Name: beam deflection under distributed load
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 16 s
Channel: JoVE

当梁承受分布载荷时，梁上任意点的剪力和弯矩可用微分形式表示。
通过进一步微分该表达式并假设抗弯刚度为常数，可得到一个三阶线性微分方程。
再次求导，得到一个四阶线性微分方程。该方程本质上决定了梁在承受分布载荷时所呈现的形状或弹性曲线。
将四阶线性微分方程乘以弯曲刚度常数并积分四次，可得到该曲线的表达式。
决定梁的挠曲线方程中的常数的是边界条件，例如支座处的转角或挠度，以及在悬臂梁的自由端剪力和弯矩是否为零，或在简支梁的两端是否仅有弯矩为零。
该技术可精确计算带有分布载荷的悬臂梁或简支梁的挠度曲线。然而，外伸梁的支座反力会导致剪力出现不连续性，因此需要采用不同的函数来完整定义整段曲线。

荷载分布的弹性曲线

<div>
 分布式荷载下梁的结构行为对于工程分析是至关重要的，工程分析的重点是预测梁在这种条件下所产生的弯曲和反应。不同类型的梁（例如悬臂梁、支撑梁或悬挑梁）在分布荷载条件下的状态各不相同。
 
 对于所有梁来说，如果要分析梁对分布荷载的反应，则需要首先了解梁的载荷与由此产生的剪力和弯矩之间的关系。最初，这种关系能够用微分方程来进行表示。
 
 <img alt="Equation 1" src="/files/ftp_upload/15668/15668_Equation_1.svg" style="text-align: center; height: 32px;">
 假设梁的柔度是恒定的，对该表达式进行进一步微分便能够得出更为复杂的微分方程，该微分方程描述了梁的挠度曲线或梁在荷载作用下的弯曲方式。
 
 <img alt="Equation 2" src="/files/ftp_upload/15668/15668_Equation_2.svg" style="text-align: center; height: 40px;">
 这个复杂的方程需要进行多次积分，以此来确定这条曲线的实际形状。每个积分步骤都会引入一个常数，该常数必须由梁的边界条件来进行定义，例如梁在其两端的支撑或连接方式。
 
 <img alt="Equation 3" src="/files/ftp_upload/15668/15668_Equation_3.svg" style="text-align: center; height: 32px;">
 边界条件根据梁的支撑方式而发生变化。例如，一端固定、另一端自由的悬臂梁，那么其两端的挠度和力将会具有不同的约束。相反，支撑梁的条件主要集中在支撑点上。其中具有挑战性的是对悬垂梁的分析，在这种情况下，梁的一部分会延伸到其支撑点之外。这些部分会承受独特的力和弯矩，需要进行不同的计算来准确描述梁在其整个长度的状态。这种详细的了解确保了结构的安全性和功能性。
 
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当梁承受分布载荷时，梁上任意点的剪力和弯矩可用微分形式表示。  
三阶……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Deflection of Beams

梁的挠度

KEY TERMS AND CONCEPTS

关键术语与概念

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 differential equation for the deflection curve, which is the shape the beam takes as it bends.

Finding the actual curve requires integrating the equation several times. Each integration adds a constant that must be determined from boundary conditions. Boundary conditions depend on how the beam is supported or connected at its ends.

Different beam types need different boundary conditions. A cantilever beam is fixed at one end and free at the other, so its deflection and force limits are set differently at each end. A supported beam is handled mainly at the support points.

Overhanging beams are more complex because part of the beam extends beyond the supports. Those outer sections experience unique forces and bending moments. They need separate calculations so the behavior along the full beam length can be described accurately.

Beam Deflection Under Distributed Load

deformation of a beam under transverse loading

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...

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 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 distributed loads create shear force, bending moment, and beam deflection. See how boundary conditions change for cantilever, supported, and overhanging beams.

Beam Deflection Under Distributed 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 Under Distributed Load

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