Name: beam deflection from loads and supports
Uploaded: 2024-05-23T13:00:01.000Z
Duration: 1 min 15 s
Channel: JoVE

Eine entscheidende Komponente bei der Bemessung von Trägern ist die Identifizierung und Messung der Durchbiegung. Ein umfassendes Verständnis der Durchbiegungen ist notwendig, wenn es um unbestimmte Balken geht.
Stellen Sie sich einen überhängenden Balken vor, der zwei konzentrierte Lasten trägt. Die Reaktionen an den Lagern werden durch Analyse des Freikörperdiagramms des Trägers berechnet.
Mit dem Freikörperdiagramm und den Reaktionslagern wird ein Biegemomentdiagramm konstruiert, das die Biegung des Balkens darstellt.
Das Biegemomentdiagramm zeigt, dass sich der Balken aufgrund eines positiven Biegemoments zwischen dem Anfang und einem bestimmten Punkt nach oben krümmt und von diesem Punkt bis zum Ende des Balkens aufgrund eines negativen Biegemoments nach unten krümmt.
Der größte Wert der Kurve oder der kleinste Krümmungsradius tritt am Lager auf, wo das Biegemoment am höchsten ist.
Für eine detaillierte Analyse und Bemessung eines Trägers ist es entscheidend, genaue Daten über die Durchbiegung und Neigung des Trägers an verschiedenen Punkten zu haben, wobei der Schwerpunkt auf dem Verständnis der maximalen Durchbiegung liegt.

verformung eines balkens unter querbelastung

Das Verständnis der Balkendurchbiegung, insbesondere bei unbestimmten Balken mit überhängenden Segmenten und mehreren konzentrierten Lasten, ist für die Gewährleistung der strukturellen Integrität und Funktionalität von entscheidender Bedeutung. Der Prozess beginnt mit der Erstellung eines genauen Freikörperdiagramms, das dabei hilft, die auf den Balken wirkenden Kräfte und Momente zu identifizieren. Dieses Diagramm ist wichtig, um zu veranschaulichen, wie sich Biegemomente entlang der Länge des Trägers ändern und seine Krümmung beeinflussen.

Die Erkenntnisse aus dem Biegemomentdiagramm reichen bis zur Analyse der Balkenkrümmung. Die Krümmung des Balkens korreliert mit den Biegemomenten entlang des Balkens und steht im umgekehrten Verhältnis zur Steifigkeit und zum Querschnitt des Balkens. Höhere Biegemomente führen zu einer stärkeren Krümmung, was für das Verständnis des Durchbiegungsprofils des Trägers von wesentlicher Bedeutung ist.

Besonderes Augenmerk liegt auf der maximalen Durchbiegung, da eine übermäßige Durchbiegung zu strukturellen Problemen oder sogar zum Versagen führen kann. Diese maximale Durchbiegung tritt typischerweise dort auf, wo die Krümmung am größten ist, oft in der Nähe von Stützen oder direkt unter den Punkten, an denen Lasten wirken. Die Ermittlung dieser maximalen Durchbiegung ist von entscheidender Bedeutung, um sicherzustellen, dass die Balkenkonstruktionen den Sicherheitsstandards entsprechen und die funktionalen Anforderungen erfüllen. Durch die Durchführung gründlicher Analysen mithilfe von Freikörper- und Biegemomentdiagrammen können Strukturen effektiv entworfen werden, die sicher und für bestimmte Belastungsbedingungen und architektonische Anforderungen geeignet sind.

Verformung eines Balkens unter Querbelastung

Eine entscheidende Komponente bei der Bemessung von Trägern ist die Identifizierung und Messung der Durchbiegung. Ein umfassendes Verständnis der Durchbiegungen ist notwendig ...

deformation-of-a-beam-under-transverse-loading

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Deflection of Beams

Durchbiegung von Balken

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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 will curve. Beam curvature is related to the bending moment and is affected by the beam’s stiffness and cross-section. When the bending moment is higher, the curvature is greater, so the deflection profile becomes easier to evaluate.

The main goal is to find the maximum deflection. Too much deflection can cause structural problems or even failure. This largest deflection often appears where curvature is greatest, such as near supports or directly under applied loads. Using free-body and bending moment diagrams helps engineers design beams that stay safe and meet functional requirements under specific loading conditions.

Beam Deflection from Loads and Supports

deformation of a beam under transverse loading

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

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 free-body diagrams and bending moment diagrams help predict beam deflection, identify maximum deflection, and check structural safety under load.

Beam Deflection from Loads and Supports | 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 from Loads and Supports

Description

Transcript

Tags