continuous beam supported on springs 1

[EngMan40]

I have multi-span beam that is fixed on one end and is supported on 4 spring supports: |------3-----3-----3----3 the beam is not on ground. I want to analyze this beam with K value of the beam is known. Is there a theory or procedure to know the reactions and displacements of the springs for statically indeterminate beams, load is hydrau-static with maximum at fixed end.

 

[IsaacStructural]

It sounds like a homework problem. You can analyze this using the Hardy Cross Moment Distribution method, but in practice I'd probably build a quick computer model in Risa or a similar program.

Licensed Structural Engineer and Licensed Professional Engineer (Illinois)

 

[EngMan40]

@saacStructural: It is not actually a homework problem, it is rather a simplified model of geotehnical problem, but since I don't have any structural analysis software i would do it by hand and that is why I am asking for a procedure that cover this aspect. thanks for your reply, i will see if Hardy Cross Moment Distribution works

 

[rb1957]

"I want to analyze this beam with K value of the beam is known." ... do you also know the string stiffness ?

"Is there a theory or procedure to know the reactions and displacements of the springs for statically indeterminate beams" ... yes, moment distribution, FEA, even hand calcs (but that'd be a chore ven using excel to do the matrix math)

"load is hydrau-static with maximum at fixed end" ... linear increasing distributed load ?

"It sounds like a homework problem. " ... does, doesn't it ?

Quando Omni Flunkus Moritati

 

[IsaacStructural]

Yeah I figured this must be a model of a retaining wall wailer system or something similar, was just noting that you described it in a homework sounding way. No problem with that. like I said, It is much easier to calculate with software, but Hardy Cross will also work, just make sure you double check your numbers, very easy to make arithmetic mistakes if you don't practice it often (as always, make sure to check your final answers with basic logic, ie. do the sum of the reactions = sum of forces, and does the reaction arrangement seem logical)

Licensed Structural Engineer and Licensed Professional Engineer (Illinois)

 

[BAretired]

I have used Hardy Cross moment distribution many times, but have never tackled the problem of spring supports using that method. Offhand, I am not sure how one would do it that way, although it may be possible.

I would be inclined to remove the four spring supports and solve the cantilever beam for deflections at each of the spring locations, then calculate the four deflections for a unit load at each point and finally solve for the four reactions using four simultaneous equations. Those equations can be solved by computer without structural analysis software.

BA

 

[IsaacStructural]

BAretired

That is a really good point, Hard cross typically deals with joint stiffness, not spring stiffness. Now I'm wondering if there is a way to model that. The distribution factor for the springs is relative to the other springs as well as to the beam stiffness.

Does your method assume that the springs are all equally stiff?

Licensed Structural Engineer and Licensed Professional Engineer (Illinois)

 

[rb1957]

you can use hardy cross with springs, i'd prefer to use the unit force method (as BA describes) if i was doing a hand calc.

it might be easier just to do trial and error ... set up a spread sheet, i'd start with the simple cantilever then add one support reaction (=k*d1) ... so far no guessing ! then with these two reactions (the FE and 1 spring) add another (initial guess = k*d2) but this'll change the deflections at both but with a little tweeking you should be able to get balanced (so that the reactions = k*d) and then the 3rd. yes, i know this sounds like a bit of a farce but it is an alternative to solving the matrix (required for the triply reducdant beam).

Quando Omni Flunkus Moritati

 

[BAretired]

Isaac,

Each spring may have a unique stiffness or they can all be equal. A unit load will deflect each spring according to its own stiffness.

rb1957,

I agree that you could solve the problem by trial and error, but the solution of a 4 x 4 matrix is trivial with a computer and not too onerous by hand.

BA

 

[rb1957]

agreed

Quando Omni Flunkus Moritati

 

[paddingtongreen]

BA, wouldn't a modified version of that paper you have (I thought I took a copy but i can't find it) for the buckling of a column work here, assume a deflected shape, from that find the moments in the beam and the forces in the springs, recalculate deflected shape...iterate to closure.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[BAretired]

Yes Michael, I think that could be done. I believe you are referring to thread507-267603. I had attached about 20 pages of notes illustrating the method but they seem to have disappeared from the thread.

BA

 

[IDS]

The spreadsheet here:

http://newtonexcelbach.wordpress.com/2012/09/06/continuous-beam-spreadsheet-with-units/

will do what you want.

It uses the method described by BARetired (i.e. calculate deflections for a cantilever then solve simultaneous equations so that:
Support deflection = -Reaction/Spring stiffness.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Sounds like just what the OP needed, Doug.

BA

 

[EngMan40]

Thank you all for the great discussion.

The actual system I am designing is soldier piles and lagging for 35 ft deep excavation that terminates on top of rock. The tiebacks have ~300 Kips/in stiffness and I would like use W 14X233 or equal steel section for soldier piles. anchoring the soldier piles into rock is also an option. The system is a continuous beam (soldier pile) which is fixed at rock socket and supported by tiebacks. My rephrased question would be: could I consider the tiebacks to be pin supports for the system or spring supports and is there a difference in tieback reactions between pin and spring (I need justified answer to present to structural engineer)? load per tieback is 150~200 Kips (mainly because of bathtub condition).

 

[BAretired]

I'm not sure how you would fix the pile to the rock, but there is a good chance the pile will not be fully fixed with zero rotation. If there is any rotation at the base of pile, the difference introduced by considering the tiebacks as pins rather than springs is likely minor by comparison to the error resulting from the assumption of full fixity.



BA

 

[EngMan40]

24" diameter rock socket 10' long. is that enough to consider it as fixed, what do you think? (rock type: Granitic Gneiss)

 

[paddingtongreen]

I would accept that as fixed, but something is still hidden, you say 150 to 200 kips but surely with bathtub conditions, this is a hydraulic load from the top, the full 35 feet? giving much higher pressure near the bottom, fading to nothing at the top. I think you may have to re-think using a fixed bottom, you may fail the pile before the tiebacks become effective. To clarify, if the pile has to deflect say 1" to develop the tieback's load, the fixed end may have to yield somewhat to allow this.



Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[BAretired]

Don't know. Is the socket filled with concrete? Where do you consider the point of fixity lies relative to the surface of the rock? How accurately do you know the position of the rock?

If the point of fixity is known, then considering each tieback as a pin instead of a spring will be somewhat conservative with respect to tieback reactions but not with respect to the fixed end moment.

BA

 

[EngMan40]

@paddingtongreen: tiebacks are usually proof tested and locked at certain load, say 100 kips, (don't confuse it with soil nails which are passive system that need to deflect to start developing its resistance). provided the excavation is done on stages and at each stage the tiebacks are installed, tested and locked before moving to next excavation stage and this allows some relaxation in soil before reaching the bottom of excavation.