Having Trouble Deriving an Expression for the Tip Deflection of an Iterized, Non-Uniform Beam
Having Trouble Deriving an Expression for the Tip Deflection of an Iterized, Non-Uniform Beam
(OP)
Some background on my project:
I'm currently on an internship assignment working on a sizing tool for mass estimation of an industry designed component. The component in reality features very complex geometry (curved surfaces, varying geometry, etc.). My assignment is to develop a scalable algorithm in python based on applied loads that would estimate the mass of the industry component. If the loads are large, the component would need to be optimized in a manner that it could support them. If the loads are small, the model must be able to resize appropriately. These constraints mean no FEA and force a simple approach geometrically that can be adjusted in size.
The loading cases are always in a cantilever arrangement, with a fixed boundary condition. An intern before me chose to model the component as an I-Beam. The previous model generated maximum stresses and iterated the I-Beam dimensions until it satisfied a safety factor requirement. This worked well, but the constant cross section caused unnecessary mass in locations of low stress and ultimately overpredicted the component mass compared to industry data. I have improved upon the model by allowing a variable cross section. My model is on the verge of violating the safety factor at nearly every location, resulting in little unnecessary material.
Now, on to my problem:
In addition to a safety factor constraint, my model must pass a tip deflection constraint. The limitations of my design environment (scripting language, no FEA) and the non-uniform cross section have made implementing this constraint difficult. I've studied some derivations for deflection of non-prismatic beams but the results I am getting are unreasonable. Here's a source I used heavily: https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&... Additionally, here's my derivation: http://imgur.com/a/S93Xw
Instituting the constraint quickly revealed the equation I derived had an error; my beam never triggered the deflection constraint even in extreme scenarios. I performed a sanity test by evaluating a beam with a constant cross section. I calculated the deflection by discretizing the beam / applying my equation and compared it with standard deflection equations (like you'd find in a textbook). The results... are terrible. Here's a comparison: http://i.imgur.com/8ak42qc.jpg My deflection equation is many orders of magnitude smaller, causing the trace to look like a line at zero. No wonder I never triggered the constraint. Here's a plot showing the components of the deflection as derived: http://i.imgur.com/EBvdxUa.jpg (sorry for the low res on these two images). I'm clearly wrong about something. The values are orders of magnitude too small and the overall shape of the deflection curve does not agree with the analytical value. Is there an error in my derivation? The approach described in the OU source? I hate to go into so much detail but I figured I'd get a lot of questions regarding the constraints of a scripting language / my overall assignment. Any help would be appreciated - I'm trying to avoid falling behind during my term here
I'm currently on an internship assignment working on a sizing tool for mass estimation of an industry designed component. The component in reality features very complex geometry (curved surfaces, varying geometry, etc.). My assignment is to develop a scalable algorithm in python based on applied loads that would estimate the mass of the industry component. If the loads are large, the component would need to be optimized in a manner that it could support them. If the loads are small, the model must be able to resize appropriately. These constraints mean no FEA and force a simple approach geometrically that can be adjusted in size.
The loading cases are always in a cantilever arrangement, with a fixed boundary condition. An intern before me chose to model the component as an I-Beam. The previous model generated maximum stresses and iterated the I-Beam dimensions until it satisfied a safety factor requirement. This worked well, but the constant cross section caused unnecessary mass in locations of low stress and ultimately overpredicted the component mass compared to industry data. I have improved upon the model by allowing a variable cross section. My model is on the verge of violating the safety factor at nearly every location, resulting in little unnecessary material.
Now, on to my problem:
In addition to a safety factor constraint, my model must pass a tip deflection constraint. The limitations of my design environment (scripting language, no FEA) and the non-uniform cross section have made implementing this constraint difficult. I've studied some derivations for deflection of non-prismatic beams but the results I am getting are unreasonable. Here's a source I used heavily: https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&... Additionally, here's my derivation: http://imgur.com/a/S93Xw
Instituting the constraint quickly revealed the equation I derived had an error; my beam never triggered the deflection constraint even in extreme scenarios. I performed a sanity test by evaluating a beam with a constant cross section. I calculated the deflection by discretizing the beam / applying my equation and compared it with standard deflection equations (like you'd find in a textbook). The results... are terrible. Here's a comparison: http://i.imgur.com/8ak42qc.jpg My deflection equation is many orders of magnitude smaller, causing the trace to look like a line at zero. No wonder I never triggered the constraint. Here's a plot showing the components of the deflection as derived: http://i.imgur.com/EBvdxUa.jpg (sorry for the low res on these two images). I'm clearly wrong about something. The values are orders of magnitude too small and the overall shape of the deflection curve does not agree with the analytical value. Is there an error in my derivation? The approach described in the OU source? I hate to go into so much detail but I figured I'd get a lot of questions regarding the constraints of a scripting language / my overall assignment. Any help would be appreciated - I'm trying to avoid falling behind during my term here





