Name: load effects away from the ends
Uploaded: 2024-03-07T13:22:38.000Z
Duration: 1 min 18 s
Channel: JoVE

Stellen Sie sich ein Bauteil mit Platten an beiden Enden vor. Wenn Lasten mittig auf die Platten aufgebracht werden, bewegen sie sich aufeinander zu, ohne sich zu drehen, wodurch sich das Bauteil mit zunehmender Breite und Dicke verkürzt.
Eine gleichmäßige Verteilung der Dehnungen und Spannungen wird erreicht, indem gerade Stab- und Ebenenquerschnitte sowie eine gleichmäßige Verformung über alle Elemente hinweg beibehalten werden.
Werden die Lasten konzentriert und direkt auf das Bauteil aufgebracht, sind Elemente in der Nähe des Lasteinleitungspunktes höheren Spannungen ausgesetzt, während entfernte Bereiche davon unberührt bleiben.
Bei Elementen, die weit von den Enden entfernt sind, neigen die Verformungen dazu, sich auszugleichen, was zu einer gleichmäßigeren Verteilung der Dehnungen und Spannungen führt.
Ab einem Abstand, der der Breite des Stabes entspricht, wird die Spannungsverteilung unabhängig von der Lasteinleitungsart. Diese Aussage ist das Saint-Venant-Prinzip.
Bei der Anwendung des Saint-Venant-Prinzips ist zu beachten, dass die tatsächliche Belastung und die Belastung, die zur Bestimmung der Spannungen verwendet wird, statisch äquivalent sein müssen und nicht zur Berechnung von Spannungen in der Nähe der Lasteinleitungspunkte verwendet werden können.

saint-venantsches prinzip

Das Prinzip von Saint-Venant postuliert, dass die Spannungsverteilung innerhalb eines Bauteils nicht von der genauen Methode der Lastaufbringung abhängt, außer in der Nähe der Lastaufbringungspunkte. Betrachten Sie ein Szenario, in dem Lasten zentral auf zwei Platten aufgebracht werden. In diesem Fall bewegen sich die Platten aufeinander zu, ohne sich zu drehen. Diese Bewegung verursacht eine Verkürzung des Mitglieds in der Länge und eine Expansion in Breite und Dicke. Eine einheitliche Verformung über alle Elemente und das Beibehalten gerader Mitglieder und ebener Abschnitte erleichtern eine konsistente Verteilung von Dehnungen und Spannungen.

Wenn jedoch Lasten konzentriert aufgebracht werden, erleiden die Elemente in der Nähe der Aufbringungspunkte große Spannungen, während jene, die weiter entfernt positioniert sind, weitgehend unbeeinflusst bleiben. Dennoch tendieren Verformungen dazu, sich bei Elementen, die weit von den Enden entfernt sind, auszugleichen, was zu einer gleichmäßigeren Verteilung von Dehnung und Spannung führt. Interessanterweise wird jenseits einer Entfernung, die der Breite des Mitglieds entspricht, die Spannungsverteilung unabhängig von der Art der Lastaufbringung, ein Schlüsselaspekt des Saint-Venantschen Prinzips. Bei der Anwendung dieses Prinzips ist es wichtig zu bedenken, dass die tatsächliche Belastung und die zur Berechnung der Spannungen verwendete Belastung statisch äquivalent sein müssen. Darüber hinaus gilt dieses Prinzip nicht für die Berechnung von Spannungen in der Nähe der Lastaufbringungspunkte.

Saint-Venantsches Prinzip

Stellen Sie sich ein Bauteil mit Platten an beiden Enden vor. Wenn Lasten mittig auf die Platten aufgebracht werden, bewegen sie sich aufeinander zu, ohne sich zu drehen, wodurch die ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Stress and Strain - Axial Loading

Spannung und Dehnung – axiale Belastung

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

Saint-Venant's Principle

load effects away from the ends

Saint-Venant's principle describes how stress changes in a structural member when a load is applied. It shows that the exact way a load is placed matters most near the point of application. Farther away from that point, the stress pattern becomes much less dependent on how the load was applied.

A simple example uses a member with plates on both ends. When the loads are applied at the center of the plates, the plates move toward each other without rotating. The member shortens in length and expands in width and thickness. When the member stays straight and its plane sections stay plane, the strain and stress are distributed more evenly.

Concentrated loads create a different response. The material close to the load application points carries large stresses, while material farther away is affected much less. As the distance from the ends increases, the deformations begin to even out. Beyond a distance about equal to the member's width, the stress distribution is no longer tied to the exact loading method.

This principle is useful only when the actual load and the simplified load used in calculations are statically equivalent. It also should not be used to find stresses near the load application points. Near those ends, the local stress pattern depends strongly on how the load enters the member.

Load Effects Away from the Ends

normal strain under axial loading

Axial Stress and Strain in a Rod

Axial stress and strain in a rod describe how a material changes length when a force is applied along its axis. The force may stretch the rod or compress it. In mechanics of materials, this deformation is called normal strain. Normal strain is found by dividing the change in length by the original length. This gives a unitless ratio that shows how much the material deforms under load. A load-deformation diagram can show elongation, but it cannot predict how another rod will deform. When a...

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Stress-Strain Limits in Materials

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Yielding and Strain-Hardening in Metals

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fatigue

Fatigue Failure in Repeated Loading

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deformation of member under multiple loadings

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Calculating Rod Stretch from Mixed Loads

Calculating rod stretch from mixed loads starts with breaking the rod into parts. This step is needed when the rod is made of different materials or has changes in cross-section. Each part has its own internal force, cross-sectional area, length, and modulus of elasticity. These values are used to find the deformation of the full rod. For a member with a variable cross-section, the strain is not constant. Instead, it changes with position along the rod. The deformation of a small element is...

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Axial Rod Reactions with Redundant Support

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Thermal Expansion of a Free Rod

Thermal strain describes how a structure changes length when temperature changes. It is important in real-world design because structures do not always stay at one temperature. Temperature shifts can affect how they behave and how long they last. A simple example is a homogeneous rod with a uniform cross-section resting freely on a flat horizontal surface. When the rod is heated, it elongates. The amount of elongation depends on the temperature change and the rod’s original length. This...

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Thermal Stress in a Composite Rod

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poisson's ratio

Material Strain Under Axial Loading

Poisson's ratio describes how a material changes shape when it is pulled or compressed along one direction. It links the strain in the direction of the force to the strain that happens sideways. When a slender bar is loaded axially, it stretches along the load and contracts laterally. Poisson's ratio is defined as the negative ratio of lateral contraction to axial elongation. The negative sign is used because the two strains usually have opposite signs. This keeps the ratio positive and makes...

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Multiaxial Stress and Strain Relation

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Volume Change Under Hydrostatic Pressure

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shearing strain

Shear Modulus and Angular Change

Shearing strain is the angular change that happens in a cubic element when it is loaded by shearing stress. Under this stress, the cube can change into an oblique parallelepiped without creating normal strains. If the cube is acted on only by shearing stress, its shape can shift strongly into a rhomboid. The amount of shearing strain is considered positive when the angle between the axes becomes smaller. This change in angle shows the direct effect of shear on the material. The deformation can...

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Axial Load Effects on Shear Strain

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saint-venant's principle

Saint-Venant's principle describes how stress changes in a structural member when a load is applied. It shows that the exact way a load is placed matters most near the point of application. Farther away from that point, the stress pattern becomes much less dependent on how the load was applied. A simple example uses a member with plates on both ends. When the loads are applied at the center of the plates, the plates move toward each other without rotating. The member shortens in length and...

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Stress at Holes and Fillets

Stress can become much higher near holes and abrupt changes in shape within a structural member. These discontinuities create localized stress that may exceed the average stress in the part. For flat bars, the stress distribution can be measured experimentally with a photoelastic method. This approach is used for bars with a circular hole or with changing widths joined by fillets. The results depend on geometric ratios, such as the hole radius to the smaller width for a circular hole. For...

residual stresses

Residual Stresses After Welding and Cooling

Residual stresses are stresses that remain in a structure after the original load or heat source is removed. They often form when different parts of a material undergo uneven plastic deformation, which means permanent shape change. A rod stretched past its yield point is one example. It will not return to its original length, and some stress stays inside even after the load is gone. These stresses can also develop during welding. Heat changes the material during the weld and can cause plastic...

key terms and concepts

Learn how stress changes near and far from a load on a structural member, and why Saint-Venant's principle works best away from the ends.

Load Effects Away from the Ends | 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

Load Effects Away from the Ends

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