Name: impact loading on a fixed rod
Uploaded: 2024-05-23T13:00:12.000Z
Duration: 1 min 19 s
Channel: JoVE

K12 Standards

Arizona Science Standards

High School

Physical Science

HS+Phy.P3

HS+Phy.P3U2.5

Impact Loading

Considérons une tige de section transversale uniforme fixée à une extrémité. Lorsqu’un autre objet en mouvement heurte l’extrémité libre de la tige, la tige se déforme et une contrainte se développe à l’intérieur de celle-ci. 
Lorsque la tige atteint la contrainte maximale, elle vibre autour de la position moyenne. Le stress qui s’est accumulé disparaît au moment du repos. Ces événements sont connus sous le nom de charge d’impact.
Ici, on suppose que le corps de frappe transfère toute son énergie à la tige, ce qui signifie qu’aucune dissipation de chaleur ne se produit et que le corps de frappe ne rebondit pas sur la tige.
Ainsi, l’énergie de déformation correspondant à la contrainte maximale est égale à l’énergie cinétique du corps de frappe.
Dans le régime élastique de la déformation, l’énergie de déformation peut être réécrite en termes de contrainte maximale et de module d’élasticité.
En réarrangeant les termes, on obtient une expression de la contrainte maximale en fonction de la vitesse du corps de frappe.
L’hypothèse utilisée ici aboutit à une conception prudente de la charge d’impact, car les hypothèses ne sont pas valides dans les systèmes réels.

chargement par impact

La charge d'impact se produit lorsqu'un objet en mouvement entre en collision avec une structure stationnaire, telle qu'une tige avec une section-transversale uniforme fixée à une extrémité. Dans ces conditions, la tige absorbe l’énergie cinétique de l’objet frappant, entraînant une déformation et le développement ultérieur de contraintes. Lorsque la tige revient à sa position d'origine et atteint sa contrainte maximale, l'énergie absorbée, initialement manifestée sous forme d'énergie cinétique, se transforme entièrement en énergie de déformation.

Dans les cas de déformation élastique, où le matériau reprend sa forme initiale sans dommage permanent, l'énergie de déformation accumulée au point de déformation maximale est équivalente à l'énergie cinétique de l'objet en mouvement. Cette équivalence suppose qu'aucune énergie n'est perdue à cause de la chaleur ou du rebond, une idéalisation que l'on ne trouve généralement pas dans les environnements pratiques. De cette relation, on peut déduire la contrainte maximale subie par la tige en fonction de la vitesse et de la masse de l'objet frappant et du module d'élasticité de la tige.

<img alt="Equation 1" src="/files/ftp_upload/15686/15686_Equation_1v2.svg" style="text-align: center; height: 40px;">

Les hypothèses formulées dans cette analyse conduisent à une approche conservatrice de la conception technique, garantissant que les structures peuvent résister à des forces inattendues. Cette approche aboutit souvent à une ingénierie excessive, intégrant des facteurs de sécurité pour tenir compte des pertes d'énergie et d'autres dynamiques non couvertes par le modèle théorique.

Chargement par impact

Considérons une tige de section transversale uniforme fixée à une extrémité. Lorsqu’un autre objet en mouvement heurte l’extrémité libre de la tige, la tige se déforme et la contrainte est ...

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impact loading on a fixed rod

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure.

As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion, and strain energy is the energy stored during elastic deformation.

For elastic deformation, the rod returns to its original shape without permanent damage. In that ideal case, the strain energy at maximum deformation is equal to the kinetic energy of the striking object. This relationship assumes that no energy is lost as heat or through rebound, which is rarely true in real situations.

From this energy balance, the maximum stress in the rod can be derived using the mass and velocity of the moving object and the rod’s modulus of elasticity. The result gives engineers a conservative design approach. It can also lead to over-engineering, with safety factors added to account for energy losses and other effects not included in the model.

Impact Loading on a Fixed Rod

strain energy

Load-Deformation and Stored Strain Energy

Strain energy is the energy stored in a material when it deforms under load. It is a key idea in materials science and structural engineering. A loaded rod fixed at one end shows this clearly. When an axial force pulls on the free end, the rod stretches and stress builds inside it. As the axial force increases, the rod elongates more. This creates a direct link between the applied force and the amount of deformation. Engineers can show this relationship with a load-deformation plot. The plot...

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Energy Stored in Loaded Rods

Energy stored in a rod under axial load helps show how a material will respond to stretching and breaking. When the rod elongates, it stores strain energy as potential energy inside the material. Strain energy density is the amount of that energy per unit volume. In the elastic region, stress and strain have a linear relationship that follows Hooke's Law. The strain energy density in this region is found from the area under the stress-strain curve up to the elastic limit. This recoverable...

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Strain Energy in Stressed Materials

Strain energy in stressed materials describes the energy stored when a material deforms under load. It is an important idea in materials science and engineering. For axial loading with a uniform stress distribution, the stress is the internal force divided by the cross-sectional area, and the strain follows from the stress through the modulus of elasticity. When the stress is uniform, the strain energy density can be found from the product of stress and strain. That density is then integrated...

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impact loading

Impact loading on a fixed rod happens when a moving object strikes a stationary rod that has a uniform cross-sectional area and is fixed at one end. The impact makes the rod deform and begin to develop stress. This setup shows how a sudden hit can store energy in a structure. As the rod moves back toward its original shape, the stress rises to a maximum. At that point, the energy from the moving object has changed from kinetic energy into strain energy. Kinetic energy is the energy of motion,...

impact loading on a cantilever beam

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Beam Deflection from Central Load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how a moving object striking a fixed rod creates stress, strain energy, and maximum stress in elastic deformation, with a conservative design model.

Impact Loading on a Fixed Rod | JoVE K12

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Impact Loading on a Fixed Rod

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