Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

Stellen Sie sich einen Balken vor, der eine verteilte Last, zwei konzentrierte Lasten und ein paar Momente erfährt. Stellen Sie die Beziehung zwischen der Schubkraft und der verteilten Last her.
Betrachten Sie zunächst einen Elementquerschnitt auf dem Träger, der frei von jeder Einzellast oder jedem Kopplungsmoment ist, und zeichnen Sie ein Freikörperdiagramm des Schnitts.
Um das Gleichgewicht zu erhalten, sollte die auf die rechte Seite des Querschnitts wirkende Querkraft um einen kleinen und endlichen Betrag erhöht werden.
Die resultierende Kraft der verteilten Last wirkt in einem Bruchteil des Abstands vom rechten Ende des Abschnitts.
Unter Verwendung der Gleichgewichtsgleichung für die Vertikalkraft erhält man eine Beziehung zwischen Schub und Last.
Als nächstes kann durch Dividieren beider Seiten der Gleichung durch Δx und Zulassen von Δx gegen Null die Steigung der Querkraft bestimmt werden, die gleich der verteilten Lastintensität ist.
Schließlich führt die Neuanordnung der Gleichung und die Integration der verteilten Last über den Elementquerschnitt zwischen zwei beliebigen Punkten zu einer Beziehung zwischen der Änderung der Schubung und der Fläche unter der Lastkurve.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seiten 370-371.</li>
	<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. Seiten 391-392.</li>
</ol>

beziehung zwischen der verteilten last und der scherung

Das Verständnis der Beziehung zwischen der verteilten Last und der Scherkraft in der Strukturanalyse ist entscheidend für die Analyse von Balken unter verschiedenen Belastungsbedingungen. Betrachten Sie den Fall eines Balkens, der einer verteilten Last, zwei konzentrierten Lasten und einem Momentpaar ausgesetzt ist. <img alt="Figure 1" src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;"> Die Verbindung zwischen der Scherkraft und der verteilten Last für den gegebenen Fall kann nach dem gegebenen Verfahren hergestellt werden. Betrachten Sie zunächst einen elementaren Abschnitt des Balkens, der frei von konzentrierter Last oder Momentpaar ist. Wir können ein Freikörperdiagramm zeichnen, um die auf diesen Abschnitt wirkenden Kräfte zu analysieren. Das Diagramm wird aus der verteilten Last entlang der Länge des Balkens, der Scherkraft V(x) auf der rechten Seite des Abschnitts und einer zunehmenden Scherkraft dV zur Aufrechterhaltung des Gleichgewichts bestehen. Um das Gleichgewicht in vertikaler Richtung zu halten, sollte die auf der rechten Seite des Abschnitts wirkende Scherkraft um einen kleinen und endlichen Betrag ΔV erhöht werden. Die resultierende Kraft der verteilten Last, w(x)Δx, wirkt in einem Bruchteil der Strecke vom rechten Ende des Abschnitts. Durch Verwendung der Gleichung des Gleichgewichts für die vertikale Kraft ergibt sich die folgende Beziehung zwischen Scherung und Belastung: <img alt="Equation 1" src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;"> Nächster Schritt ist die Division beider Seiten der Gleichung durch Δx und das Annähern von Δx an Null: <img alt="Equation 2" src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;"> Diese Gleichung zeigt, dass die Steigung der Scherkraft der Intensität der verteilten Last entspricht. Schließlich können wir die Gleichung umstellen und die verteilte Last über den elementaren Abschnitt zwischen zwei beliebigen Punkten Q und R integrieren: <img alt="Equation 3" src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;"> Diese Integralgleichung zeigt die Beziehung zwischen der Änderung der Scherkraft und der Fläche unter der Lastkurve. Der Unterschied in der Scherkraft zwischen den Punkten Q und R entspricht der Fläche unter der verteilten Lastkurve zwischen diesen beiden Punkten.

Beziehung zwischen der verteilten Last und der Scherung

Das Verständnis der Beziehung zwischen der verteilten Last und der Querkraft in der Strukturanalyse ist entscheidend für die Analyse von Trägern, die verschiedenen ...

relation-between-the-distributed-load-and-shear

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Relation Between the Distributed Load and Shear

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 force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

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Beam Shear Force Diagram

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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

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

relation-between-the-shear-and-bending-moment

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 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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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...

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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.

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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

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | JoVE K12

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Shear Change Under Distributed Loading

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