# Do inertia affect stiffness matrix of beam?

Refering to a pdf I uploaded at here. Why does these stiffness matrix displacement do not need to consider the inertia (determines the shape of cross-sectional area of each element) ? I have always thought that the inertia will affect the displacement too. Thank you for reading and have a nice day :)

• You mean $I$ as in second moment of area, and not mass moment of inertia. Apr 21, 2022 at 14:52
• @JAlex, yes sorry. Its the moment of inertia that affects the amount of displacement. However, why does the stiffness matrix does not have that? Does that mean any shape of cross-sectional area will produce the same amount of defection caused by the same load in stiffness matrix? Apr 21, 2022 at 14:59
• Stiffness matrix is dependent on $I$ if bending is involved. The link to personal drives is blocked in many cases. Please consider embedding images as needed, and type the equations out using math formatting MathJAX. Apr 21, 2022 at 15:09
• @JAlex, but wouldnt applying load will cause bending haha? Apr 21, 2022 at 15:12
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Apr 21, 2022 at 15:14

Indeed the stiffness matrix for problems with bending is dependent on the area moment $$I$$ as well as the elastic modulus $$E$$_.

Take the example of a cantilever beam.

Page 2 of this pdf lecture for example, shows the following

As you can see the stiffness matrix depends on $$E I$$ (shown outside the matrix as a common factor) where $$E$$ is the modulus of elasticity and $$I$$ is are second moment of area of the beam.

For example a rectangular beam has $$I = b h^3/12$$.

But for problems with only axial deflection (like the truss made of rods the question describes), then the stiffness matrix only depends on the axial stiffness of the rod which is $$k = \frac{E A}{L}$$

• Hi, I have updated a picture above. It does not involve any inertia at all Apr 21, 2022 at 15:23
• This pdf is a force on a truss system analysis which involves tension and compression. But, tension and compression is caused by bending Apr 21, 2022 at 15:28
• @catherinetan - The truss members are two-force members that carry only axial load, and no side loading / moment. So each member can only compress or expand and cannot bend. This is what $I$ is not involved here. Apr 21, 2022 at 15:30
• Hi Alex, so you are saying the final total displacement of truss caused by the dispalcement of (tension compression ) of truss member and the displacement caused by bending moment of the entire truss structure (if we consider the moment of inertia of the entire truss structure ? :) Apr 21, 2022 at 15:35
• @catherinetan - yes, the deformed shape is due to individual member tensions/compression. You might describe the overall deflection as "bending" only when the structure approximates a beam in shape. But a structure made of rods in the shape of a sphere for example won't "bend". Apr 21, 2022 at 15:41