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continuous beam supported on springs

continuous beam supported on springs

continuous beam supported on springs

(OP)
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.

RE: continuous beam supported on springs

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)

RE: continuous beam supported on springs

(OP)
@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

RE: continuous beam supported on springs

"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

RE: continuous beam supported on springs

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)

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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)

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

agreed

Quando Omni Flunkus Moritati

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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

BA

RE: continuous beam supported on springs

Sounds like just what the OP needed, Doug.

BA

RE: continuous beam supported on springs

(OP)
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).

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

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

RE: continuous beam supported on springs

(OP)
@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.

RE: continuous beam supported on springs

(OP)
BAretired: Concrete/neat grout (5000 PSI). I am not sure where the fixity point would be to be honest (FEM analysis showed moment inflection point below top of socket but not at fixed point with respect to that top). subsurface investigation was carried out and rock breaks are 8,000 PSI strength. what do you think?

RE: continuous beam supported on springs

I would need to think about it a bit more. In the final analysis, the precise point of zero rotation of the pile and the variable load between that point and the top of rock is going to be a rather crude estimate at best. Perhaps that justifies the assumption of tiebacks as pins instead of springs.

BA

RE: continuous beam supported on springs

I would not approach this using springs, either. The pile has fixity, in my opinion. The tiebacks are propping a cantilever. Assuming you develop the strength of the tieback without soil mobilization, it is a pinned connection.

RE: continuous beam supported on springs

Ron - what do you mean by "develop the strength of the tieback without soil mobilization"?

I'd suggest doing the analysis with upper bound anchor stiffness (i.e. pinned supports) and lower bound stiffness, as a starting point.

If this is modelling loads due to excavation, with anchors placed as excavation proceeds, the results of any single stage analysis are going to be substantially different from what occurs in the actual structure, but if you cover the envelope of possible loads you will be OK.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: continuous beam supported on springs

(OP)
The construction sequence is going to be: (1) drilling for the soldier piles and socket them into rock (2) install the lagging ( in this case sheet piles) (3) excavating the first stage and installing first row of tiebacks (4) excavation of subsequent stages and installing tiebacks. Since the socket is installed before excavation then The question is: am I conservative if I design the rock socket using results obtained from structural analysis of continuous beam fixed at one end with three or four pins? should I consider the effect of excavation and installation sequence in the analysis and increase the moment accordingly ?

RE: continuous beam supported on springs

calling these props "pins" mean they rotate but don't compress, yes?

modelling as springs mean you're accounting for compression, yes? ... why not ?

fixity to the rock is one thing, fixity at the horizontal beam is something not being talked about (maybe 'cause you guys know) ... is it s single point attmt (ie pinned) or a multiple fastener (ie some moment capability) ?

if the choice is critical to the structure ... why not choose the most conservative ? by that i mean analysis the various models, and pick the most conservative result at each point (ie not consistently fixed or pinned, but which ever model produces the local maximum load) ?

Quando Omni Flunkus Moritati

RE: continuous beam supported on springs

rb1957,
Tiebacks are usually cables anchored into the soil or rock and in this case, prestressed to 100 kips. They can be horizontal or angled downward.

Where the cable intersects the pile, it is hinged but the cable will strain with additional load. The total magnitude of strain in inches is expected to be P/300 where P is the cable tension in kips. Cables should be modeled as springs.

BA

RE: continuous beam supported on springs

The later information now makes some sense of the original question but I still have little experience of this type of work. The problem, of course, is that most of us are used to precise placement of members and loads, where this is an imprecise black art.

Just trying to think my way through this.

Drill holes, drop soldier piles in holes, place concrete in the rock to embed the pile.
Pile is not in contact with undisturbed soil, it is a cantilever sticking up from the rock. Is the hole flooded?
Excavate to first tieback level, place and pre-tension the tieback.
The pile is a cantilever being pulled towards the soil it is to retain. This soil must be dewatered or else it would pour in during excavation.

When the second level of tiebacks are tensioned, it will relieve the installed tension in the first level, and so on with each level. They probably go back and adjust with a torque wrench.

The retained soil still is not against the lagging but when dewatering stops, the gap will fill with water, I think.

I know this doesn't answer the question.

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

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