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

Кривизна плоской кривой в точке кривой может быть выражена с помощью выражения, включающего первую и вторую производные кривой.
Наклон незначителен для упругой кривой балки, поэтому определяющее уравнение для упругой кривой выражается в виде линейного дифференциального уравнения второго порядка, учитывающего жесткость на изгиб.
Если жесткость на изгиб изменяется вдоль луча, она выражается как функция x. Однако для призматической балки жесткость на изгиб остается постоянной.
Интегрирование этого уравнения дает угол, образованный касательной к упругой кривой в точке с горизонталью.
Этот малый угол при интегрировании дает нам отклонение луча в любой точке. Граничные условия опор балки определяют константы в уравнениях.
Опорные, нависающие и консольные - это три типа балок, которые рассматриваются в первую очередь. Для опорных и нависающих балок прогиб в точках опоры равен нулю.
Как прогиб, так и уклон в точке опоры равны нулю для консольных балок. Эти условия позволяют вычислять константы.

Уравнение упругой кривой

Equation of the Elastic Curve

Понятие кривизны плоских кривых, имеет важное значение в проектировании конструкций, определяет, насколько резко балка изгибается под нагрузкой. Эта кривизна определяется с использованием первой и второй производных кривой. 

Рассмотрим консольную балку с точечной нагрузкой на свободном конце (например, трамплин). При анализе прогиба балки с небольшим наклоном ключевое значение приобретает форма упругой кривой балки. Основное уравнение для этого анализа включает изгибающий момент и изгибную жесткость балки, которая является произведением модуля упругости и момента инерции поперечного сечения балки. 

<img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style="text-align: center; height: 32px;">

Для призматических балок, у которых поперечное сечение остается постоянным, анализ упрощается, делая изгибную жесткость постоянной по всей длине балки. Интегрирование основного уравнения позволяет рассчитать угол, образованный касательной к кривой в любой точке, что при дальнейшем интегрировании дает отклонение балки в этой точке. 

<img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style="text-align: center; height: 28px;">

Граничные условия на опорах балки имеют важное значение для выполнения этих расчетов. Опорные, нависающие и консольные балки — это распространенные типы балок, каждая из которых имеет свои граничные условия. Например, прогиб и наклон в точке опоры консольной балки равны нулю, что важно для расчета констант уравнений прогиба. 

Точное прогнозирование отклонения балки имеет решающее значение для обеспечения безопасности и функциональности конструкции. Чрезмерный прогиб может привести к разрушению конструкции или проблемам с эксплуатацией, что подчеркивает важность понимания поведения балки под нагрузкой.

Кривизна плоской кривой в точке кривой может быть выражена с помощью выражения, включающего первую и вторую производные кривой.
Тем...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Deflection of Beams

Прогиб балок

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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