Name: 3d truss force analysis with joint equilibrium
Uploaded: 2023-09-22T11:49:24.000Z
Duration: 1 min 29 s
Channel: JoVE

Рассмотрим штатив, состоящий из четырехгранной пространственной фермы с шаровым соединением в точке C.
Если даны высота и длина горизонтальных и вертикальных стержней, то какая сила действует на стержни BC и BA с учетом того, что растягивающая сила приложена в точке D?
Строится диаграмма свободного тела, включающая все силы реакции в точках А, В и С.
Применяется условие равновесия момента в соединении С. Расстояния выражаются в терминах векторов положения в трех измерениях.
Упрощая дальше и используя условия равновесия сил, векторные компоненты вдоль i дают FAy. Аналогично, приравнивание коэффициентов k к нулю, дает FBx. 
Наконец, уравнив коэффициенты j, получим FAx. 
Теперь для расчета сил FBC и FBA рассматривается диаграмма свободного тела в соединении B.
Силы FBD, FBC и FBA могут быть выражены с помощью векторов положения.
Применение условия равновесия сил в соединении B и приравнивание коэффициентов единичных векторов i, j и k к нулю дает силы вдоль BC и BA.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 301.</li>
	<li>Бир, Ф..; Джонстон, Э.Р.; Мазурек, Д.Ф; Кромвель,.Д. и Селф, Б.. (2019). Векторная механика для инженеров \u2012 Статика и динамика. Нью-Йорк: Макгроу-Хилл. С. 306</li>
	<li>Мериам, Д.Л., Крейдж, Л.Г., и Болтон, Д.Н. (2020). Инженерная механика \u2012 Статика. Хобокен, Нью-Джерси: Джон Уайли. С. 193</li>
</ol>

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Пространственная трасса: Решение задачи

Пространственная трасса - это трехмерный аналог плоской трассы. Эти структуры состоят из элементов, соединенных в их концах, часто используя шарниры для создания стабильной и универсальной рамы. Благодаря своей адаптивности и способности выдерживать сложные нагрузки, пространственные трассы широко используются в различных строительных проектах.

Рассмотрим треножник, состоящий из тетраэдрической пространственной трассы с шарниром в точке C. Предположим, что известны высота и длины горизонтальных и вертикальных элементов.

<img alt="Figure 1" src="/files/ftp_upload/14289/14289_PT_Fig_1v2.svg" style="text-align: center; height: 350px;" />

Предполагая, что растягивающая сила приложена в точке D, можно нарисовать схему свободного тела, включающую все реакционные силы в точках A, B и C, чтобы определить силу, действующую на элементы BC и BA.

Состояние равновесия момента в точке C применяется, учитывая расстояния, выраженные в положительных векторах в трех измерениях.

<img alt="Equation 1" src="/files/ftp_upload/14289/14289_PT_Equation_1v3.svg" style="text-align: center; height: 15px;" />

Далее упрощая и используя условия равновесия сил, векторные компоненты вдоль i дают значение FAy равное 6 Н, а вдоль k дает значение FBx равное -7.2 Н. Наконец, приравнивая коэффициенты j, получаем значение F_Ax равное 6 Н.

Теперь рассмотрим схему свободного тела в точке B, чтобы вычислить силы FBC и FBA. Силы FBD, FBC и FBA можно выразить с помощью позиционных векторов. Применяется условие равновесия сил в точке B.

<img alt="Equation 2" src="/files/ftp_upload/14289/14289_PT_Equation_2v2.svg" style="text-align: center; height: 13px;" />

Приравнивая коэффициенты единичных векторов i, j и k к нулю получаем нулевые силы вдоль BC и BA.

Пространственная ферма является трехмерным аналогом плоской фермы. Эти структуры состоят из стержней, соединенных на концах, часто с использованием ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Structural Analysis

Структурный анализ

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Space Trusses: Problem Solving

3d truss force analysis with joint equilibrium

A space truss is a three-dimensional truss structure made of members connected at their ends. It often uses ball-and-socket joints to form a stable and versatile framework. Because it can handle complex loads, this type of structure is used in many construction projects.

The example begins with a tripod made from a tetrahedral space truss. A ball-and-socket joint is located at C. The height and the lengths of the horizontal and vertical members are known. A tensile force is applied at joint D. A free-body diagram is then drawn to include the reaction forces at joints A, B, and C.

To find the forces in members BC and BA, moment equilibrium is applied at joint C. The distances are written as position vectors in three dimensions. After the equations are simplified, the i component gives F Ay as 6 N, and the k component gives F Bx as -7.2 N. The j component then gives F Ax as 6 N.

The analysis continues with a free-body diagram at joint B. The forces F BD, F BC, and F BA are written using position vectors. Force equilibrium is applied at joint B. When the i, j, and k unit-vector coefficients are set to zero, the forces in BC and BA are both found to be zero.

3D Truss Force Analysis with Joint Equilibrium

3d-truss-force-analysis-with-joint-equilibrium

introduction to structures

Structure Types and Force Paths

Structures are systems of interconnected members that support or transfer forces and handle the loads placed on them. Engineers can find the internal forces in a structure by breaking it apart and drawing free-body diagrams for individual members or groups of members. This helps show how structural elements behave and how the whole system stays stable under load. There are three main categories of structures: trusses, frames, and machines. Trusses are stationary structures that stay rigid when...

simple trusses

Simple Trusses and Triangular Frames

Simple trusses are planar trusses built from slender members joined at points in one flat plane. They are designed to carry external loads while using less material and keeping weight low. A truss is a structural framework, and the simple truss is one of its most basic forms. The most basic simple truss uses three members arranged in a triangle. That triangular shape is naturally stable and rigid because of its geometry. It gives students a clear starting point for understanding how larger...

method of joints

Truss Forces Using the Method of Joints

The method of joints is used to find the forces in structural trusses. It relies on equilibrium, which means the forces at a joint must balance. In this model, the truss members are connected by frictionless pins. For a plane truss, the members lie in the same plane. That means each joint has a coplanar and concurrent force system. The analysis begins by identifying and labeling all the joints in the truss. Next, each joint is isolated and shown in a free-body diagram. This diagram includes...

method of joints: problem solving i

Truss Joint Force Analysis by Equilibrium

The method of joints uses equilibrium to find forces in structural trusses. It treats the truss members as being joined by frictionless pins. For each joint, the forces acting there must balance. That balance makes it possible to solve for the member forces one joint at a time. Here, a truss has 20 N acting at joint C and 10 N acting at joint D. The method of joints is used to find FCB, FDC, FDB, and FAD. The analysis starts with a free-body diagram of joint C. Free-body diagram means drawing...

method of joints: problem solving ii

Truss Force Analysis at Joints A to E

Truss force analysis at joints A to E uses the method of joints to find the force in each member. The truss has frictionless joints, a wall support, and a roller support. A 150 N force is applied at joint A, and the member forces are solved from joint to joint. First, the angle between members AC and AB is found using the inverse tangent of the ratio of BC to AB. This gives an angle of 67.38°. A free-body diagram of joint A is then drawn, and each unknown force is split into vertical and...

zero-force member

Finding Zero-Force Members in Trusses

Trusses are frameworks made of slender members that connect at their ends by joints. Engineers and architects use them to stabilize and strengthen structures such as bridges, roofs, and towers. Truss members carry loads through tension and compression so the structure can handle external forces. A key part of truss analysis is spotting zero-force members. A zero-force member is a truss member that carries no stress under loading conditions. During analysis, some members may show zero force,...

method of sections

Truss Member Force Analysis by Sections

Truss member force analysis by sections finds the force in selected members of a truss. In this problem, forces F1 and F2 act at joints B and D. The goal is to determine the forces in members EF, DC, and DF. The method uses the idea that a truss in equilibrium also has each of its sections in equilibrium. A free-body diagram of the whole truss is drawn first. The moment equation about joint A is then used to estimate the support reaction at point E. Next, a cut is made along a sectional plane...

method of sections: problem solving i

Truss Member Force Calculation by Sections

Truss member forces can be found with the method of sections in a symmetrical roof truss. The structure here includes vertical, diagonal, and horizontal members. Each horizontal member is 4 m long. The vertical members FB and HD are 4 m long, and member GC is 6 m long. The roof truss carries loads at several joints. Joints F, G, and H each have a 2 kN load. Joints A and E each have a 1 kN load. To start the analysis, moment equilibrium is used at point A. The force and distance values are...

method of sections: problem solving ii

Truss member force analysis by sections finds the forces in selected members of a wall-mounted truss. The structure here includes diagonal, vertical, and horizontal members. The known loads and member lengths are used to solve for the forces in CB, GB, and GH. A cut is made through members CB, GB, and GH. The right side of the truss is then isolated as a free-body diagram. Next, the moment equilibrium equation is taken about point G. This step gives the force in member CB as 4 kN, and the...

space trusses

Space Truss Forces and Stability

A space truss is a three-dimensional truss made from members joined at their ends. It is the 3D version of a planar truss. These frameworks often use ball-and-socket joints, which help the structure stay stable and flexible under load. Space trusses are used in many construction projects because they can adapt to complex forces. Their basic building block is the tetrahedron. A tetrahedron has six members and four triangular faces. Engineers can build larger space trusses by adding one joint...

space trusses: problem solving

A space truss is a three-dimensional truss structure made of members connected at their ends. It often uses ball-and-socket joints to form a stable and versatile framework. Because it can handle complex loads, this type of structure is used in many construction projects. The example begins with a tripod made from a tetrahedral space truss. A ball-and-socket joint is located at C. The height and the lengths of the horizontal and vertical members are known. A tensile force is applied at joint D.

frames

Frame Analysis in Structural Systems

Frames are important parts of many mechanical and structural systems used every day. They are valued for stability and for their ability to bear heavy loads. A frame is built from two-force members and multi-force members, and the parts are connected with pin joints. Trusses are different because they are made entirely of two-force members. Frames appear in many practical settings. They can support beams and columns in buildings. They are also used in automobile chassis construction. In the...

frames: problem solving i

Jib Crane Reaction Forces in Frames

Jib crane reaction forces in frames are found by analyzing the pin joints, the cable tension, and the load carried by the pulley. The structure in this problem has an external load suspended from the pulley, and the member sizes are shown in the figure. Because the pulleys are frictionless, the force balance can be worked out step by step. The frame has two main parts. Member BD is a two-force member, which means it is a straight part loaded only at its two ends, D and B. Those end forces are...

frames: problem solving ii

Hydraulic Hoist Force Analysis

Hydraulic hoist force analysis shows how to find the forces in key members of a simple frame. The load is 1 kN, and the structure is represented with known member dimensions. In this setup, BF and BD act as two-force members. EFG and EDC are multi-force members, so each one is analyzed with a free-body diagram. The first step is to study member EFG. The inclined force in BF is split into vertical and horizontal parts. Then the moment equilibrium condition is applied at joint E. Using this...

machines

Cutting Pliers as a Force-Multiplying Machine

Cutting pliers are a machine that helps transmit force through movable, pin-connected members. In this example, the handles and cutting edges work together to cut wire by turning an input force into a larger output force at the jaws. When equal and opposite forces are applied to the handles, the cutting edges move toward each other. They then apply equal and opposite reaction forces to the wire. These reaction forces are greater than the forces applied to the handles, which is why the wire can...

machines: problem solving i

Toggle Clamp Force Analysis

A toggle clamp can be analyzed to find its clamping force from a known handle force. In this example, a 200 N force is applied at the handle. The clamp is used in holding and clamping tasks in woodworking, metalworking, and assembly operations. The toggle clamp is a machine structure made of movable, pin-connected multi-force members. These members form a stabilized system that transmits force. Member CD is a two-force member, which means the forces at its ends are equal in magnitude and...

machines: problem solving ii

Lifting Tong Force Analysis

Lifting tong force analysis shows how forces move through a pin-connected machine. The structure uses movable sections DAF and CBG with member AB linking them together. Because these parts work as connected members, the load can be studied with equilibrium methods. The example starts with a 100 kg load at point E. The load weight is found by multiplying mass by gravity, which gives 981 N. The cable tension acts upward and balances that weight. Since the structure is symmetrical at E, the...

key terms and concepts

Analyze a space truss using free-body diagrams, position vectors, and joint equilibrium to find reaction forces and determine member forces in a 3D structure.

3D Truss Force Analysis with Joint 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

3D Truss Force Analysis with Joint Equilibrium

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