Name: beam deflection using moment-area rules
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 17 s
Channel: JoVE

Теорема о площади момента обеспечивает геометрические свойства упругой кривой для определения прогиба и наклона в любой точке балки, поддерживающей пол здания.
При построении графика по длине балки отношение M/EI создает диаграмму, аналогичную диаграмме изгибающего момента.
Теорема о площади момента выводит уравнение наклона и тангенциального отклонения из диаграммы луча. Первая теорема о площади момента приравнивает угол между касательными в точках P и Q к площади между этими точками под диаграммой M/EI.
Вторая теорема о площади момента связывает касательное отклонение точки P от Q до первого момента той же площади относительно вертикальной оси через P.
Эта теорема также выражается в терминах произведения площади и расстояния от центроида площади до вертикальной оси через P.
Тангенциальное отклонение — это вертикальное расстояние от Q до касательной в точке P. Он вычисляется путем умножения площади под диаграммой M/EI на расстояние от ее центроида до вертикальной оси через Q.

Теоремы о моменте и площади

Теорема о моменте площади имеет важное значение в проектировании конструкций для анализа изгиба балок, особенно в таких приложениях, как опоры перекрытий зданий. Эта теорема использует геометрические свойства кривой упругости, которая показывает, как балка деформируется под нагрузкой, чтобы упростить расчеты прогибов и наклонов.

Теорема состоит из двух частей. Первая часть связывает угол между касательными в любых двух точках кривой упругости балки с площадью под кривой, полученной путем построения графика величины M/EI (где M — изгибающий момент, E — модуль упругости, а I — момент инерции) от прогиба балки по ее длине. Площадь под этой кривой напрямую соответствует общему вращению, происходящему между этими двумя точками.

Вторая часть теоремы касается тангенциального отклонения—или вертикального смещения—между любыми двумя точками, возникающего в результате изгиба балки. В нем говорится, что это отклонение эквивалентно первому моменту площади под кривой M/EI относительно вертикальной оси, проходящей через одну из этих точек, что обеспечивает меру смещения сегмента балки от его исходного положения. Эти теоремы эффективно определяют наклон и прогиб в различных точках балки, что важно для обеспечения структурной безопасности и работоспособности конструкции под нагрузкой.

Теорема о площади момента обеспечивает геометрические свойства упругой кривой для определения прогиба и наклона в любой точке балки, поддерживающей ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Deflection of Beams

Прогиб балок

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 between two points on the elastic curve.

The second theorem deals with tangential deviation, or vertical displacement, between two points. It says this distance equals the first moment of the area under the M/EI curve about a vertical axis through one of the points. Together, the two theorems make it easier to find slope and deflection at different points along a beam. That information helps engineers check structural safety and performance under load.

Beam Deflection Using Moment-Area Rules

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

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 moment-area theorems use the elastic curve and M/EI diagram to find beam slope, rotation, and deflection in structural engineering.

Beam Deflection Using Moment-Area Rules | 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 Using Moment-Area Rules

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