Name: beam deflection from loads and supports
Uploaded: 2024-05-23T13:00:01.000Z
Duration: 1 min 15 s
Channel: JoVE

Важнейшим компонентом при проектировании балок является идентификация и измерение прогиба. Всестороннее понимание прогибов необходимо при работе с неопределенными балками.
Рассмотрим нависающую балку, которая несет две сосредоточенные нагрузки. Реакции на опорах рассчитываются путем анализа диаграммы свободного тела балки.
С помощью диаграммы свободного тела и реакционных опор строится диаграмма изгибающего момента, которая показывает, как изгибается балка.
Диаграмма изгибающего момента показывает, что луч изгибается вверх между началом и определенной точкой из-за положительного изгибающего момента, и он изгибается вниз от этой точки до конца луча из-за отрицательного изгибающего момента.
Наибольшее значение кривой или наименьшего радиуса кривизны происходит в опоре, где изгибающий момент наиболее высок.
Для детального анализа и проектирования балки крайне важно иметь точные данные об отклонении и наклоне балки в различных точках, уделяя особое внимание пониманию максимального прогиба.

Деформация балки при поперечной нагрузке

Понимание отклонения (прогиба) балки, особенно балок неопределенной формы с нависающими сегментами и множеством сосредоточенных нагрузок, имеет решающее значение для обеспечения структурной целостности и функциональности. Процесс начинается с построения точной диаграммы свободного тела, которая помогает определить силы и моменты, действующие на балку. Эта диаграмма важна для визуализации того, как изгибающие моменты изменяются по длине балки, влияя на ее кривизну.

Результаты диаграммы изгибающих моментов распространяются и на анализ кривизны балки. Кривизна балки коррелирует с изгибающими моментами вдоль балки и обратно пропорциональна жесткости и поперечному сечению балки. Более изгибающие моменты приводят к большей кривизне, что важно для понимания профиля отклонения балки.

Особое внимание уделяется максимальному отклонению, поскольку чрезмерный прогиб может привести к структурным проблемам или даже поломке. Это максимальное отклонение обычно происходит там, где кривизна наибольшая, часто рядом с опорами или непосредственно под точками приложения нагрузок. Определение этого максимального отклонения имеет решающее значение для обеспечения соответствия конструкции балки стандартам безопасности и функциональным требованиям. Проведя тщательный анализ с использованием диаграмм свободного тела и изгибающих моментов, можно эффективно спроектировать конструкции, которые безопасны и соответствуют конкретным условиям нагрузки и архитектурным требованиям.

Важнейшим компонентом при проектировании балок является идентификация и измерение прогиба. Необходимо всестороннее понимание прогибов...

deformation-of-a-beam-under-transverse-loading

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Deflection of Beams

Прогиб балок

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 will curve. Beam curvature is related to the bending moment and is affected by the beam’s stiffness and cross-section. When the bending moment is higher, the curvature is greater, so the deflection profile becomes easier to evaluate.

The main goal is to find the maximum deflection. Too much deflection can cause structural problems or even failure. This largest deflection often appears where curvature is greatest, such as near supports or directly under applied loads. Using free-body and bending moment diagrams helps engineers design beams that stay safe and meet functional requirements under specific loading conditions.

Beam Deflection from Loads and Supports

deformation of a beam under transverse loading

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

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 free-body diagrams and bending moment diagrams help predict beam deflection, identify maximum deflection, and check structural safety under load.

Beam Deflection from Loads and Supports | 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 from Loads and Supports

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