Name: beam deflection and level points
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 17 s
Channel: JoVE

Analiza podpartej belki pod niesymetrycznymi obciążeniami wymaga stycznej odniesienia o znanym nachyleniu w celu zidentyfikowania punktu poziomu.
Nachylenie stycznej odniesienia oblicza się, określając przesunięcie styczne między końcami.
Nachylenie belki w dowolnym punkcie można wyznaczyć, korzystając z twierdzenia o powierzchni pierwszego momentu, a następnie pisząc równanie dla tego nachylenia.
Odległość pionową od punktu do stycznej odniesienia, znaną również jako odchyłka styczna tego punktu względem podpory, można wyznaczyć za pomocą twierdzenia o obszarze drugiego momentu.
Odchylenie to jest równe pewnemu segmentowi i oznacza odległość pionową od punktu do stycznej odniesienia. Ugięcie w punkcie wskazuje jego odległość w pionie od linii poziomej.
Ponieważ to ugięcie jest równe określonemu rozmiarowi segmentu, można je zdefiniować jako różnicę między dwoma segmentami.
Badając podobne trójkąty, można określić jeden z tych segmentów. Biorąc pod uwagę konwencje znakowe dotyczące ugięć i odchyłek stycznych, odnotowuje się ugięcie w punkcie.

belki z niesymetrycznymi obciążeniami

Analiza belki podpartej poddanej niesymetrycznym obciążeniom jest niezbędna w inżynierii konstrukcyjnej, aby zrozumieć, w jaki sposób belki reagują na różne rozkłady sił. Analiza ta polega na obliczeniu ugięcia i identyfikacji punktów, w których nachylenie belki wynosi zero, które są kluczowe dla zapewnienia stabilności i funkcjonalności konstrukcji.

Pierwsze twierdzenie o powierzchni momentu określa nachylenie w dowolnym punkcie belki. Twierdzenie to wskazuje, że zmiana nachylenia między dwoma punktami belki odpowiada obszarowi pod wykresem momentów w tym przedziale. Styczną odniesienia, która pomaga w pomiarze odchyleń, identyfikuje się na podstawie jej znanego nachylenia, obliczonego na podstawie przesunięć stycznych pomiędzy końcami belki.

Drugie twierdzenie o powierzchni momentu oblicza następnie odchylenie pionowe dowolnego punktu od tej stycznej odniesienia, znane jako odchylenie styczne. Pomiar ten jest niezbędny do zrozumienia zachowania belki przy zginaniu pod obciążeniem i określenia maksymalnych punktów ugięcia krytycznych ze względów projektowych.

Wreszcie, po określeniu nachylenia w pożądanym punkcie za pomocą pierwszego twierdzenia, drugie twierdzenie mierzy, jak daleko ten punkt odbiega pionowo od linii odniesienia. To odchylenie określa maksymalne dopuszczalne ugięcie, zapewniając belce spełnienie standardów bezpieczeństwa i eksploatacji, zapobiegając awariom konstrukcyjnym lub nadmiernym odkształceniom. Dzięki takim analizom można zaprojektować belki, które zachowują integralność pod obciążeniami eksploatacyjnymi i spełniają wymagania bezpieczeństwa.

Belki z niesymetrycznymi obciążeniami

Analiza podpartej belki pod niesymetrycznymi obciążeniami wymaga stycznej odniesienia o znanym nachyleniu w celu zidentyfikowania punktu poziomu.
Nachylenie ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Deflection of Beams

Ugięcia belek

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 that interval. A reference tangent is then chosen. This line has a known slope and is used to measure how the beam bends away from it.

The second moment-area theorem gives the vertical deviation from the reference tangent. This deviation is called the tangential deviation. It helps identify the level point, where the slope of the beam is zero, and it also shows the maximum deflection point.

Finding slope and tangential deviation is essential for checking structural stability and function. These values help engineers confirm that the beam stays within the maximum allowable deflection. That reduces the risk of failure or excessive deformation and supports safe design under operational loads.

Beam Deflection and Level Points

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 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 moment-area theorems are used to find beam slope, tangential deviation, and level points for a supported beam under unsymmetrical loading.

Beam Deflection and Level Points | 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 Level Points

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