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

The stress components of a body subjected to multiple loading are analyzed by considering a small cube centered at point O.
The cube has the normal and shearing stress components that act on faces perpendicular to the corresponding axes.
The hidden faces also experience these stresses that are equal and opposite to the ones on the visible faces. So, equilibrium is maintained.
Although the stresses on the cube&#39;s faces differ slightly from those at point O, this small error disappears as the cube&#39;s side length reduces to zero.
Considering the cube&#39;s free-body diagram, the normal and shearing forces acting on the faces are determined by multiplying the corresponding stress components with the area of each face.
Then, by applying the equilibrium equations, relations among the shearing stress components are derived, which implies that only six stress components are required to define the stress condition at a given point.
Also, the shear stress cannot occur just in one plane. So, an equal shearing stress must be exerted on another plane perpendicular to it.

components of stress

Stress analysis under multiple loading conditions is intricate, necessitating a comprehensive grasp of normal and shearing stresses. Consider a small cube at point O, subjected to stress on all six faces, visible or not. Normal stress components &#963;x, &#963;y, &#963;z act perpendicularly to the x, y, and z axes. Shearing stress components &#964;xy and &#964;xz are exerted on faces perpendicular to these axes.
Interestingly, the hidden cube faces also experience these stresses, equal and opposite to those on the visible faces, ensuring equilibrium. As the cube&#39;s side length nears zero, the difference between the stresses at O and those on the cube&#39;s faces becomes negligible.
Examining the forces acting on the cube leads to crucial relations among shearing stress components: &#964;xy equals &#964;yx, &#964;yz equals &#964;zy, and &#964;zx equals &#964;xz, indicating that only six stress components, not nine, define the stress condition at a point.
Importantly, shear cannot occur in only one plane. An equal shearing stress must be applied on another plane perpendicular to the first. Finally, the interpreted stress situation can vary depending on the orientation of the considered element, highlighting the complexity of stress analysis under multiple loading conditions.

Components of Stress

The stress components of a body subjected to multiple loading are analyzed by considering a small cube centered at point O.
The cube has the normal and ...

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Mechanical Engineering

Unit 1

Concept of Stress

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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...

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 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.

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Stress Components on a Cube at a Point

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