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

四面体スペーストラスとCのボールとソケットのジョイントで構成される三脚を考えてみましょう。
水平部材と垂直部材の高さと長さが与えられている場合、Dに引張力が加えられることを考慮して、部材BCとBAにはどのような力が作用していますか?
A、B、Cのすべての反力を含むフリーボディダイアグラムが描画されます。
ジョイントCにおけるモーメント平衡条件が適用されます。距離は、3 次元の位置ベクトルで表されます。
さらに単純化し、力の平衡条件を使用すると、i に沿ったベクトル成分は FAy を生成します。同様に、k 係数を 0 と等しくすると、FBx が得られます。
最後に、j係数を等しくして、FAxを求めます。 
ここで、ジョイントBの自由体線図は、力FBCとFBAを計算するために考慮されます。
FBD、FBC、およびFBAの力は、位置ベクトルを使用して表すことができます。
ジョイントBに力の平衡条件を適用し、i、j、k単位ベクトルの係数をゼロに等しくすると、BCとBAに沿った力が得られます。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。ニュージャージー州ホーボーケン:ピアソンプレンティスホール。301ページ。</li>
	<li>ビール、FP;ジョンストン、ER;マズレック、D.F;クロムウェル、PJおよびセルフ、BP(2019)。エンジニアのためのベクトル力学 \u2012 静力学と動力学。ニューヨーク:マグロウヒル。306ページ</li>
	<li>メリアム、JL、クレイジ、LG、およびボルトン、JN(2020)。エンジニアリングメカニクス \u2012 静力学。ニュージャージー州ホーボーケン:ジョン・ワイリー。193ページ</li>
</ol>

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スペーストラス：問題解決

スペーストラスは、平面トラスの三次元バージョンです。これらの構造は、末端で接続されたメンバーから成り、しばしばボールアンドソケットジョイントを利用して安定かつ多目的なフレームワークを作ります。その適応性と複雑な荷重に耐える能力から、スペーストラスはさまざまな建設プロジェクトで広く使用されています。

ボールアンドソケットジョイントを持つ四面体型のスペーストラスからなる三脚を考えてみましょう。水平および垂直メンバーの高さと長さが既知であるとします。

<img alt="Figure 1" src="/files/ftp_upload/14289/14289_PT_Fig_1v2.svg" style="text-align: center; height: 350px;" />

点Dで引張力が作用していると仮定すると、点A、点B、および点Cにおける反力を含む自由物体図を描いて、メンバーBCおよびBAに作用する力を決定することができます。

位置ベクトルで表される距離を考慮し、点Cにおけるモーメントの平衡条件が適用されます。

<img alt="Equation 1" src="/files/ftp_upload/14289/14289_PT_Equation_1v3.svg" style="text-align: center; height: 15px;" />

簡素化し、力の平衡条件を使用すると、ベクトル成分沿いに、FAyは6 N、FBxは-7.2 Nとなります。最後に、係数jを等しくすることで、FAxの値は6 Nとなります。

次に、点Bにおける自由物体図を考慮して、力FBCとFBAを計算しましょう。FBD、FBC、およびFBAは、位置ベクトルを使用して表現することができます。点Bにおける力の平衡条件が適用されます。

<img alt="Equation 2" src="/files/ftp_upload/14289/14289_PT_Equation_2v2.svg" style="text-align: center; height: 13px;" />

i、j、およびkの単位ベクトルの係数をゼロに等しくすることで、BCとBAに沿った力はゼロとなります。

スペーストラスは、平面トラスの 3 次元対応物です。これらの構造は、端で接続されたメンバーで構成されており、多くの場合、...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Structural Analysis

構造解析

KEY TERMS AND CONCEPTS

主要用語と概念

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

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mechanical engineering

3D Truss Force Analysis with Joint Equilibrium

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