Superposition to find deflection of beam with multiple point loads?
Superposition to find deflection of beam with multiple point loads?
(OP)
I know the moment and shear force can be found via superposition. If I have a simply supported beam with 5 point loads, can the deflection be found using superposition of 5 beams with a single point load? I have been adding all 5 point loads together and treating it as a single center point load but the deflection from this is conservative.






RE: Superposition to find deflection of beam with multiple point loads?
simplifying to a single point load is extremely conservative. applying an equivalent UDL (sum forces over the length of the beam)probably isn't too bad.
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RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
the double integration isn't That difficult, I could write it out in an email ...
M(x) = (Pb/L)*x ... 0<x<a
M(x) = (Pb/L)*x-P*(x-a) ... a<x<b
or M(y) = (Pa/L)*y ... 0<y<b
EI*v(x) = ...
another day in paradise, or is paradise one day closer ?
RE: Superposition to find deflection of beam with multiple point loads?
EI*d(x) = Pb*x^3/(6L)+Cx, 0<x<a and
EI*d(y) = Pa*y^3/(6L)+Dy, 0<y<b
C = Pab/(6L)*(2b-a)
D = Pab/(6L)*(2a-b)
for 5 different loading ... different P, a, b = L-a
evaluate at different x and y
summ
if you really want to, from the summ you could further refine your x and y to home in on the maximum
another day in paradise, or is paradise one day closer ?
RE: Superposition to find deflection of beam with multiple point loads?
Live long and prosper!
RE: Superposition to find deflection of beam with multiple point loads?
Given a simple beam with point loads, it is quite likely that the max. deflection will occur, at one of the point loads. Use a numerical integration method, such as Newmark’s Method, and put nodes at least at each of the loads, and you should come up with a pretty good answer. The Conjugate Beam and Moment-Area Methods would also work well on these types of problems. Of course, superposition will work, but you must calc./plot the deflections caused by each load at each of the other load points (nodes), and then repeat this process for each load.
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
This could be done in Excel also.
RE: Superposition to find deflection of beam with multiple point loads?
it doesn't give you the generalised equation. ok, with some smarts you can modify the expression for "dx, x<a", i think.
the math is relatively easy, perhaps the trick is setting up the equations to simplify the integration.
another day in paradise, or is paradise one day closer ?
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
(1/E*I)*[(P/6)*<x-a>^3 - (P*(L-a)/L)*(x^3/6) + (P*a*(2*L^2 - 3*L*a + a^2)/6*L)*x]
the <x-a>^3 is the singularity function which equals 0 when the term in the <> is negative, then it acts just like a normal (x-a)^3
Programing it in something like Mathcad isn't bad, probably hard in excel, and a true pain if done by hand.
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
Definitely not for long term deflections for concrete or other inelastic materials.
RE: Superposition to find deflection of beam with multiple point loads?
btw, there's an error in my calc, i had one of the boundary conditions wrong (and checked with a = L/2 which didn't find it). not unitll i tried to find max deflection (ie zero slope) ...
now i think ...
C = -Pab/6L*(a+2b)
D = -Pab/6L*(2a+b)
at least this seems to give zero slope where i expect it (on the "b" span)
another day in paradise, or is paradise one day closer ?
RE: Superposition to find deflection of beam with multiple point loads?
another day in paradise, or is paradise one day closer ?
RE: Superposition to find deflection of beam with multiple point loads?
If you are attempting to find the maximum deflection of the beam and at one point it occurs, then draw the shear diagram for the loads on the beam and where the shear force is zero the maximum bending moment location will be found.
I would then proceed to calculate that moment based on the beam loads, this can be followed by integration to find the maximum deflection value.
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
RE: Superposition to find deflection of beam with multiple point loads?
https://newtonexcelbach.wordpress.com/2013/12/03/m...
will analyse any number of point loads on a single span or continuous beam, or you can define a vehicle as a series of point loads and find the position that gives the maximum deflection (or any other output value).
And its free even if you are not a member of the AISC.
As noted by others, if you are dealing with concrete allow for cracked stiffness (if the beam is cracked under maximum loads), and allow for creep, shrinkage and differential temperature effects for long term loads.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Superposition to find deflection of beam with multiple point loads?
http://www.assakkaf.com/courses/enes220/lectures/l...