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Buckling restraint forces eg: top chord of thru truss

Buckling restraint forces eg: top chord of thru truss

Buckling restraint forces eg: top chord of thru truss

(OP)
HI All

Just wondering if somebody can provide a process for design of through-trusses (i.e without top chord lateral bracing). In my case, the truss is simply supported and the top chord is in compression due to vertical acting loads (i.e. maximum in middle…reducing to zero at each ends).

My issue is in regard to the U-Frames (spaced at 1m longitudinally) that are “supposedly” meant to brace the top chord. My procedure below summaries my question:

1) Thru-truss input into analysis software “Microstran”.
2) Elastic critical buckling analysis is run.
3) Computer output indicates that the top chord members need to be designed with an effective length factor of ky=3.0 (the display also shows a typical sinusoidal buckling shape - when viewed in plan)


Question…does this mean than that if the “sinusoidal” plot is viewed (with the ky = 3.0)…then the out-of-plane buckling forces should theoretically be placed at every 3rd U-frame ...AND..should the buckling forces alternate in direction??

The Eurocode provides a “buckling load” for braces "qd" but only applies it in one direction and does not give guidance on how to design the member to be braced. In other words it provides a good procedure on how to determine forces in the bracing members (instead of using the traditional 2% or 2.5% rules). The Canadian code implies that the forces should alternate in direction – which is contrary to the Eurocode procedure!!

Whist the loads to restrain buckling can be determined (from either method)..there does not appear to be a readily available procedure on how to design the member being braced – in my case the top chord.

In other words – I have an effective length from the ECL…and… a buckling restraint load (from either eurocode or Canadian code). BUT where to go from this point??:

A) Should the top chord be loaded by the lateral buckling forces in one direction only? Or should the lateral load alternate in direction ...regardless of how the ECL displays the buckled shape
B) Should the top chord be loaded by the lateral buckling forces that closely follow the profile of the ECL outputted.
C) If (B) option is considered would this mean that in order to determine the bracing loads from eurocode bracing method…the lengths to be considered would need to be between contra-flexure points?


This is such an important topic..yet I don’t think there is any good guidance on the matter in terms of a design process – I would appreciate some high level though on this complex – yet frequent problem,

regards and thanks

RE: Buckling restraint forces eg: top chord of thru truss

Is the Buckling over 3 frames though? If you post a sketch or analysis outputs it will help to understand your exact arrangement. I am guessing what you are referring to is two trusses linked at the bottom chord so they form a U shape, with continuity in the U?

Personally I would work out what effective length I required to give the capacity and then allocate the Bracing forces to the u frame (I.E. 2.5% of axial load or whatever proportion is required by your steel code) to ensure you can achieve this effective length by providing a load path for the restraint forces back to the truss ends. Also consider the stiffness of the restraining system, if it deflects too far then it's not really considered as a restraint as it's not forcing a higher mode of buckling. Additionally if there is no bracing or diaphragm in plan (at bottom chord level for example), your effective length may still be the full length of the top chord as there is no load path for the restraint forces except for relying on the truss chords bending out of plane. This argument then suggests a deflected shape in bending which is similar to the compression buckling over the full length of the truss top chord!

It would be conservative to consider all the loads in one direction. In reality they could all be in one direction if you consider an initial imperfection in the top chord, between braced nodes it wants to buckle over the full length. The intermediate restraints if considered as a series of springs resist this buckling by applying a restoring force all in the same direction in this scenario (in my mind at least!)

There are a few similar threads here on Lateral Bracing of beams that might prove useful. A paper that is referenced in some of these threads 'Fundamentals of Beam Bracing' by JOSEPH A. Yura may prove useful in understanding how 'lean on' Bracing systems work.

RE: Buckling restraint forces eg: top chord of thru truss

A few references:

Guide to Stability Design Criteria for Metal Structures (4th Ed.) - Galambos
Theory and Design of Steel Structures - Ballio and Mazzolani
Buckling Strength of Metal Structurs - Bleich

Each book has a chapter on members with elastic lateral restraints. The general design procedure laid out is to determine the lateral stiffness you have (they show hand methods, I use point loads in RISA), and then based on the number of U-frames, the ultimate load in the top chord, and the distance between U-frames, you can determine an effective K value for the top chord. Using that K, determine the buckling capacity of the top chord.

As far as the required strength of the lateral restraint, I haven't seen much agreement or standardization. AASHTO recommends 300 pounds per foot (~4.4kN/m) applied laterally, though this is likely excessive if you're not dealing with a traffic bridge. Ballio and Mazzolani suggest a minimum of 1% of the axial force.

In regards to your specific questions:
A) If by alternate, you mean alternate the direction of the force as you walk down the chord, I would advise to follow Agent66's advice and put them all in the same direction.
B) Unsure. I'm not sure how to relate the theory laid out in the references to an elastic critical buckling analysis.
C) Again, unsure.

I hope this helps, and I agree, there's not much in the way of official guidance on these structures.

RE: Buckling restraint forces eg: top chord of thru truss

Quote (geoffstruct)

A) Should the top chord be loaded by the lateral buckling forces in one direction only? Or should the lateral load alternate in direction ...regardless of how the ECL displays the buckled shape
B) Should the top chord be loaded by the lateral buckling forces that closely follow the profile of the ECL outputted.
C) If (B) option is considered would this mean that in order to determine the bracing loads from eurocode bracing method…the lengths to be considered would need to be between contra-flexure points?

In the Canadian code,
(A) The top chord is not loaded with brace forces. The top chord produces brace forces which must be resisted by a bracing system strong enough and stiff enough to be considered a brace. The brace forces alternate in direction. The equation for the stiffness of the brace was derived on the premise that the braces force the member to buckle into a series of half sine waves of length L. If the brace forces were all in the same direction, the magnitude of the brace force would be much smaller.

(B) The top chord is not loaded with brace forces.

(C) I am not familiar with the eurocode method. In the CSA, nodes are assumed to occur at the brace points, so the top chord is designed for the factored compression, Cf over length L where L is the spacing of braces.

BA

RE: Buckling restraint forces eg: top chord of thru truss

Just from first principles, buckling restraint is a function of the straightness of the compression member. If you ignore the lateral bending strength of the member, an out of straight of L/300 requires a buckling restraint of 1.3% of axial force. Its a very simple little free body diagram. I have seen studies suggesting that 0.6% is conservative.

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
HI All

Thanks for the responses so far. I should have provided a sketch and will do this shortly - hopefully to fully understand the issue.

Again thanks for the ideas so far - I will provide sketches in a moment.

cheers

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
HI All..

Please find attached some sketches and extracts that may assist with the understanding of the issue.

Please also note that my comments etc.. so far also apply to the "2.5% rule in the AS4100". In other words, should one chose to use this code instead of the Euro or canadian approaches...would the loads alternate? would they be applies at the mid-contraflexure points OR would they be applied to every U-Frame but in an alternating horizontal direction (or in one direction regardless of the ECL shape from software)???

Many questions I know...but it will be good to finally get a proper understanding of this for all to benefit.

RE: Buckling restraint forces eg: top chord of thru truss

You seem to be considering it round the wrong way by considering the restraint loads on the top chord at point of Max deflection in the chords, the restraint forces are applied at the points of in flexion to the structure that is providing the restraint (in your case the u frames). You need to apply the 2.5 % restraint forces to the u frame uprights and design the system for this load taking into account the stiffness/deflection of the restraint point (too much deflection then it's hard to argue that it's really a restraint).

In reality you have a square frame with some vierendeel action as you are showing a member across the top. BARetired mentioned the correct way to consider the design for the restraint load path.

Is there something stopping you from bracing the top chord level? Then kl of 1.0 could be considered and you might even be able to reduce the chord size.

Because you have more frames than the buckled shape assumed, AS4100 does allow you to distribute the restraint force over several frames if the restraints are spaced closer than that required for the design. For example if you have sized the top chord based on buckling over the three bays of frames then the 2.5% force can be spread between several frames.

RE: Buckling restraint forces eg: top chord of thru truss

You do seem to be looking at this backwards. but I'll leave that to the others.

I want to point out that, you will get some interesting results because of the horizontal truss at the bottom chords. When the chords are stretched in tension, the diagonals are also stretched! the nodes are moving apart. This induces forces in the diagonals and their lateral components will be kinking the chords left and right along the length of the trusses and affecting the top chords through the "U" shape rigid frames.

I don't know where the "three frames" comes from, these look like full span with no outside lateral restraint. It begins to look like a pony truss. I don't know if this helps:Paper on Pony Truss design

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: Buckling restraint forces eg: top chord of thru truss

geoffstruct,

Buckling of a bar on an elastic foundation under distributed axial loads is treated briefly in "Theory of Elastic Stability" by Timoshenko and Gere. There are several papers available on the internet on the subject of pony truss design in addition to the excellent one cited by paddingtongreen.

I don't think a design of this complexity can be properly executed using existing codes alone. The effective length of the compression chords for out of plane buckling is largely dependent on the stiffness of the "U" shaped braces. To get a proper understanding of the behavior of pony truss bridge members requires careful study of available papers and books dealing with the subject, some of which have been mentioned in earlier posts.

BA

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
Hi all
Thanks for all the great advice so far. It seems as though i have a consensus to alternate the load direction (rather than show them in one direction as does the Eurocode method I attached). The altenating 2.5% lateral load (with stiffness criteria) makes more sense to me as it then follows suit to the ECL buckling output..

BUT...could someone just elaborate on:

1) If the buckling occurs over multiple u-frames (recall i have ky=3)...how is the 2.5% distributed to top chord (i.e is it a UDL "smeared" over the same half sine wave length shown from my ECL..or.. discrete point loads at u-frames). Agent666.. I note your latest comments above regarding inflection...but i cant see how that laeral load can be distributed(as permitted by AS4100 and other codes) without being on the half sine wave???

2) Notably in using alternating lateral loads as in (1), the sum of the horizontal forces must always equal. EG for "even" numbers of intermediate braces the lateral loads would always cancel out (because they alternate in direction and are equal in magnitude). An odd number of intermediate braces must mean that the end reactions take out the difference??


Thanks to BAretired for pointing out that yes...the top chord produces bracing forces...a simple and very true point. Thanks:)

Please feel free to mark up a sketch if u feel that a picture paints a thousand words.

again..thanks to all

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
sorry !!!

2) Notably in using alternating lateral loads as in (1), the sum of the horizontal forces must always equal 0 . EG for "even" numbers of intermediate braces the lateral loads would always cancel out (because they alternate in direction and are equal in magnitude). An odd number of intermediate braces must mean that the end reactions take out the difference??

RE: Buckling restraint forces eg: top chord of thru truss

They aren't a real load per say that needs to be applied to the chord, the load is derived historically/theoretically from providing a suitable restraint stiffness at the point of restraint. Real bracing systems designed for the notional 2.5% loads satisfy the stiffness requirements for practical braces, enabling that point to be classified as a 'lateral restraint'. If you read the paper I referenced it goes through the reasoning on this. It's probably covered in the other texts referenced.

Your previous steel code AS1250 used to have a stiffness requirement in addition to the 2.5% load. It was removed as practical braces always meet the requirements (if you think about the axial deformation of an angle fly brace for example between the bottom flange of a rafter and a purlin you can see that over such a short member that the change in length is going to be very small). I can dig out the old clause but I think it was something like the point of restraint shouldn't deflect laterally by more than the member span/400 for it to be considered as a point of restraint for the member design.

Regarding your point 1.
You need to get away from the notion that the loads are somehow applied to the top chord, they are applied to the u frame at the points you have chosen as points of lateral restraint in your top chord design. You design the u frame separately for these forces. If for example you showed that the top chord just had sufficient capacity considering its effective length for compression over 4 bays say, then if you have a restraining u frame everything second bay you can design for distributing the 2.5% restraint force over the two frames (effectively half the 2.5% because you are providing more bracing than is required for the loads you have).

At least that's my understanding of the intent of AS4100 clauses relating to restraints for bending and axial compression (I'm from NZ and our code is almost identical apart from having a lot more specific seismic provisions) .

Another way to think about it if you are still in doubt is that the point of inflexion is restrained, the chord wants to buckle, the resulting reaction at the restraint point is the 2.5% load which you are applying out in the chord.

I would still be considering them in the same direction, AS4100 has a clause relating to parallel restrained elements where you need to consider the cumulative restraint force and provide a load path for this equal to 1 element at 2.5% + up to 6x1.25%. While this strictly applies to several beams in a row beside each other, indirectly I consider it to apply to a number of restraints along the member. It's open to interpretation obviously which is why there is conflicting opinions on it even in codes it seems!

Remember structures won't always behave exactly as per your assumptions, in the area of providing a load path for restraint forces I would consider it more prudent to follow a conservative approach. Real behaviour would be somewhere between the alternating loads vs all loads acting in the same direction.


I have not read the paper from Paddington, but I suspect that it will cover exactly what you are after.

RE: Buckling restraint forces eg: top chord of thru truss

@geoffstruct, Consider a slender rod held so that it had enough compressive force to make it bend like a bow. Have a friend push it in the middle till it is back on the original centerline. Theoretically, that friend's push would go to zero, but nothing's perfect so we assume it is off slightly and the force remains. This remaining force would be resisted at the reaction points. When the center is back in place, the rod bends into two bows in opposite directions forming a letter "S", now ask two more friends to push the quarterpoints back in line....you would then have four bows alternating in direction. They would not cancel out at the nodes because the maximum force is in the chords towards the center.

Two things are stuck in my mind at the moment; one is the overall stability of the bridge as a single section with a long span, it is a "C" shape, not good in torsion. The other is related, what if the tolerances all go one way and you start with a bridge with a single curve on plan? Should that be a case on a long span narrow bridge. I haven't a good feel for the proportions of 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: Buckling restraint forces eg: top chord of thru truss

(OP)
hi agent666

Thanks for your thorough explaination.

** Sorry to labour this point..but are you implying a seperate model for the u-frame with the lateral load? if so why?? is there no merit in modelling the lateral loads within the 3d model??

->> the way i see it...in my model (that has ky = 3), apply 2.5% lateral load (and check stiffness criteria) at each contraflexure point(BUT alternating in direction at each contraflexure point). In my mind i cant see how applying the lateral loads in 'one direction' is correct since the ECL provides an euler bucking shape that is mathematically correct in utilising relevant structural stiffnesses.

Thus... theoretically, and as you pointed out, only the u-frames at the contraflexure points need to provide "the restraint" or else less than 2.5% can be spread to adjacent u-frames (if one wished to engage them for retraint). Would that be correct?



RE: Buckling restraint forces eg: top chord of thru truss

(OP)
hi paddingtongreen.

That paper you linked is very good and I have had a quick flick thru it. Well written,,,but(like the others ive found) it seems to address "effective length" values (and required lateral stiffness Creq).... rather than appoach it from "the other angle" by having to work out the buckling restraint values and how to apply them. Not being to harsh...but Microstran can give me the ECL and Pcrit already!

In other words these researh papers deal with stiffnesses,effective lengths and how to determine Pcrit!!! (which i already have), but beyond this...the current codified design requirments are to have the restraint to chord member satisy a force(+ lateral stiffness) criteria - a clear disconnect!

RE: Buckling restraint forces eg: top chord of thru truss

It is not a clear disconnect at all. The codes are saying that if a compression member is too long to safely resist its load, bracing may be added to reduce the effective column length. If braces are spaced at L then, for design purposes, the column may be considered of length L provided that the braces have the strength and stiffness requirements as set out in the code.

In your case, bracing is provided at a spacing much less than required to meet the above criteria. Your bracing behaves like a series of springs, each with the same spring constant. If the springs are very flexible, the chords will buckle over their full length, that is the primary buckling mode. If the springs provide greater restraint, the chord will buckle in two half waves with an inflection point in the middle. If the springs become stiffer and stiffer, the length of half wave decreases further.

If you have input the stiffness of the members correctly, your software should have taken this into account. The lateral force between chord and bracing will be variable along the chord because the deflection is variable but none would be as high as 2.5% of chord compression. I'm not sure that you need to know the magnitude of the force, but if you do, you can calculate it from the deflection of each leg of the U-frame.

BA

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
Sorry BAretired. I disagree fully with your last comment.

Most papers including the one listed by Paddingtongreen(many thanks for that Padditngtongreen) provide analysis techniques to determine stiffness and effective lengths..they fall short of providing guidance on how to utilize the analysis results to enable designers to complete a design (such as my case with top chord buckling). I would go so far as saying as that many codes and research articles explored thus far almost have a fixation on "Le" and "Creq". Have a read of AS5100 Bridge Code under the "Steel Section". You will see that, it only goes as far as "Creqd and Le". Also, the paper that was graciously uploaded by paddintongreen...again... only has a token comment on AASHTO's lateral force to apply to U-frames, and, does not explore this in any further detail in reference to mode shapes, direction etc.. We have to now evolve from this!

Strand 7, Spacegass and Microstran are fully capable of determining effective lengths based on member and structural stiffness. But of course, we are fortunate to have the works prepared by Holt et al to cross-check...but that is besides my point.

Thus, whilst the Eurocode, Canadian code (extracts in my previous post) plus many other codes have explored methods of determining "restraint forces" (based on assumed imperfection etc...) there is limited research and advice on how to do this based on the analysis results (ECL, Le etc... that is provided by analysis). The "gap" for designers needs to be filled in.

As I explained in my post above. My sinusoidal ECL profile (from computer analysis) does not seem to match current recommended design techniques (except for Canadian code perhaps?? which at least which "applies" load in alternating directions..contrary to Eurocode approach.). There is an argument that the 2.5% rule from AS4100 should be alternating also (but a single direction application produces worse results!)

Yes..there is a very strong disconnect on this matter that many designers are unsure of how to proceed. The company of 50 engineers I work for had very different opinions on this and most of these guys are Phd/masters ! - Why is there a multitude of questions (this forum + others), research articles and so on .. that delv into this issue. If it were clear, no such queries would be warranted.

Note... it is important to distinguish that I am not criticizing past research..I am merely stating that the research needs to evolve to the next level to gel the analysis and the design, which in my opinion is clearly missing.

If you do have any methodologies/opinions etc.. I certainly welcome these! alternatively, you may agree with my last post above...Or... even a sketch to convey your points.

all the best!

RE: Buckling restraint forces eg: top chord of thru truss

The problem of elastic buckling of a member on an elastic foundation can be solved by first principles using an iterative approach. Newmark's Numerical Procedure has been used to solve many buckling problems with known boundary conditions and is quite easy to apply.

Choose a trial chord section restrained by a series of springs of known stiffness and the critical buckling load can be calculated. Whether the chord buckles in a single arc or a series of shorter half waves depends on the restraint provided by the springs. If there are no springs, the chord buckles over its entire length in a half sine wave just as any column buckles. If the springs are very flexible, the member may still buckle in a single arc but its critical load will increase because buckling engages the springs which do work in deflecting to meet the buckled shape of the chord.

If stiffer springs are selected, they provide greater restraint and the buckled shape may have one, two, three or more inflection points. Stiffer springs increase the critical buckling load of the chord.

I am familiar with the section of the Canadian code which you have cited and am of the opinion that it does not apply to your situation. If you chose to limit the effective length of your chord to 3m, you could brace the chord at 3m centers. Each brace would then be required to comply with Article 9.2.5 Simplified Analysis, Article 9.2.6.1 Second Order Method or Article 9.2.6.2 Direct Method. In other words, it would satisfy the strength and the displacement requirement of CSA S16-01. The chord could then be considered a simply supported column of length 3m.

That is not what you are doing and perhaps not something you can do economically or aesthetically because your braces are flexible. You are bracing the chords at one meter centers and calculating the critical buckling load of the compression chord based on elastic theory.


BA

RE: Buckling restraint forces eg: top chord of thru truss

Aerodynamic behavior is another important consideration. Wind blowing across the bridge bends all chords one way which tends to cause the top chords to buckle over their full length, particularly if the deck is narrow relative to the span. Wind forces applied eccentric to the deck will cause twisting of the bridge as a whole. Displacement of bracing systems at the brace point is difficult to determine because the U-frame braces are all moving relative to each other.

The problem is quite complex and not specifically addressed in building codes.

BA

RE: Buckling restraint forces eg: top chord of thru truss

(OP)
Hi BArequired..

Thats good sound advice and thankyou for that! :)

RE: Buckling restraint forces eg: top chord of thru truss

Hi Geoff

I think its conservative to apply the restraint forces to a separate model only so far as to design the frame for these forces and ensure sufficient stiffness. If you dont some of the load will potentially be carried in bending by the chords, we dont have any idea of the scale of the frame or loads or stiffness so its a bit hard to understand relative stiffnesses and magnitude of loads.

I think you need to consider the global behaviour when adding in the restriant loads in your global model, I can see that if they all act in one direction that the structure you have appears quite torsionally flexible. If this is the case then you are back to needing to consider buckling over a longer length.

You also need to remember your structural model doesnt have any imperfections, so buckling critical load is the theoretical load.

I believe once you have this theoretical buckling load (which I assume you are getitng from the analysis you are doing) you apply the code provisions to clause 6.3.4(b) of AS4100 to work out a modified slenderness ratio based on P_crit being N_om. The design then proceeds like any other compression member design to clause 6.3.3. If you read the commentary to 6.3.4 it suggests that this is how it is done unless I am missing something? Have you seen this before, discounte dit for any reason?




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