Name: finding cable shape under distributed load
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 24 s
Channel: JoVE

Bei Hängebrücken erfahren die Hauptseile verteilte Lasten, die auf die Stütztürme und Verankerungen übertragen werden können. Bestimmen Sie die Form und Spannung des Kabels.
Es wird davon ausgegangen, dass das Kabel für die Analyse flexibel, nicht dehnbar und mit vernachlässigbarem Gewicht ist.
Betrachten Sie ein kleines Segment des Kabels und zeichnen Sie das Freikörperdiagramm des Segments, das die verteilte Last und die Änderungen der Zugkraft zeigt.
Wenden Sie als nächstes die Gleichgewichtsgleichungen an und durch Ersetzen der horizontalen und vertikalen Kräfte und Momente, die auf das Segment wirken, erhalten Sie einen Satz von Gleichungen.
Dividiert man jede Gleichung durch Δx und nimmt man jeden Grenzwert, der gegen Null tendiert, erhält man einen neuen Satz von drei Gleichungen.
Die Integration der ersten Gleichung ergibt die horizontale Komponente der Zugkraft an einem beliebigen Punkt entlang des Kabels.
Integrieren Sie als Nächstes die zweite Gleichung. Dividiert man ihn durch die Komponente der Zugkraft und setzt man den Wert von tanθ ein, erhält man den Ausdruck für die Steigung des Seils.
Schließlich bestimmt eine zweite Integration der abgeleiteten Gleichung die Kurve des Kabels.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seiten 384-386.</li>
</ol>

kabel unter einwirkung einer verteilten last

Die Analyse von Hängebrücken ist ein komplexer und kritischer Prozess, der mehrere Faktoren umfasst, darunter die Form und Spannung der Hauptkabel. Die Hauptkabel von Hängebrücken sind einer verteilten Last ausgesetzt, was zu Veränderungen der Zugkräfte und Verformung des Kabels führt. Diese Lasten müssen sorgfältig berücksichtigt werden, um sicherzustellen, dass die Brücke sicher ist und das Gewicht unterschiedlicher Lasten tragen kann. <img alt="Figure 1" src="/files/ftp_upload/14308/14308_Figure_1v2.svg" style="text-align: center; height: 350px;"> Um ein Kabel unter Einwirkung einer verteilten Last zu analysieren, betrachten Sie ein kleines Segment des Kabels und zeichnen Sie ein Kräftediagramm, das die verteilte Last und die Veränderungen der Zugkraft zeigt. Durch Anwendung von Gleichgewichtsgleichungen und Substitution der horizontalen und vertikalen Kräfte und Momente, die auf das Segment wirken, erhält man eine Reihe von Gleichungen. Durch Teilung jeder Gleichung durch Delta Δx und Annäherung an null erhält man eine neue Reihe von drei Gleichungen. Die erste Gleichung, die aus dieser Analyse gewonnen wird, hilft dabei, die horizontale Komponente der Zugkraft an jedem Punkt entlang des Kabels zu bestimmen. <img alt="Equation 1" src="/files/ftp_upload/14308/14308_Equation_1v2.svg" style="text-align: center; height: 13px;"> Sie ermöglicht auch die Berechnung der Spannung des Kabels an jedem Punkt entlang seiner Länge. Der Ausdruck für die Steigung des Kabels wird erhalten, indem man die zweite Gleichung, die die vertikale Komponente der Zugkraft ist, durch die horizontale Zugkraftkomponente teilt und den Wert von tan θ substituiert. <img alt="Equation 2" src="/files/ftp_upload/14308/14308_Equation_2v2.svg" style="text-align: center; height: 35px;"> Schließlich bestimmt eine zweite Integration der abgeleiteten Gleichung die Krümmung des Kabels. <img alt="Equation 3" src="/files/ftp_upload/14308/14308_Equation_3v2.svg" style="text-align: center; height: 35px;"> Somit hilft uns der Prozess, das Verhalten des Kabels bei Einwirkung von verteilten Lasten zu verstehen. Er ist auch nützlich bei der Gestaltung des Tragwerks der Brücke und stellt sicher, dass die Kabel das Gewicht des Verkehrs und anderer Belastungen tragen können.

Kabel unter Einwirkung einer verteilten Last

Die Analyse von Hängebrücken ist ein komplexer und kritischer Prozess, der mehrere Faktoren umfasst, darunter die Form und Spannung der Hauptseile. ...

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Cable Subjected to a Distributed Load

finding cable shape under distributed load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation.

The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations are then applied to the horizontal force, vertical force, and moments acting on the cable piece.

Next, each equation is divided by delta x, and the limit is taken as delta x approaches zero. This gives a new set of three equations. The first equation helps determine the horizontal component of tensile force at any point along the cable, and it can also be used to find the cable tension along its length.

The second equation is used to find the slope of the cable. It relates the vertical component of tensile force to the horizontal component. By substituting tan theta, the cable angle can be linked to the shape of the cable. A second integration of the final equation gives the curve of the cable. This analysis helps engineers understand cable behavior under distributed loads and design bridge support systems that can handle traffic weight and other stresses.

Finding Cable Shape Under Distributed Load

finding-cable-shape-under-distributed-load

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Beam Types and Support Conditions

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shear diagram

Beam Shear Force Diagram

Beam shear force diagrams show how shear changes along the length of a supported beam. In beam mechanics, they help students track the internal shear force caused by perpendicular loads. The beam AB in this example is supported at both ends, so the first step is to draw a free-body diagram that shows the external forces and internal reactions. The reaction forces at supports A and B are found with equilibrium equations for force and moment. For the beam shown, the vertical reaction at A is 24...

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Beam Load Analysis with Moment Diagrams

Beam load analysis uses a bending moment diagram to show how bending changes along the length of a beam. This diagram helps engineers and designers check whether a structure can handle applied forces. It is also useful for parts such as shelving arms, which must resist forces and moments from a load. To build the diagram, start by finding the reactive forces and any couple moments on the beam. In some cases, especially when a beam is inclined, these forces must be resolved into components that...

relation between the distributed load and shear

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Shear Change Under Distributed Loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

relation between the shear and bending moment

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Shear and Moment Diagrams in Beam Loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

shear and bending moment diagram: problem solving

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Beam Load Analysis with Shear Diagrams

Beam load analysis uses shear and bending moment diagrams to track the internal forces in a beam. These diagrams are important when a beam carries concentrated loads and a distributed load. They help show how the beam responds so structures can be designed safely and efficiently. The first step is to draw a free-body diagram of the whole beam. A free-body diagram shows the concentrated loads, the distributed load, and the reaction forces at the supports. Use the equilibrium equations, where...

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Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force. To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium...

cable subjected to a distributed load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation. The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations...

cable subjected to its own weight

Cable Sag Under Its Own Weight

Overhead power transmission cables sag under their own weight, and that shape affects how they carry electricity across long distances. To study this behavior, the cable is analyzed with a generalized loading function. This function treats the load along the cable's arc length instead of its projected length, which gives a more accurate picture of the cable's response. The analysis then focuses on a small cable segment. A free-body diagram is drawn to show the forces acting on that segment.

cable: problem solving

Cable Tension Under Uniform Loading

Cable tension under uniform loading is found by analyzing a cable fixed between two supports. The problem starts with the support conditions and the load on the cable. From there, the cable’s shape must be described with an equation that matches the curve formed by the load. The next step is to integrate the shape equation to get the cable’s shape function. This makes it possible to solve for constants in the equation. The boundary condition at the origin gives one constant. The slope, which...

key terms and concepts

Analyze how a suspension bridge cable responds to distributed loads using free-body diagrams, equilibrium equations, slope, tension, and cable curve.

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Finding Cable Shape Under Distributed Load

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