever and other beam types.

Beam Deflection and Boundary Conditions | 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 Boundary Conditions

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