Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

Stellen Sie sich einen einfach gelagerten Balken PQ der Länge L vor, der in der Mitte eine Punktlast F trägt. Wie groß ist die Ablenkung in der Mitte des Strahls?
Hier sind die Reaktionen an beiden Enden des Balkens gleich und betragen jeweils die Hälfte der zentralen Last.
Betrachten Sie das Segment PC, und wählen Sie einen beliebigen Punkt im Abstand x vom Ende P. Zu diesem Zeitpunkt ist das Moment, das durch die Reaktion am Punkt P entsteht, die Last am Punkt P multipliziert mit der Entfernung. Eine partielle Differenzierung der Momentengleichung in Bezug auf die Last am Ende P ist halb von x. 
Eine ähnliche Analyse kann für das Segment QC des Trägers durchgeführt werden.
Unter Verwendung des Satzes von Castigliano wird die Durchbiegung im Punkt C durch die partiellen Ableitungen der Dehnungsenergie aufgrund der aufgebrachten Last bestimmt.
Diese Gleichung kann vereinfacht werden, indem die beiden Segmente des Strahls PC und der QC berücksichtigt werden.
Wenn Sie die Integration über die Hälfte der Länge des Balkens durchführen, erhalten Sie einen Ausdruck für die Ablenkung in der Mitte des Balkens.

castiglianos theorem: problemlösung

Die Durchbiegung eines einfach gestützten Balkens, der eine zentrale Punktlast trägt, kann mithilfe von Prinzipien der Strukturmechanik analysiert werden, insbesondere durch Anwendung des Castigliano-Theorems. Dieses Theorem setzt die Verschiebung am Lastangriffspunkt mit den partiellen Ableitungen der Dehnungsenergie in der Struktur in Beziehung. Der einfach gestützte Balken mit einer Punktlast in seiner Mitte hat symmetrische Reaktionskräfte an den Stützen, die jeweils die Hälfte der Last tragen. Das Biegemoment an jedem Punkt entlang des Balkens wird aus diesen Reaktionen abgeleitet, berechnet über den Abstand von der nächsten Stütze.

Der Satz von Castigliano besagt, dass die Durchbiegung am Punkt, an dem die Last angewendet wird, durch Differenzieren der Dehnungsenergie des Balkens in Bezug auf die Last bestimmt wird. Die Dehnungsenergie wird basierend auf dem Biegemoment entlang des Balkens berechnet, das über seine Länge integriert wird. Für diesen Balken wird die Dehnungsenergie aufgrund der Biegung aus dem Quadrat des Biegemomentausdrucks berechnet, das über die halbe Balkenlänge integriert wird.

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="text-align: center; height: 106px;">

Da der Balken symmetrisch ist, wird dieser Wert verdoppelt, um den gesamten Balken zu berücksichtigen, und die Durchbiegung in der Mitte des Balkens wird berechnet. Sie hängt von der Lastgröße und der dritten Potenz der Balkenlänge ab und ist umgekehrt proportional zum Produkt aus Trägheitsmoment und Elastizitätsmodul des Balkenquerschnitts.

<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="text-align: center; height: 42px;">

Castiglianos Theorem: Problemlösung

Stellen Sie sich einen einfach gelagerten Balken PQ der Länge L vor, der in der Mitte eine Punktlast F trägt. Wie groß ist die Ablenkung in der Mitte des Strahls?
Hier ist die ...

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Castigliano&#39;s Theorem: Problem Solving

beam deflection from central load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam.

For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support.

Castigliano's theorem says the displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

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Using Strain Energy to Find Deflection

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load. The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point...

castigliano&#39;s theorem: problem solving

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how Castigliano's theorem is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central Load | 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.

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Beam Deflection from Central Load

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