# Will the second beam have higher or lower stiffness in X direction than the first beam?

Attached is a picture of two beams. The second beam has exactly the same properties and dimensions as the first one, but is rotated, say, by 45 degrees. I want to know if the second beam will have higher or lower stiffness in the X direction than the first beam. And an exlpanation for the answer will be appreciated.

The reason for asking this question is that when I was going over the Geometric Non-linearity effect inside a FEA software (like ANSYS), I read that if Large-Rotations are expected inside the model during solution, then Geometric Non-linearity should be turned ON. When I further researched that why does Large-Rotation need to have Geometric Non-linearity turned ON, it was discovered that the stiffness matrix changes because "If an element's oreintation changes (rotation), then the transformation of its local stiffness into global components will change", [as cited in Practical Finite Element Analysis, 1st Edition, Nitin].

I just want to understand the reason that why would changing the orientation (by rotating the beam) will change the local stiffness in a specific direction or axis, like X axis.

• The answer depends on the material properties., the end constraints, and probably other things as well. (For example if the beam buckles, do you consider that as "stiffness?") I would guess your question is making some unstated assumptions, possibly based on a "strength of materials" notion of what a "beam" is. Oct 2 at 11:46
• I would add loading also as a prominent factor that affects "stiffness" Oct 2 at 13:14
• B is shorter in the X axis than A. Oct 2 at 13:23
• Maybe I didn't clarify my question clearly. If you think in terms of stiffness matrix only (in FEA), then it is only dependent upon the material properties and dimensions. Thats why I wrote that these properties are the same. What kind of constraints, loads, etc generally don't control the stiffness matrix. So apparently, I was thinking that the stiffness should also be the same for both of these beams. However, I discovered later that the stiffness of these two beams is not the same in X direction. I couldn't understand the reason behind it. Oct 2 at 20:47
• @SolarMike, so it means that the Body B is more stiff than A in X axis? Moreover, the cross sectional area of Body B is also going to be higher than A if a perpendicular cut to X axis is taken. These things are overall making me believe that Body B is more stiff than A along X axis. Oct 2 at 20:50

The stiffness is the same for both, as the properties of each are identical along its longitudinal (x) axis, and $$(\dfrac{EI}{L})_A = (\dfrac{EI}{L})_B$$.

However, for the physical length is the same, given the same loading, beam A will experience a higher moment and deflect more, as its vertical projected length (the loading span) is longer ($$L_B = L_A cos\theta$$).

ADD: The sketch below reflects the point I was making in response to your latest comment/question. The beams are assumed to have the same span length (horizontal dimension in the global "X" axis.

Note that for the case of the inclined beam, the force component normal to the beam axis "x" is ignored, since it has no significance in comparing the rigidity of the beams, but represents a stability concern.

Final Note: My last piece for you to think about is whether your question on the relative rigidity of a horizontal beam and an inclined beam based on the global X-axis (horizontal) has a legit theoretical base that merits extensive discussion.

• I know the properties are the same along its longitudinal axis. Therefore each of the beam's stiffness should be the same along its longitudinal axis. That is evident in the question. That is why I was asking about the stiffness along X axis only. Regarding your second para, yeah so it is understood that the stiffness of Body B is higher than that of A. But what you are talking about is subjective to loading condition. I was, in general, asking about the stiffness of the beam along X axis only. Oct 2 at 20:56
• For any beam, the reactions are assumed to act on the beam centerline using the Cartesian coordinate system, which ties the cross-sectional properties of the beam and forms the baseline for the development of beam equations and calculations of beam properties and reactions. Strictly to say, for an inclined beam, the calculations based on the global coordinate system violates all beam/mechanics theories, therefore they are meaningless. Once you've rotated everything on the global coordinate system (axes) to conform to the angle of inclination, then you will find the results are the same.
– r13
Oct 2 at 23:53
• I have edited my original question in case if it was not clear enough. Please go through it again. Oct 3 at 21:44
• The short answer to your update is that the beam theory was developed for small deflection, for which linear elastic behavior is holding valid. As I have shown in the sketches when rotating an object to inclination, there is an eccentricity and additional component force that causing the object to rotate in space and result in large displacement that is beyond the scope of linear elastic behavior. If you feel this answer does not answer your question, I suggest opening a new post with the latest update to draw new responses.
– r13
Oct 3 at 22:12