Name: finding resultant loads on a beam
Uploaded: 2023-09-22T11:49:22.000Z
Duration: 1 min 29 s
Channel: JoVE

Ein Träger der Länge L, der einer variierenden Last ausgesetzt ist, die eine Kombination aus parabolischer und trapezförmiger Lastverteilung entlang der x-Achse ist.
Betrachten Sie bei der Parabellast ein Differenzelement der Kraft dR über die Länge dx. Die resultierende Last wird durch Integration des Differentialelements über die Länge der Belastung erhalten. Sie befindet sich am Schwerpunkt der parabolischen Fläche, und die x-Koordinate des Schwerpunkts wird nach dem Momentenprinzip bestimmt.
Der trapezförmige Bereich kann in rechteckige und dreieckige Bereiche unterteilt werden, wobei die resultierende Belastung an deren Schwerpunkt wirkt. Diese Schwerpunkte werden auf der halben Länge des Rechtecks und auf einem Drittel der Basislänge von der vertikalen Seite des Dreiecks positioniert.
Die resultierende Last für den trapezförmigen Bereich erhält sich durch Addition der einzelnen resultierenden Lasten und verläuft durch den durch das Momentenprinzip bestimmten Schwerpunkt der Fläche.
Die gesamte verteilte Last ist die Summe der resultierenden Lasten der trapezförmigen und parabolischen Flächen, die durch den Schwerpunkt der kombinierten Fläche verlaufen, bestimmt durch das Momentenprinzip.

<ol>
	<li>Bier, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, P.J. und Self, B.P. (2019). Vektormechanik für Ingenieure \u2012 Statik und Dynamik. New York: McGraw-Hill. Seite 264</li>
	<li>Meriam, J.L.; Kraige, L.G. und Bolton, J.N. (2020). Technische Mechanik \u2012 Statik. Hoboken, New Jersey: John Wiley. Seite 273</li>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seite EM-2</li>
</ol>

laden einer einzelnen achse

In der Baustatik ist die Analyse von Balken, die unterschiedlichen Lasten ausgesetzt sind, ein kritischer Aspekt zum Verständnis des Verhaltens und der Leistung dieser strukturellen Elemente. Ein häufiges Szenario beinhaltet einen Balken, der einer Kombination verschiedener Lastverteilungen ausgesetzt ist. Betrachten Sie einen Balken der Länge L, der einer variierenden Last ausgesetzt ist, die eine Kombination aus parabolischer und trapezförmiger Lastverteilung entlang der x-Achse darstellt. In diesem Fall ist es wesentlich, die resultierenden Lasten, ihre Standorte und den Schwerpunkt der kombinierten Fläche zu bestimmen, um die Reaktion des Balkens unter diesen Belastungsbedingungen vorherzusagen. Zuerst betrachten Sie die parabolische Lastverteilung. Betrachten Sie ein differentielles Kraft-Element dR, das über eine kleine Länge dx wirkt. Um die resultierende Last für den parabolischen Bereich zu bestimmen, wird das differentielle Element dR über die gesamte Länge der Last integriert. Der Standort dieser resultierenden Last befindet sich im Schwerpunkt des parabolischen Bereichs. Das Momentenprinzip wird angewendet, um die x-Koordinate des Schwerpunkts zu finden, die besagt, dass das erste Moment der Fläche um eine Achse gleich dem Produkt aus der Fläche und dem Abstand ihres Schwerpunkts von der Achse ist. Als Nächstes wird die trapezförmige Lastverteilung analysiert, indem sie in zwei rechteckige und dreieckige Bereiche unterteilt wird. Die resultierenden Lasten für diese einzelnen Bereiche wirken in ihren jeweiligen Schwerpunkten. Für den rechteckigen Bereich liegt der Schwerpunkt in der Mitte der Länge des Rechtecks. Für den dreieckigen Bereich befindet sich der Schwerpunkt ein Drittel der Basislänge vom vertikalen Rand des Dreiecks entfernt. Durch das Hinzufügen der einzelnen resultierenden Lasten für die rechteckigen und dreieckigen Bereiche kann die resultierende Last für den gesamten trapezförmigen Bereich bestimmt werden. Der Standort der resultierenden Last verläuft durch den Schwerpunkt des trapezförmigen Bereichs,der ebenfalls mit dem Momentenprinzip bestimmt werden kann. Ähnlich kann durch das Addieren der resultierenden Lasten der parabolischen und trapezförmigen Bereiche die gesamte resultierende Last, die auf den Balken wirkt, bestimmt werden. Der Standort der gesamten resultierenden Last verläuft durch den Schwerpunkt der kombinierten Fläche, der weiterhin mit dem Momentenprinzip bestimmt werden kann.Diese Analyse ermöglicht es, die Reaktion des Balkens unter der gegebenen Lastverteilung genau zu verstehen und vorherzusagen. Durch die Bestimmung der resultierenden Lasten und ihrer Standorte können Ingenieure die Auswirkungen dieser Lasten auf die Struktur vorhersagen und geeignete Maßnahmen zur Sicherstellung der strukturellen Integrität und Sicherheit ergreifen. Die Kenntnis des Schwerpunkts der Lastverteilung ist auch entscheidend für die Auslegung der Unterstützungen und Verbindungen des Balkens, um ein Versagen zu vermeiden und eine optimale Leistung zu gewährleisten.

Laden einer einzelnen Achse

Im Hochbau ist die Analyse von Trägern, die unterschiedlichen Belastungen ausgesetzt sind, ein kritischer Aspekt, um das Verhalten und die Leistung dieser Balken zu verstehen.

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Force System Resultants

Resultierende eines Kraftsystems

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Load along a Single Axis

finding resultant loads on a beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted.

The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the full length of the load. The resultant load for the parabolic area acts at the centroid of that area. The x-coordinate of the centroid is found with the moment principle, which says the first moment of an area about an axis equals the area times the distance to its centroid.

The trapezoidal load is then split into two simpler shapes. One part is rectangular, and the other is triangular. Their resultant loads act at their own centroids. For the rectangle, the centroid lies halfway along its length. For the triangle, the centroid is one-third of the base length from the vertical side.

After those parts are found, their loads are added to get the resultant load of the full trapezoidal area. That load passes through the centroid of the trapezoid, which can also be found with the moment principle. Finally, the resultant loads from the parabolic and trapezoidal areas are combined. Their sum gives the total load on the beam, and its location passes through the centroid of the combined area.

Finding Resultant Loads on a Beam

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Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

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key terms and concepts

Learn how parabolic and trapezoidal loads on a beam are broken into simpler areas, then combined to find the resultant force and centroid.

Finding Resultant Loads on a Beam | JoVE K12

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mechanical engineering

Finding Resultant Loads on a Beam

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