Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

ضع في اعتبارك شعاعا مدعوما ببساطة PQ بطول L ، يحمل حمولة نقطة F في المركز. ماذا سيكون الانحراف في مركز الحزمة؟
هنا ، تكون التفاعلات على طرفي الحزمة متساوية ، وكل منها نصف الحمل المركزي.
ضع في اعتبارك قطعة الكمبيوتر واختر أي نقطة على مسافة x من النهاية P. عند هذه النقطة، تكون اللحظة الناتجة عن التفاعل عند النقطة P هي الحمل عند النقطة P مضروبا في المسافة. التمايز الجزئي لمعادلة العزم بالنسبة للحمل في النهاية P يساوي نصف x. 
يمكن إجراء تحليل مماثل لجزء مراقبة الجودة من الحزمة.
باستخدام نظرية كاستيجليانو ، يتم تحديد الانحراف عند النقطة C من خلال المشتقات الجزئية لطاقة الإجهاد بسبب الحمل المطبق.
يمكن تبسيط هذه المعادلة من خلال النظر في جزأين من شعاع PC و QC.
يعطي إجراء التكامل على مدى نصف طول الحزمة تعبيرا عن الانحراف في مركز الحزمة.

نظرية كاستيجليانو: حل المسألة

<div style="direction: rtl;">
 يمكن تحليل انحراف عارضة مدعومة ببساطة والتي تحمل حمولة نقطية مركزية باستخدام مبادئ الميكانيكا الهيكلية، وخاصة من خلال تطبيق نظرية كاستيجليانو. تربط هذه النظرية الإزاحة عند نقطة تطبيق الحمل بالمشتقات الجزئية لطاقة الانفعال في الهيكل. إن العارضة المدعومة ببساطة مع حمل نقطة في مركزها لها قوى رد فعل متناظرة عند الدعامات، كل منها يحمل نصف الحمل. يتم اشتقاق عزم الانحناء عند أي نقطة على طول العارضة من هذه التفاعلات، ويتم حسابها على المسافة من أقرب دعم.
 
 تشير نظرية كاستيجليانو إلى أن الانحراف عند النقطة التي يتم فيها تطبيق الحمل يتم تحديده عن طريق التمييز بين طاقة إجهاد العارضة فيما يتعلق بالحمل. يتم حساب طاقة الانفعال بناءً علىعزم الانحناء على طول العارضة ، المتكاملة على طولها. بالنسبة لهذه العارضة ، يتم حساب طاقة الانفعال الناتجة عن الانحناء من مربع تعبير عزم الانحناء، المتكامل على طول نصف طول العارضة.
 
 <img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="text-align: center; height: 106px;">
 وبما أن العارضة متماثلة، تتم مضاعفة هذه القيمة لحساب العارضة بأكملها، ويتم حساب الانحراف في مركز العارضة. ويعتمد على حجم الحمل، ومكعب طول العارضة ، ويتناسب عكسيا مع منتج عزم القصور الذاتي ومعامل المرونة للمقطع العرضي للعارضة.
 
 <img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="text-align: center; height: 42px;">
</div>

ضع في اعتبارك شعاعا مدعوما ببساطة PQ بطول L ، يحمل حمولة نقطة F في المركز. ماذا سيكون الانحراف في مركز الحزمة؟
هنا ، ...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

Energy Methods

طرق الطاقة

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الرئيسية

Castigliano&#39;s Theorem: Problem Solving

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 displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

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...

strain-energy density

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...

elastic strain energy for normal stresses

elastic-strain-energy-for-normal-stresses

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...

elastic strain energy for shearing stresses

elastic-strain-energy-for-shearing-stresses

Shear Energy Stored in Twisted Shafts

Shear energy stored in twisted shafts depends on elastic strain energy in a material. This energy is stored when a solid is deformed within its elastic limit. It comes from internal work done against the cohesive forces inside the material. When a material carries shearing stress, it also develops shearing strain. The strain energy density is the energy stored per unit volume. Within the elastic limit, stress is proportional to strain through the modulus of rigidity, so the strain energy...

impact loading

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,...

impact loading on a cantilever beam

Dropped-Weight Bending of a Cantilever Beam

Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact. After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the...

castigliano&#39;s theorem

Using Strain Energy to Find Deflection

Castigliano's theorem uses strain energy to find deflection and rotation in elastic structures. Strain energy is the work done by external loads during elastic deformation. The theorem links that energy to the movement caused by the load. The key idea is simple. The partial derivative of the total strain energy with respect to a specific force gives the displacement at the point where that force acts. The same method also works for a moment, where the derivative gives the rotation at the point...

castigliano&#39;s theorem: problem solving

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 Castigliano's theorem is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central Load | 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 Central Load

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