Name: axial rod reactions with redundant support
Uploaded: 2024-03-07T13:22:37.000Z
Duration: 1 min 16 s
Channel: JoVE

2本の円筒形の棒、1本は鋼鉄、もう1本は真鍮で、点Bで接合され、点AとCで剛体支持体によって拘束されていると考えてみよう。
ポイントAとCでの反応を決定します。また、点Bでの変形を求めます。
ここで、ロッド構造は、平衡状態に必要な支持体よりも多くの支持体を持ち、平衡方程式に対して未知の反応が過剰に発生するため、静的に不確定であると考えられています。
したがって、ポイントCでの反応は冗長と見なされ、サポートから解放されます。これは追加の負荷として扱われます。
次に、重ね合わせ法を使用して、ロッド構造の各セクションの変形を決定し、組み合わせて全体の変形を決定します。
 全変形表現、ロッド構造の全変形がゼロに等しいこと、およびすべての荷重の合計がゼロに等しいことを考慮すると、未知の反力が決定されます。
点Bでの変形は、ロッド構造の点Bより前のセクションの変形を合計することによって計算されます。

 不静定問題の解決

Statically Indeterminate Problem Solving

不静定問題とは、静力だけでは内部力や内部反力を決定できない問題です。 鋼と真鍮で作られた 2 本の円筒形のロッドで構成される構造を考えてみましょう。 これらのロッドは点 B で結合され、点 A と C で硬い支持体によって拘束されています。ここで、点 A と C での反力と点 B でのたわみを求める必要があります。 この棒構造は、平衡を維持するのに必要以上の支持体を構造が有しており、利用可能な平衡方程式に対して未知の反応が余剰となるため、静的に不定として分類されます。

静的不確定性は、点 C での反応を冗長とみなしてそのサポートから解放することによって解決されます。 この冗長な反応は追加の負荷として扱われます。 次に、重ね合わせの原理を導入して、ロッド構造の各セクションの変形を測定します。 これら個々の変形を組み合わせることにより、構造物全体のトータルの変形表現が導き出されます。 式を考慮すると、ロッド構造の全変形はゼロに等しく、すべての荷重の合計はゼロに等しく、未知の反力が決定されます。 最後に、点 B におけるたわみは、点 B より前のロッド構造セクションの変形を合計することによって計算されます。

2本の円筒形の棒、1本は鋼鉄、もう1本は真鍮で、点Bで接合され、点AとCで剛体支持体によって拘束されていると考えてみよう。
決定する ...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Stress and Strain - Axial Loading

応力とひずみ - 軸方向荷重

KEY TERMS AND CONCEPTS

主要用語と概念

axial rod reactions with redundant support

Axial rod reactions with a redundant support are found by combining equilibrium with deformation. The structure here uses two cylindrical rods, one steel and one brass, joined at point B and held by rigid supports at points A and C. The goal is to determine the reactions at A and C and the deflection at B.

This setup is statically indeterminate because the supports create more unknown reactions than the equilibrium equations can solve. Statics alone is not enough to find every internal force or support reaction. To handle this, the reaction at point C is treated as a redundant reaction, meaning it is released from its support and used as an added load.

The superposition method is then used to find the deformation in each section of the rod structure. These separate deformations are combined to form the total deformation expression for the full rod system. When the total deformation is set equal to zero, and the sum of the loads is also zero, the unknown reaction forces can be determined.

After the reactions are found, the deflection at point B is calculated by adding the deformations in the sections of the rod structure before point B. This approach shows how compatibility of deformation helps solve a problem that equilibrium alone cannot complete.

Axial Rod Reactions with Redundant Support

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normal strain under axial loading

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Axial stress and strain in a rod describe how a material changes length when a force is applied along its axis. The force may stretch the rod or compress it. In mechanics of materials, this deformation is called normal strain. Normal strain is found by dividing the change in length by the original length. This gives a unitless ratio that shows how much the material deforms under load. A load-deformation diagram can show elongation, but it cannot predict how another rod will deform. When a...

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Ductile Material Stress-Strain Stages

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Stress-Strain Limits in Materials

Stress-strain limits in materials explain how solids respond when force is applied. Stress is the applied force per area, and strain is the change in shape or size that results. In the elastic range, strain stays directly proportional to stress. The constant that connects them is called the modulus of elasticity, or Young's modulus. This linear behavior continues only up to the proportional limit. After that point, the stress-strain relationship becomes nonlinear. For ductile materials, this...

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Yielding and Strain-Hardening in Metals

Yielding and strain-hardening describe how a material changes when stress goes beyond the elastic range. Under elastic behavior, the stress disappears when the load is removed, and the material returns to its original shape. Once stress exceeds the yield point, yielding begins and permanent deformation, or plastic set, starts. The switch from elastic to plastic behavior depends on the peak stress and on how long the load stays in place before removal. A specimen that is loaded, unloaded, and...

fatigue

Fatigue Failure in Repeated Loading

Fatigue failure happens when materials rupture after repeated or fluctuating loads. The stress can be far below the material’s static breaking strength. Even ductile materials can fail in a brittle way under fatigue. This behavior matters in machines and structural parts that face repetitive or changing loads. One example is an unbalanced pump impeller that creates vibrations. Another is a thin steel rod or wire bent back and forth in the same spot until it breaks. The number of loading...

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Calculating Rod Stretch from Mixed Loads

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statically indeterminate problem solving

Axial rod reactions with a redundant support are found by combining equilibrium with deformation. The structure here uses two cylindrical rods, one steel and one brass, joined at point B and held by rigid supports at points A and C. The goal is to determine the reactions at A and C and the deflection at B. This setup is statically indeterminate because the supports create more unknown reactions than the equilibrium equations can solve. Statics alone is not enough to find every internal force...

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Thermal Expansion of a Free Rod

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Thermal Stress in a Composite Rod

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Multiaxial Stress and Strain Relation

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Volume Change Under Hydrostatic Pressure

Bulk modulus describes how much a material resists a uniform squeeze. It links a change in pressure to the relative change in volume. For high school students, it is easiest to think of it as a measure of how hard it is to compress a material evenly from all sides. The idea becomes clearer when an isotropic material element is treated as a cube with unit volume. When normal stresses act on the cube, it deforms into a rectangular parallelepiped and its volume changes. The difference between the...

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Shear Modulus and Angular Change

Shearing strain is the angular change that happens in a cubic element when it is loaded by shearing stress. Under this stress, the cube can change into an oblique parallelepiped without creating normal strains. If the cube is acted on only by shearing stress, its shape can shift strongly into a rhomboid. The amount of shearing strain is considered positive when the angle between the axes becomes smaller. This change in angle shows the direct effect of shear on the material. The deformation can...

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Axial Load Effects on Shear Strain

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saint-venant's principle

Load Effects Away from the Ends

Saint-Venant's principle describes how stress changes in a structural member when a load is applied. It shows that the exact way a load is placed matters most near the point of application. Farther away from that point, the stress pattern becomes much less dependent on how the load was applied. A simple example uses a member with plates on both ends. When the loads are applied at the center of the plates, the plates move toward each other without rotating. The member shortens in length and...

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Stress at Holes and Fillets

Stress can become much higher near holes and abrupt changes in shape within a structural member. These discontinuities create localized stress that may exceed the average stress in the part. For flat bars, the stress distribution can be measured experimentally with a photoelastic method. This approach is used for bars with a circular hole or with changing widths joined by fillets. The results depend on geometric ratios, such as the hole radius to the smaller width for a circular hole. For...

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Residual Stresses After Welding and Cooling

Residual stresses are stresses that remain in a structure after the original load or heat source is removed. They often form when different parts of a material undergo uneven plastic deformation, which means permanent shape change. A rod stretched past its yield point is one example. It will not return to its original length, and some stress stays inside even after the load is gone. These stresses can also develop during welding. Heat changes the material during the weld and can cause plastic...

key terms and concepts

Learn how a steel-brass rod system with rigid supports uses redundant reactions, superposition, and zero total deformation to find support forces and deflection at point B.

Axial Rod Reactions with Redundant Support | JoVE K12

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mechanical engineering

Axial Rod Reactions with Redundant Support

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