when a structure works as two springs

huangyhg

when a structure works as two springs
in a recent calculation review, i encountered a structural problem which triggered quite different opinions. i'd like to post in the forum. you may find it very simple, but it's definitely beneficial to some of us.
how to compute the spring constant when part of structure works as two springs to remaning structure?
it is a mining conveyor system. the primary conveyor truss is supported by the drive station (like a building structure) at one end. we use staad to model the truss and simplify the drive station to two springs as the restraints at the interface (a beam in the drive station supporting the truss bottom chords). in the calculation of the spring constant, the originator (also my supervisor) used the spring constant obtained from the drive station model using a single point load and applied to the two springs in truss model. when i pointed out that this is not quite right the dispute started. i was very confident based on: there is only one drive station, if the interface is a single spring the approach is right. when the drive station is simplified to two springs, the spring constant from the single point load approach cannot be applied twice... (i save my argument and my dealing approach here in case i am wrong). what induced my confidence is two more engineers (including the lead engineer) joined the discussion and agreed with the originator. they pointed out only single point load can be used in the drive station model according to the "definition" of spring constant ... (i save their further argument here in case they are wrong).
i further considered the thing and explained to them. but none of them agreed up to now.
am i wrong???
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j1d -
i think i can visualize your structure - but just to be sure - you have a twin truss conveyor (two trusses parallel to each other and spanning some distance).
the two trusses each are supported at one end by a common beam, perpendicular to the trusses.  the beam spans some distance between, i presume, some support columns or something similar.
the other ends of the trusses are supported by something else, not pertinent to this discussion.
is this correct?
if so, the concept of "replacing" the support beam with two springs of constant k (kips/in) would be accomplished by first applying a point load (1 kip) to the point on the support beam that holds truss 1 (call it point 1).  the deflection at this point on the beam is d11.  
the deflection of the beam where truss 2 connects (point 2) would be d21.  your spring for the truss 1 support would be 1/(d11+d21).
similarly, you then place another 1 kip load at the point where truss 2 is supported and get a deflection at that point, d22.  also, you get the deflection at point 1 due to the unit load at point 2, d12.  the spring used to support truss 2 would be 1/(d22+d12).
what you are doing is finding the response of the beam to a load at a point on the beam.  this response (so many inches of deflection for a certain load) is affected by the load that occurs at the other truss point.
build a quick model of the two trusses (just use a single member) and a connecting/support   
jae,
you visualized the structure accurately.
the conveyor truss is a 3d, box-shape truss. the beam (perpendicular to the truss) supporting the bottom chords of the 3d truss (the top chords merged to the bottom at the support) is supported by columns in the drive station. set truss span direction as x, the transverse direction as y. the task was to find out the spring constants in x and y directions, cx and cy. the way you described for cx is correct, it is also a confusing point to the engineers. in their view, cx = unit force at point 1 (or point 2)/deflection at the loading point, cx obtained this way is the constant of the two springs in the truss model. my view is: with two support points on the beam, we should apply two point loads on the beam, and use the cx (two constants, may not be the same) to the springs respectively.
it is less confusing when we consider cy.
i don't believe using two unit loads to determine a spring constant is correct.  you use one unit load and get the deflections as indicated above.
jae,
thanks for your question and in-depth thought.
using two unit loads in computing the spring constants seems inconsistent with the definition. but using the spring constant from the single unit force to two springs is totally wrong since this ignores d12 (or d21) as you pointed out. let鈥檚 simply put it this way: imagine the number of connections between the truss and the beam is 200 instead of 2, we model the drive station as 200 springs to the truss model, can we use cx=1/d11 (the spring constant when only one connection between them presents) 200 times? as a whole, then, the drive station will be 200 times stiffer.
point 1 and point 2 are symmetrical w.r.t. the beam.  if we simplify the whole drive station to a beam and apply two unit loads, you can find d11=d22 and d12=d21 and cx=1/(d11+d21). in the linear elastic range, applying unit loads separately obtains the same cx.
importantly, the single beam doesn鈥檛 represent the drive station structure at all. the beam is a very stiff  36鈥漻36鈥?box shape member. the flexibility is mainly from rest of the drive station structure.
it is less confusing when we consider cy. we don鈥檛 have the hassle of d21, d12, etc. to the truss, the drive station works as an upstanding spring.
spring constant is a function of number of springs. if co=1/d (d=deflection with single unit load in a given direction), c=co/n (n=number of connections or number of springs to be used in the truss model) approximately for this calculation case.
i hope this discussion continues, thanks for the input!