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

考虑一根长度为 L 的简支梁 PQ，在其中心作用有集中载荷 F。梁中心处的挠度是多少？
此处，梁两端的支座反力相等，且各自均为中心载荷的一半。
考虑线段 PC，并在距 P 端距离为 x 处任选一点。在该点处，由 P 点反力引起的弯矩等于 P 点处的荷载乘以距离。对弯矩方程关于 P 端荷载进行偏微分，结果为 x 的一半。
可以对梁的 QC 段进行类似的分析。
利用卡斯蒂利亚诺定理，通过应变能对所施加载荷的偏导数来确定点 C 处的挠度。
通过考虑梁的两个部分 PC 和 QC，该方程可以简化。
对梁长度的一半进行积分，可得到梁中心处挠度的表达式。

卡斯提利亚诺定理：解决问题

Castigliano&#39;s Theorem: Problem Solving

<div>
 通过利用结构力学原理（特别是应用卡斯提利亚诺定理），则能够用来分析位于简支梁承载中心处点荷载的挠度。该定理能够将荷载施加点处的位移与结构中应变能的偏导数联系起来。简支梁的中心有一个点载荷，并在支撑处具有对称的反作用力，每个反作用力都会承受一半的荷载。位于梁上任意一点处的弯矩均来自于这些反作用力，并根据与最近支撑之间的距离来进行计算的。
 
 卡斯提利亚诺定理表明了，通过对梁上相对于荷载的应变能进行微分，便能够确定施加荷载处的挠度。应变能是根据梁上的弯矩来计算的，并在其长度上进行积分。对于梁来说，由弯曲所引起的应变能是根据弯矩表达式的平方计算得出的，并将其沿梁上长度的一半来进行积分。
 
 <img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="text-align: center; height: 106px;">
 由于梁是对称的，因此将得出的值进行加倍来对整个梁进行分析，并计算位于梁中心的挠度。它取决于荷载的大小、梁长度的立方，并且与梁横截面上的惯性矩和弹性模量的乘积成反比。
 
 <img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="text-align: center; height: 42px;">
</div>

考虑一根长度为 L 的简支梁 PQ，在其中心承受一个集中载荷 F。梁中心处的挠度是多少？  
此处，……

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JoVE Core

Mechanical Engineering

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Unit 1

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Energy Methods

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KEY TERMS AND CONCEPTS

关键术语与概念

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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