Name: axial load effects on shear strain
Uploaded: 2024-03-07T13:22:38.000Z
Duration: 1 min 15 s
Channel: JoVE

受轴向载荷作用的细长杆件会在轴向和横向方向发生变形。这种变形会影响其内部的立方单元。
根据其取向，立方体将转变为长方体或菱形，从而产生剪切应变。
该单元上的轴向载荷导致剪切应变和正应变的组合。
施加轴向载荷会在与载荷轴线成45度角取向的单元上引发正应力和剪切应力。
立方单元与对角平面相交时，形成一个棱柱形单元。该棱柱形单元的内部角度和边长会根据载荷产生的应变成比例地发生变化。
通过应用两角差的正切公式，可确定最大剪应变与轴向应变之间的关系。
根据胡克定律，可得到常数之间的关系。其中一个常数可由另外两个常数确定。

泊松比、弹性模量和刚性模量之间的关系

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

当轴向载荷施加到细长的杆上时，便会在轴向和横向方向上发生变形。这种变形通常会影响到杆内的立方体元素，通过将其转变为长方体或菱形，能够得到其具体的方向。该转变过程会产生剪切应变，轴向载荷则会引起剪切应变和法向应变。其中施加的轴向载荷会在与载荷轴成45°角的元件上产生相等的法向应力和剪切应力。值得注意的是，剪切应变在这个特定方向能够达到最大水平。

当对角平面与立方体单元进行相交时，就会形成棱柱形元件。该棱柱形元件会根据负载所引起的应变并按比例来改变其内角和侧面。该变形的过程阐明了应力对元件结构完整性的影响。最大的剪切应变和轴向应变之间的关系可以通过应用两个角度差的正切表达式来进行得出。

<img alt="Equation 1" src="/files/ftp_upload/15603/15603_Equation_1v2.svg" style="text-align: center; height: 24px;">
这种数学关系提供了对于轴向载荷条件下这两种应变类型之间所产生的相互作用提供了深入的了解。胡克定律与常数之间存在关联，这是用来研究材料在负载的条件下能够发生变法的关键原则。通过这种相关性，可以从其他两个常数中来确定一个常数，从而丰富了我们对材料与应力和应变响应的理解。

细长杆件在承受轴向载荷时，会在轴向和横向方向发生变形。这种变形会影响杆件内部的立方微元……

relation-between-poisson-s-ratio-modulus-elasticity-modulus

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Stress and Strain - Axial Loading

应力与应变 - 轴向加载

KEY TERMS AND CONCEPTS

关键术语与概念

axial load effects on shear strain

Axial load effects on shear strain are shown by how a slender bar deforms under tension or compression. When the bar is loaded along its axis, it changes in both the axial direction and the transverse direction. A small cubic element inside the bar can turn into a rectangular parallelepiped or a rhombus, depending on how it is oriented.

This change in shape creates shearing strain, which is the distortion caused when angles in a material change. Axial loading produces both normal strain and shearing strain. For an element placed at 45° to the load axis, the normal stress and shearing stress are equal. The shearing strain is also at its maximum at this orientation.

If a diagonal plane cuts through the cubic element, a prismatic element is formed. This element changes its sides and internal angles in response to the load-induced strains. The result shows how stress affects the structural form of the element under axial loading.

The maximum shearing strain can be related to axial strain by using the tangent of the difference between two angles. This relationship helps explain how the two strain types interact. Hooke's law links the material constants, so one constant can be found if the other two are known. That connection is key to understanding how a material responds to stress and strain.

Axial Load Effects on Shear Strain

normal strain under axial loading

Axial Stress and Strain in a Rod

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...

stress-strain diagram

Tensile Test Data and Material Behavior

A stress-strain diagram shows how a material responds during a tensile test. It records the mechanical behavior of a cylindrical specimen as it is pulled by an increasing centric load. The specimen is carefully prepared before testing. Two gauge marks are inscribed on the central part of the specimen. The distance between these marks is the gauge length. As the load increases, the gauge length also increases. This change in length is called elongation. The elongation is measured for each load...

stress-strain diagram - ductile materials

Ductile Material Stress-Strain Stages

Ductile materials such as structural steel and aluminium show a multi-stage stress-strain response. When a specimen is loaded, its length first increases linearly at a slow rate. This produces a steep straight line on the stress-strain diagram. That initial part shows elastic deformation, which means the material returns to its original shape after unloading. When the stress reaches a critical value, plastic deformation begins. In this stage, the material changes shape a lot while the applied...

stress-strain diagram - brittle materials

Brittle Material Failure Under Tension

Brittle materials such as glass, cast iron, and stone fail with very little change in elongation. Their breaking strength and ultimate strength are nearly the same, and the strain at rupture is low. In these materials, failure is usually driven by normal stress, so the break surface is typically perpendicular to the applied load. Brittle materials also show little or no necking. Necking is the local narrowing of a material’s cross section under stress. Another important feature is that many...

true stress and true strain

Stress-Strain Curves Beyond Yield

Stress-strain curves beyond yield show how ductile materials behave after they start to deform permanently. Engineering stress is found by dividing the load by the original, undeformed cross-sectional area. It is a useful approximation, but it can lose accuracy when strain becomes large. True stress gives a more exact picture of the same specimen under load. It is calculated by dividing the applied load by the instantaneous cross-sectional area during deformation. As the specimen stretches,...

hooke's law

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...

plastic behavior

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...

deformation of member under multiple loadings

deformation-of-member-under-multiple-loadings

Calculating Rod Stretch from Mixed Loads

Calculating rod stretch from mixed loads starts with breaking the rod into parts. This step is needed when the rod is made of different materials or has changes in cross-section. Each part has its own internal force, cross-sectional area, length, and modulus of elasticity. These values are used to find the deformation of the full rod. For a member with a variable cross-section, the strain is not constant. Instead, it changes with position along the rod. The deformation of a small element is...

statically indeterminate problem solving

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...

thermal strain

Thermal Expansion of a Free Rod

Thermal strain describes how a structure changes length when temperature changes. It is important in real-world design because structures do not always stay at one temperature. Temperature shifts can affect how they behave and how long they last. A simple example is a homogeneous rod with a uniform cross-section resting freely on a flat horizontal surface. When the rod is heated, it elongates. The amount of elongation depends on the temperature change and the rod’s original length. This...

temperature dependent deformation

Thermal Stress in a Composite Rod

A steel-and-brass composite rod can build stress when temperature changes while both ends are restrained. The rod is nonhomogeneous, so its two sections respond differently. To find the compressive load, the problem is treated as statically indeterminate, which means the force cannot be found from equilibrium alone. The analysis begins by disconnecting one support so the rod can expand or contract freely under the temperature change. An unknown force is then applied at the free end. That force...

poisson's ratio

Material Strain Under Axial Loading

Poisson's ratio describes how a material changes shape when it is pulled or compressed along one direction. It links the strain in the direction of the force to the strain that happens sideways. When a slender bar is loaded axially, it stretches along the load and contracts laterally. Poisson's ratio is defined as the negative ratio of lateral contraction to axial elongation. The negative sign is used because the two strains usually have opposite signs. This keeps the ratio positive and makes...

generalized hooke's law

Multiaxial Stress and Strain Relation

Multiaxial stress and strain describe how an isotropic material responds when forces act along more than one direction. A cube-shaped sample under normal stresses on the three coordinate axes can change into a rectangular parallelepiped. Even after this deformation, the sides remain equal, and normal strain appears along the coordinate axes. The strain components are found from the stress components by studying each stress effect one at a time. The separate effects are then combined using the...

bulk modulus

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...

shearing strain

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...

relation between poisson's ratio, modulus of elasticity and modulus of rigidity

Axial load effects on shear strain are shown by how a slender bar deforms under tension or compression. When the bar is loaded along its axis, it changes in both the axial direction and the transverse direction. A small cubic element inside the bar can turn into a rectangular parallelepiped or a rhombus, depending on how it is oriented. This change in shape creates shearing strain, which is the distortion caused when angles in a material change. Axial loading produces both normal strain and...

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...

stress concentrations

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...

residual stresses

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

Explore how axial loading changes a bar’s shape, creates normal and shearing strain, and links maximum shear strain to Hooke’s law and material constants.

Axial Load Effects on Shear Strain | 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

Axial Load Effects on Shear Strain

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