Name: constraints in structural equilibrium
Uploaded: 2023-09-22T11:49:23.000Z
Duration: 1 min 26 s
Channel: JoVE

系统的平衡不仅由其方程定义，还通过约束条件来确定。约束是指对系统运动的限制。
如果一个系统由确保平衡所需的最少数量的约束来维持，则称该系统为静定的。
对于此类系统，可利用平衡方程估算未知的反力约束。
如果向系统中添加了额外的冗余支撑以维持平衡，则该系统将变为静不定。这意味着作用在系统上的约束数量超过了可用于求解的平衡方程数量。
另一方面，考虑一个杆件 AB，在点 C 和 D 处连接两个铰支座，使得约束反力的作用线在点 C 处相交。
施加的载荷将导致梁处于部分约束状态，因为在此加载条件下无法满足平衡方程。
类似地，当所有约束反力相互平行时，会导致约束不当。

<ol>
	<li> 希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。第 251-254 页。 </li>
</ol>

约束和静态确定性

Constraints and Statical Determinacy

在结构工程中，系统的平衡不仅由其平衡方程来进行决定，而且还借助约束的条件来进行确定。约束是指对系统运动的限制。合适的约束组合可以使维持系统机械平衡所需的约束总数最小化。当这种情况发生时，系统被称为静态确定的。对于这样的系统，可以使用平衡方程来估计未知的反力支撑。 静态确定的系统最容易进行分析和设计的，因为可以使用特定数量的平衡方程来确定系统上的所有作用力。通常，平衡的方程涉及到系统上所有作用力的总和与所有力矩的总和等于零。 然而，如果向系统添加额外的冗余支撑以维持平衡，则系统将会变为静态不确定的。这意味着在系统上起作用的约束数量超过了可用于解决其平衡方程的数量。在静态不确定的系统中，额外的支撑限制了系统的运动，但同时也创建了额外的方程。这些额外的方程通常是通过系统的变形来获得的，这是在材料的力学下进行处理的。超过静止确定的系统所面临的挑战是找到添加到系统中正确数量的冗余支撑，以便在不发生过度约束系统的情况下保持其平衡。 此外，不适当的约束是指所有的反力要么在一个点处是相交的，要么是通过一个公共的轴线，要么是使其变得平行。这种情况可能会导致结构变得不稳定，并应该避免其在工程的实践中得到应用。当反作用力在一个相交的点处或沿着一个公共的轴线进行运动时，它们会使物体失去平衡并导致其倒塌。同样的，相互平行的力可能无法维持其平衡状态，便会导致其结构发生破坏。避免不适当的约束是为了防止有不稳定的状态和潜在的危险。

在结构工程中，一个系统的平衡不仅由其平衡方程决定，还需借助约束条件来确定。

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Equilibrium of a Rigid Body

刚体的平衡

KEY TERMS AND CONCEPTS

关键术语与概念

constraints in structural equilibrium

Constraints in structural engineering control how a body can move and help keep it in mechanical equilibrium. A constraint is a restriction on motion. When the right number of constraints is used, the system can stay balanced with the fewest supports needed.

A system with the proper number of supports is called statically determinate. In that case, the unknown reaction forces can be found from equations of equilibrium. These systems are the easiest to analyze and design because the forces can be solved from a fixed set of equilibrium equations.

The usual equilibrium equations include the sum of all forces equal to zero and the sum of all moments equal to zero. If extra, redundant supports are added, the system becomes statically indeterminate. That means there are more constraints than equilibrium equations, so the extra reactions cannot be found from equilibrium alone.

For statically indeterminate systems, the extra equations usually come from deformation. This topic is handled in mechanics of materials. The main challenge is choosing the correct number of redundant supports so the structure stays in equilibrium without becoming overconstrained.

Improper constraints can also create instability. This happens when all reactive forces meet at a single point, pass through a common axis, or become parallel. In those cases, the structure may topple or fail to stay balanced, so engineers avoid these conditions in practice.

Constraints in Structural Equilibrium

conditions of equilibrium

Rigid Body Force Balance in Equilibrium

Rigid body equilibrium depends on a balance of external forces and moments. A rigid body is in equilibrium when it has no translational motion and no rotational motion. This happens when the net force and the net couple acting on the body are both zero. Internal forces are not used when checking equilibrium. These forces act in equal and opposite pairs inside the body, so they cancel each other. Because of that, they do not affect the overall motion or change in motion of the rigid body. The...

support reactions

Types of Structural Supports

Types of structural supports control how a coplanar force system moves and reacts under load. A coplanar force system is a set of forces that all lie in the same plane. The supports create reactions at the point of contact. These reactions help keep a structure stable under external forces. Supports work by stopping translation with an equal and opposite force. They can also stop rotation by applying a couple moment in the opposite direction. In structural design, these reactions are important...

internal forces and center of gravity

Forces Inside a Body and Balance

Internal forces and the center of gravity help explain how objects stay in balance. These mechanics ideas matter for structures, machines, and vehicles that must remain stable under different loads and conditions. Internal forces are created inside a body when its particles interact. They can appear as tension, compression, or shear forces. These forces act in collinear pairs with equal magnitude and opposite direction. Between connected parts, they cancel each other and help keep the body in...

alternative sets of equilibrium equations

alternative-sets-of-equilibrium-equations

Equilibrium Conditions for a Supported Plate

Equilibrium conditions for a supported plate are used to describe when forces and moments balance. In this state, the plate has no net movement and no rotation. Engineers use these equations to study how structures behave under external loads. In many cases, the standard equilibrium equations are enough. But some systems need an alternative set of equations to match the geometry of the support points. A rectangular plate supported at points A, B, and C is one example. For this plate, the...

two force member

Two-Force Body Equilibrium

A two-force body is a rigid body acted on by only two external forces. In equilibrium, those two forces must have the same magnitude, the same line of action, and opposite directions. This balance is what keeps the body at rest. For a plate with forces F1 and F2 applied at points A and B, equilibrium means the sum of moments about any axis is zero. If moments are summed about point A, the moment from F2 must be zero. That means the line of action of F2 must pass through point A. The same idea...

three force member

Solving a Three-Force Member Example

A three-force member is a rigid body acted on by three forces at three points. The forces must have concurrent lines of action. If the forces are parallel, their lines of action stay parallel instead. A dumpster provides the example used here. It is connected to a pin support at point A and to a pin on a hydraulic cylinder at point B. The hydraulic cylinder acts as a two-force member in equilibrium. The dumpster weight acts through the center of gravity, and the support reaction acts at point...

equations of equilibrium in three dimensions

equations-of-equilibrium-in-three-dimensions

Checking Force Balance in 3D Structures

Checking force balance in 3D structures is a key step when a system is at rest. The vector and scalar equations of equilibrium are used to test whether a body is stable and will not collapse or fall apart. These equations give a clear way to study the forces acting on a body. The vector equations of equilibrium require the vector sum of all external forces on a body to equal zero. They also require the vector sum of the couple moments and the moments of all forces about a point to equal zero.

support reactions in three dimensions

3D Support Types for Stability and Motion

Support reactions in three dimensions keep structures stable and in equilibrium. They stop unwanted translation and rotation, so a design can handle external forces safely and work as intended. These reactions are important in many structures and systems. A ball and socket joint is one support that allows free rotation about any axis. It does not provide a moment that resists turning. At the same time, it restricts motion in space with three reaction forces along the three dimensions. A...

constraints and statical determinacy

Constraints in structural engineering control how a body can move and help keep it in mechanical equilibrium. A constraint is a restriction on motion. When the right number of constraints is used, the system can stay balanced with the fewest supports needed. A system with the proper number of supports is called statically determinate. In that case, the unknown reaction forces can be found from equations of equilibrium. These systems are the easiest to analyze and design because the forces can...

key terms and concepts

Learn how constraints affect structural equilibrium, including statically determinate and indeterminate systems, redundant supports, and unstable constraint patterns.

Constraints in Structural Equilibrium | 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

Constraints in Structural Equilibrium

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