Name: stress components on a cube at a point
Uploaded: 2024-03-07T13:22:33.000Z
Duration: 1 min 23 s
Channel: JoVE

複数の荷重を受ける物体の応力成分は、点Oを中心とする小さな立方体を考慮して解析されます。
キューブには、対応する軸に垂直な面に作用する法線応力成分とせん断応力成分があります。
隠れた面も、目に見える面のストレスと等しく、反対のこれらのストレスを受けます。したがって、均衡が維持されます。
立方体の面の応力は点Oの応力とはわずかに異なりますが、立方体の辺の長さがゼロに減少すると、この小さな誤差はなくなります。
立方体の自由体図を考慮すると、面に作用する法線力とせん断力は、対応する応力成分に各面の面積を掛けることによって決定されます。
次に、平衡方程式を適用することにより、せん断応力成分間の関係が導出されます。これは、特定の点での応力条件を定義するために必要な応力成分が 6 つだけであることを意味します。
また、せん断応力は1つの平面だけで発生するわけではありません。したがって、等しいせん断応力をそれに垂直な別の平面に加える必要があります。

応力の構成要素

複数の応力条件下での応力解析は複雑であり、垂直応力とせん断応力を包括的に把握する必要があります。 目に見えるかどうかに関係なく、6 つの面すべてに応力がかかる、点 O にある小さな立方体を考えてみましょう。 法線応力成分 σ_x、σ_y、σ_z は、x、y、z 軸に対して垂直に作用します。 せん断応力成分 τ_xy および τ_xz は、これらの軸に垂直な面に作用します。

興味深いことに、立方体の隠れた面にも、表示されている面と同じか反対の応力がかかり、平衡が保たれています。 立方体の辺の長さがゼロに近づくと、O での応力と立方体の面の応力の差は無視できるほどになります。

立方体に作用する力を調べると、せん断応力成分間の重要な関係がわかります: τ_xy は τ_yx に等しく、τ_yz は τ_zy

に等しく、τ_zx は τ_xz に等しい。これは、9 つではなく 6 つの応力成分だけが点での応力状態を定義することを示しています。

重要なのは、せん断は 1 つの平面のみで発生することはできないということです。 最初の平面に垂直な別の平面に等しいせん断応力を適用する必要があります。 最後に、解釈される応力状況は考慮される要素の方向に応じて変化する可能性があり、複数の応力条件下での応力解析の複雑さが浮き彫りになります。

複数の荷重を受ける物体の応力成分は、点Oを中心とする小さな立方体を考慮して解析されます。
立方体には法線と...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Concept of Stress

ストレスの概念

KEY TERMS AND CONCEPTS

主要用語と概念

Components of Stress

stress components on a cube at a point

Stress components on a small cube help describe how a body behaves under multiple loading. The cube is centered at point O and has stress acting on all six faces, whether the faces are visible or hidden. Normal stress components σx, σy, and σz act perpendicular to the x, y, and z axes. Shearing stress components also act on the cube faces.

The visible faces carry shear stresses such as τxy and τxz. The hidden faces carry equal and opposite stresses. This balance keeps the cube in equilibrium. As the side length of the cube becomes very small, the difference between the stress at point O and the stress on the cube faces becomes negligible.

Looking at the forces on the cube also shows key links between shear components. The shear stresses satisfy τxy = τyx, τyz = τzy, and τzx = τxz. Because of these relations, only six stress components are needed to define the stress condition at a point, not nine. Shear cannot act in only one plane. An equal shearing stress must also act on a perpendicular plane.

The stress picture can change when the cube is rotated. That means the interpreted stress state depends on the orientation of the element being studied. This makes stress analysis under multiple loading conditions more complex.

Stress Components on a Cube at a Point

normal stress

Normal Stress in Axial Loading

Normal stress acts perpendicular to a material’s cross-sectional area. It appears when a force is applied along the axis of an object, which is called axial loading. Bridge truss members are a practical example of this kind of loading. When a rod carries axial load, internal forces develop across the section. Because these forces act normal to the plane of the section, the stress is called normal stress. Stress here means the average stress over the whole cross-section, not the value at one...

shearing stress

Shear Forces in Bolts and Rivets

Shearing stress is caused by transverse forces acting on an object. These forces create internal forces in the plane where the external forces are applied. The total internal force in that section is called the shear. Average shearing stress is found by dividing the shear by the cross-sectional area. This value is only an average, because shear stress is not spread evenly across the section. It can be zero at the surface and much higher than the average in other areas. Bolts, pins, and rivets...

bearing stress

Bolt-to-Plate Contact Pressure

Bearing stress is the contact pressure between two separate bodies. A common example is a bolt pushed through a plate. The bolt applies a force to the plate, and the plate pushes back with an equal and opposite force. That force is not a single point load. It is made up of many smaller forces spread across the contact surface between the bolt and the plate. Because these microforces are complex, engineers use an average value called bearing stress. Bearing stress is found by dividing the...

applications of stress

Boom and Rod Stress Analysis

Boom and rod stress analysis shows how a pinned support structure carries a load. The boom and the rod are connected by a pin and held in place by brackets and pins. The structure is then separated from its supports so the forces can be studied clearly. A free-body diagram is drawn to identify every force on the boom, the rod, and the load. The reaction forces on both members are found using equilibrium equations. At point P, the reaction causes compression in the boom. At point R, the...

stress on an oblique plane

Axial Load Stress on Inclined Sections

Axial loading creates stress in a material along its central axis. This kind of load can cause compression or elongation, and it produces normal stress. Normal stress acts perpendicular to the material area and can be either tensile or compressive. When the section is not perpendicular to the load, it becomes an oblique plane. On this inclined plane, the stress state includes both normal stress and shearing stress. Shearing stress acts parallel to the area and changes shape without changing...

stress: general loading conditions

Stress Components on a Loaded Body

Stress components on a loaded body can be found by cutting the body through a point with a plane parallel to the xy plane. That section carries the original loads, along with normal forces and shearing forces. The result is a simple way to describe how a structure responds under combined loading. The shearing force in the section can point in different directions within the plane. It is split into two parts that run parallel to the x and y axes. Each part is divided by the area, and as that...

components of stress

Stress components on a small cube help describe how a body behaves under multiple loading. The cube is centered at point O and has stress acting on all six faces, whether the faces are visible or hidden. Normal stress components σx, σy, and σz act perpendicular to the x, y, and z axes. Shearing stress components also act on the cube faces. The visible faces carry shear stresses such as τxy and τxz. The hidden faces carry equal and opposite stresses. This balance keeps the cube in equilibrium.

design consideration

Safety Factors in Structural Design

Safety factors in structural design help engineers choose a material and loading limit that can be used safely. The process begins with a material’s ultimate strength. This value comes from tests that increase force until the material reaches its breaking point or limit. The ultimate load is then divided by the original cross-sectional area to find the ultimate normal stress, or strength. Ultimate shearing stress is another key value used in design. These limits show how much stress a material...

key terms and concepts

Learn how normal and shearing stresses act on a small cube at a point, why only six stress components are needed, and how orientation changes stress analysis.

Stress Components on a Cube at a Point | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Stress Components on a Cube at a Point

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