Beam Bending: Centroidal Axis Rotation

In summary, in a beam bending problem, the beam cross section bends around the centroidal axis due to the moment generated by externally applied forces. This is necessary for the beam to satisfy statics and be stable. The location of the centroidal axis can vary, but for symmetrical and homogeneous beams, it is the axis about which bending occurs. This is because the sum of the areas above and below this axis multiplied by the corresponding stress must be equal for equilibrium. In the case of a tube bending with a concave downward shape, there will be tension stresses on the outer top surface, compression on the top inner surface, tension on the lower inner surface, and compression on the lower outer surface.
  • #1
chandran
139
1
how do we say that in a beam bending problem the bean cross section bends(rotates)around the centroidal axis. Why not about any other axis?
 
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  • #2
The beam has to satisfy statics. A moment is generated in the beam due to externally applied forces, and this moment results in internal forces which also have to be in equilibrium for the beam to be stable.

Looking at the cross sectional internal forces, there will be forces causing compression on the cross section with equal and opposite tension forces on the section in order to balance an external moment (all compression (or tension) stresses will be above the neutral axis and the opposite will be below the neutral axis).

The centroidal axis is not always the axis bending is about. But this assumption does apply for a symmetrical, homogeneous beams. In this case, the sum of all the areas above an axis passing through the section multiplied by the stress in each area has to equal the sum of all the areas below the same axis passing through the section multiplied by the stress on all these elements. You will find the location of this axis to be the centroidal axis.
 
  • #3
Hello,

If you have a tube under load and bending with concave say downward, would you have a tension stress on the outer top surface, then compression on the top inner surface then tension on the lower inner surface and finally compression on the lower outer surface?

Thank you.
 

Related to Beam Bending: Centroidal Axis Rotation

1. What is beam bending and why is it important?

Beam bending is the process by which a beam is subjected to an external load and its shape changes as a result. This change in shape is known as deflection. It is important because it helps us analyze the strength and stability of a beam, as well as its ability to withstand external forces.

2. What is the centroidal axis and how does it relate to beam bending?

The centroidal axis is an imaginary line that runs through the centroid (geometric center) of a beam. It is important in beam bending because it is the axis about which the beam rotates when subjected to an external load. The rotation of the beam is known as centroidal axis rotation.

3. How does the shape of a beam affect its centroidal axis rotation?

The shape of a beam plays a critical role in its centroidal axis rotation. Beams with different cross-sectional shapes (e.g. rectangular, circular, I-beam) have different moments of inertia, which determine how much the beam will deflect when subjected to an external load. The higher the moment of inertia, the less the beam will rotate.

4. What factors affect a beam's centroidal axis rotation?

There are several factors that can affect a beam's centroidal axis rotation. These include the type of material the beam is made of, the shape and size of the beam, the magnitude and direction of the external load, and the beam's support conditions (e.g. fixed, pinned, cantilever).

5. How is centroidal axis rotation calculated?

Centroidal axis rotation can be calculated using the principles of mechanics, specifically the equations for bending moment and deflection. The exact calculation will depend on the specific beam and loading conditions, but it typically involves determining the beam's moment of inertia and using it to calculate the deflection at various points along the beam's length.

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