Name: locating the beam’s highest deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 13 s
Channel: JoVE

Рассмотрим поезд, движущийся по мосту. В этом случае несимметричная нагрузка прикладывается к опорной балке, где максимальный прогиб обычно не происходит посередине.
Максимальное отклонение луча вычисляется путем определения точки O на кривой, где касательная луча горизонтальна.
Наклон касательной в точке X определяется путем вычисления тангенциального отклонения между опорами и деления его на расстояние между ними.
Поскольку наклон в точке O равен нулю, наклон между O и X равен отрицательному наклону в точке X.
Первая теорема о площади момента используется для определения точки O путем измерения площади под диаграммой M/EI, равной отрицательному наклону в опоре X.
Максимальный прогиб равен тангенциальному отклонению опоры X вокруг точки О. Это значение можно получить, вычислив первый момент, относящийся к вертикальной оси через X области между X и O.

Максимальный прогиб (отклонение)

При анализе балок под несимметричными нагрузками, например, при движении поезда по мосту, важно точно определить точки максимального напряжения и прогиба. Процесс предполагает определение максимального прогиба балки, который не всегда возникает в ее средней точке из-за неравномерного распределения нагрузки.

Максимальный прогиб происходит в определенной точке, известной как точка O, где касательная к кривой отклонения горизонтальна. Чтобы найти точку О, необходимо определить наклон касательной в любой заданной точке X вдоль балки. Наклон в точке X можно рассчитать, учитывая тангенциальное отклонение между опорами и разделив его на расстояние между ними. Этот наклон равен нулю в точке О, что указывает на место максимального прогиба.

Первая теорема о площади момента играет ключевую роль в определении точки O. Согласно этой теореме, площадь под диаграммой изгибающих моментов между любыми двумя точками вдоль балки соответствует изменению наклона между этими точками. Точку О можно определить, рассчитав эту площадь до отрицательного наклона в опоре Х.

После определения точки O максимальный прогиб рассчитывается путем анализа тангенциального отклонения опоры X относительно точки O. Этот подход обеспечивает систематический метод оценки структурного поведения балок при несимметричной нагрузке, обеспечивая безопасность и устойчивость таких конструкций, как железнодорожные мосты.

Рассмотрим поезд, движущийся по мосту. Здесь несимметричная нагрузка прикладывается к опираемой балке, где максимальный прогиб обычно не происходит в ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Deflection of Beams

Прогиб балок

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 is examined.

The slope at point X can be found from the tangential deviation between the supports divided by the distance between them. When this slope becomes zero, point O has been reached. The First Moment-Area Theorem helps with this step. This theorem says the area under the bending moment diagram between two points equals the change in slope between those points.

Point O is identified by using that area up to the negative slope at support X. After point O is found, the maximum deflection is calculated by studying the tangential deviation of support X about point O. This method gives a clear way to analyze beams under unsymmetrical loading and helps support the safety and stability of structures like railway bridges.

Locating the Beam’s Highest 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

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

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 unsymmetrical loads shift the point of maximum deflection in a supported beam and how the First Moment-Area Theorem helps locate it.

Locating the Beam’s Highest Deflection | JoVE K12

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mechanical engineering

Locating the Beam’s Highest Deflection

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