Name: moment-area method for cantilever deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 15 s
Channel: JoVE

Метод Moment-area может быть реализован на консольной балке под действием сосредоточенной нагрузки и момента для определения наклона и прогиба.
Процесс начинается с рисования диаграммы свободного тела пучка и расчета реакций на неподвижном конце.
Затем строится диаграмма изгибающего момента, чтобы определить точку, где момент равен нулю.
Затем строится диаграмма M/EI для определения областей, соответствующих отрезкам AO и OB, с присвоением положительных или отрицательных знаков на основе их расположения относительно оси x.
Первая теорема о площади момента применяется для вычисления угла между касательными, а вторая теорема о площади момента используется для определения тангенциального отклонения, которое равно первому моменту вокруг вертикальной оси через B от общей площади между A и B.
Наконец, определяется прогиб в точке В, который равен тангенциальному отклонению. Затем отклоняемая форма луча набрасывается для визуального понимания.

Балки с симметричными нагрузками

Beams with Symmetric Loadings

Метод момент-площадь — это аналитический инструмент, используемый в проектировании конструкций для определения наклона и прогиба балок под различными нагрузками. Рассмотрим консоль с сосредоточенной нагрузкой и моментом на свободном конце. Первым шагом является построение диаграммы-свободного тела для расчета реакций на неподвижном конце. Затем строится диаграмма изгибающего момента, чтобы визуализировать, как изгибающий момент изменяется по длине балки, уделяя особое внимание точкам, где изгибающий момент равен нулю.

Затем рисуется диаграмма M/EI, где M — изгибающий момент, E — модуль упругости, а I — момент инерции поперечного-сечения балки. На этой диаграмме определяются области под кривой для отрезков между точками, где изгибающий момент равен нулю, и областям присваиваются положительные или отрицательные знаки в зависимости от их положения относительно оси-X.

Первая теорема о моменте и площади применяется для расчета наклона в любой точке балки путем интегрирования площади под диаграммой M/EI между двумя точками. Вторая теорема о моменте и площади используется для нахождения прогиба балки, приравнивая его к первому моменту площади под диаграммой M/EI относительно вертикальной оси, проходящей через конечную точку.

Наконец, форма прогиба балки зарисовывается, обеспечивая визуальное представление аналитических результатов и иллюстрируя изгиб балки в условиях приложенной нагрузки. Этот метод дает точное представление о структурных характеристиках консольных балок.

Метод Moment-area может быть реализован на консольной балке под действием сосредоточенной нагрузки и момента для определения наклона и прогиба.
Процесс...

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 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, the M/EI diagram is constructed. In this diagram, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam cross-section. The areas under the curve are identified between the zero-moment points. Each area is given a positive or negative sign based on its position relative to the x-axis.

The first moment-area theorem is then used to find the slope at a point on the beam. It does this by using the area under the M/EI diagram between two points. The second moment-area theorem is used to find deflection. It equals the first moment of the area under the M/EI diagram about a vertical axis through the endpoint.

The final step is to sketch the deflected shape of the beam. This gives a clear picture of how the beam bends under the applied load and moment. The method provides precise insight into the structural behavior of cantilever beams.

Moment-Area Method for Cantilever Deflection

moment-area-method-for-cantilever-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

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

Use the moment-area method to analyze a cantilever beam with a load and end moment. Learn how free-body diagrams, M/EI diagrams, slope, and deflection are found.

Moment-Area Method for Cantilever Deflection | JoVE K12

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mechanical engineering

Moment-Area Method for Cantilever Deflection

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