Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Прогиб балки в конструкции крыши может быть определен с помощью метода интегрирования, при условии, что изгибающий момент может представлять одна аналитическая функция.
Однако, если нагрузка на балку требует нескольких функций для представления изгибающего момента, потребуются дополнительные константы и уравнения, что приведет к длительным расчетам.
Эту сложность можно упростить с помощью функций сингулярности. Рассмотрим призматическую балку, поддерживаемую на концах и несущую эксцентриковую нагрузку.
Функция поперечной силы этой балки может быть представлена с помощью соответствующей функции сингулярности. Функция изгибающего момента может быть получена путем интегрирования этой функции поперечной силы.
Наклон и прогиб балки могут быть получены путем интегрирования этой функции момента. Константы в уравнениях могут быть определены из граничных условий.
Использование функций сингулярности избавляет от необходимости в дополнительных константах и уравнениях, упрощая вычисления. В результате, функции сингулярностей обеспечивают эффективный способ расчета наклона и прогиба балки при сложных нагрузках.

Прогиб балки

Точное определение прогиба и наклона балки при различных условиях нагрузки в проектировании конструкций имеет важное значение для обеспечения безопасности и целостности конструкции. Функции сингулярности предлагают упрощенный подход к анализу балок, особенно когда несколько функций нагрузки усложняют уравнение изгибающего момента. 

Функции сингулярности, описанные в одном из предыдущих уроков, являются мощными математическими инструментами, которые представляют собой разрывы внутри функции, часто встречающиеся в сценариях нагрузки конструкций. Эти функции помогают компактно выразить уравнения поперечной силы и изгибающего момента даже при сложных или множественных нагрузках. 

<img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="text-align: center; height: 32px;">
<img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="text-align: center; height: 32px;">

Для призматической балки, обычно равномерной по длине и с опорой на обоих концах, эксцентричная нагрузка представляет особые проблемы. Поперечную силу в любой точке такой балки можно смоделировать с помощью функций сингулярности. Эти функции легко справляются с разрывами, вызванными нагрузками, приложенными в определенных точках или через определенные интервалы, например нагрузкой, представленной в виде ступенчатой ​​функции на диаграмме поперечной силы. 

Изгибающий момент, полученный путем интегрирования функции поперечной силы, имеет решающее значение для оценки характеристик балки. Этот момент влияет на распределение напряжений балки и общий прогиб. Прогиб балки определяется путем двойного интегрирования функции изгибающего момента и применения граничных условий балки для определения констант интегрирования. Использование функций сингулярности для моделирования поперечных сил и изгибающих моментов устраняет необходимость использования множества дополнительных констант и сложных уравнений, упрощая расчеты и повышая их эффективность. Этот метод позволяет легко оценить различные условия нагрузки на прогиб балки и распределение напряжений, что важно для безопасного проектирования и обслуживания конструктивных систем.

Прогиб балки в конструкции крыши может быть определен с помощью метода интегрирования, при условии, что одна аналитическая функция может представлять ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Deflection of Beams

Прогиб балок

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

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 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 singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | 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 Slope and Deflection by Singularity

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