Using classical methods to solve statically indeterminate beam.
Using classical methods to solve statically indeterminate beam.
(OP)
The propped cantilever in the attachment has a rigid rod supporting the free end, a uniform distributed load and a given deflection at the free end. Assuming the rod does not elongate (infinite elastic modulus) is there a way to determine the moment produced at the wall without utilizing table 3-23 in AISC 325?
I should include that this isn't coursework, is a problem I ran across and was unable to solve.
I should include that this isn't coursework, is a problem I ran across and was unable to solve.






RE: Using classical methods to solve statically indeterminate beam.
1) Treat the beam as a cantilever and calculate the amount of force that would be required at the tip to produce the known deflection. Multiply this force times the length of the beam to get this component of the wall moment (M1).
2) Treat the beam as fixed at one end and simply supported at the other end. Apply the uniform load by itself and work out this component of the moment at the wall (M2).
3) M_total = M1 + M2
If this is a real problem, you've gotta tell us what the the heck it is. Grade beam settlement?
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Using classical methods to solve statically indeterminate beam.
Calculate the tip deflection that you'd get from just the uniform load on a cantilever beam. Draw out the moment diagram for that case.
Calculate the load that would be required at the end of the cantilever to to produce a negative tip deflection of the same magnitude. Draw out the moment diagram for this 2nd case.
Just super-impose those two moment diagrams together.
RE: Using classical methods to solve statically indeterminate beam.
M = wL2/2 - FL
BA
RE: Using classical methods to solve statically indeterminate beam.
RE: Using classical methods to solve statically indeterminate beam.
Concepts still the same though.
RE: Using classical methods to solve statically indeterminate beam.
BA
RE: Using classical methods to solve statically indeterminate beam.
for the propped cantilever, the easiest (IMHO) is unit force method ...
solve as a cantilever, calculate the tip deflection, DL
apply a unit load at the tip, calculate the unit load deflection, du
Prop reaction is DL/du*unit load.
apply prop reaction and re-solve the fixed end reactions.
another day in paradise, or is paradise one day closer ?
RE: Using classical methods to solve statically indeterminate beam.
After the advice from the folks here, my approach was to calculate the deflection at the tip due to the UDL as if the beam is cantilevered. Subtracting the existing deflection from this total deflection we have the deflection restrained by the rod from which we can calculate the force in the rod.
Is the total moment at the wall then just the moment at the wall caused by the UDL acting downwards on the cantilever minus the moment at the wall created by the rigid rod pulling upwards? This seems too simple, especially given the strange shape I expect the beam to take: |--\__
RE: Using classical methods to solve statically indeterminate beam.
Yeah, I believe so.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Using classical methods to solve statically indeterminate beam.
BA
RE: Using classical methods to solve statically indeterminate beam.
"my approach was to calculate the deflection at the tip due to the UDL as if the beam is cantilevered." ... ok, that's the first step in unit load method.
"Subtracting the existing deflection" ... what existing deflection ?
"... from this total deflection we have the deflection restrained by the rod from which we can calculate the force in the rod." ... that's not how I do unit force method.
another day in paradise, or is paradise one day closer ?
RE: Using classical methods to solve statically indeterminate beam.
As far as the beam knows, it has a moment and vertical reaction at one end, a udl along its length, and a vertical reaction at the other end, so the deflected shape is not very strange. Starting from zero deflection and slope at the left we have a fourth order polynomial curve due to the UDL, minus a cubic polynomial due to the end point load.
It would be quite easy to set up a spreadsheet to calculate moment, curvature, slope and deflection at points along the beam, and confirm that the end deflection is as expected.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Using classical methods to solve statically indeterminate beam.
but now I'm re-reading (for about the 3rd time) your posts, and instead of a propped cantilever you have a cantilever with an elastic support (my bad). the solution is much the same, with a propped cantilever you have zero net deflection at the prop; with an elastic support you have a finite net deflection, x = F/k.
for a rod PL/EA = x
1) calc the deflection of the tip of the cantilever due to the loads; D
2) calc the deflection at the tip due to a unit load, u; d
3) for a rigid prop, the prop reaction, P = D/d*u, ie D-d*P/u = 0
4) but you want D-d*P/u = PL/EA ... P = D/(L/EA+d/u)
5) then adjust the cantilever fixed end reactions to account for reaction P.
another day in paradise, or is paradise one day closer ?