Beam in Class 3 compression and 2 bending 8

[HanStrulo]

Hi everyone.

To my Canadian friends out there, I am new to designing using the CISC and i have a weird question.
I am designing a beam in combined bending and compression. the beam is class 2 in bending and class 3 in compression.

My head has no idea how to reconcile these two classes. Do you take the worst one? are they independent from each other? What is going on?

for a bit of a background, I want to design a beam and reuse it. the beam is W610x155 so i am using the elastic section modulus to determine the bending capacity.

Thank you

 

[KootK]

 

[Settingsun]

Thanks for posting the CSA snippets.

I agree, of course, but I don't understand why you felt compelled to add that in response to my paraphrasing somebody else's argument that I don't agree with. Can you elaborate?

I thought that part of the discussion came from your musing on why beams-columns can be designed to the plastic limit if that would result in instability, with Enable saying things (loads) shouldn't get that far. It just seemed as though you two were taking ULS to mean slightly different things: the last point where the structure is stable vs the point it collapses. Very fine difference but a significant one in the context. Maybe that's not the reason for the different opinions.

The 1.18 factor (and maybe/probably the 0.85 factor) is an approximation to the plastic section capacity interaction curve determined from first principles for I-sections. See the image below. Since full plasticity in a beam requires infinite curvature (in principle), strain hardening would have commenced before full plasticity but is disregarded on the safe side.

Edit/additional: I don't think you can take the U factor as preventing plasticity. It's a real effect that causes greater stress in the beam-column than determined by an analysis that disregards P-smalldelta. Which I'm sure you know, so presumably I've misunderstood your argument.

 

[Settingsun]

And this makes great sense for columns because, if you allowed a column to go plastic under axial load, that column would loose it's flexural stiffness at mid-height and buckle straight away. A column really cannot be allowed to go plastic at any location where it is not laterally braced.

On further reflection, I think this only applies to additional load. But there isn't any because you already applied the ULS design load. Buckling, up to and including the ULS load, is accounted for by the many section and member slenderness checks. The code procedure satisfies the lower bound theorem: equilibrium is satisfied without exceeding the design yield stress, without buckling, and with ductility.

 

[KootK]

I don't think you can take the U factor as preventing plasticity. It's a real effect that causes greater stress in the beam-column than determined by an analysis that disregards P-smalldelta. Which I'm sure you know, so presumably I've misunderstood your argument.

I'm going to back this up a little bit as it was never my intention to spend 400 posts obsessing over the U-factor.

Reboot

1) Were there to be no beam-column provisions in existence and I found myself forced to fudge my own, I would find it prudent and natural to preclude flexural section plastification in columns with a provision that read something like this:

KOOTK FUDGE CODE: IF [Axial Load > 10% Axial Capacity] THEN [P/A + U x M / Sx < 1.00] END IF

2) Since the CSA provision is not like my proposal, I wonder what is inherent in either the theory or the method to address my concern for flexural section platification in columns. Where is there some possible margin to cover this if indeed it needs covering?

3) One of several possible sources of "margin" that I've considered is the U-factor. The U-factor derivation is an approximation to an infinite series solution that is only truly valid for a particular loading circumstance. As such, there's some "margin" baked into that. And the higher the axial load relative to the Euler load, the more margin there would be. That's it; that's my entire argument with the U-factor. It's not a magical guarantor of non-plastification.

4) If our 0.85 factor is meant to reflect strain hardening and, thus, another possible source of "margin", that's an even better explanation as far as I'm concerned.

Since full plasticity in a beam requires infinite curvature (in principle), strain hardening would have commenced before full plasticity but is disregarded on the safe side.

In the Aussie code discussion of the 1.18 business, I note that they limit their discussion to what is referred to as "zero length members". Does that mean that it's in consideration of a cross sectional stress check and does not necessarily capture buckling concerns? That's how a lot of the plastic design literature in the part of the world handles [axial + bending]: effectively a reduction of available plastic bending strength in the presence of axial. I'm really interested in the reverse: the available axial in the presence of a section already fully plastified in flexure.

With regard to the infinite curvature business, I note that is in reference to a fully plastified cross section. I would argue that, as soon as you start yielding any of your extreme flexural fibers, you begin to lose some of the flexural cross section stiffness upon which buckling capacity depends. How that factors into things, I do not know.

 

[KootK]

On further reflection, I think this only applies to additional load. But there isn't any because you already applied the ULS design load.

I disagree. I would agree with that IF there was something explicit in our checking procedure that prevented the cross section from plastifying under [axial + bending) when bending is designed to Mp. Something like my KOOTK FUDGE CODE. All that I do see, however, is potential sources of "margin" as previously discussed. As a result, it seems to me that even though ULS loads are considered at design, they may still be producing cross sectional stresses that plastify the section. At the least, I so far see no guarantee that this is not the case. The setup doesn't need any "additional" load to push the stresses into the plastic range in my opinion. And really, that's the fundamental issue that concern me.

The code procedure satisfies the lower bound theorem: equilibrium is satisfied without exceeding the design yield stress, without buckling, and with ductility.

1) I don't see that the code procedure does rigorously prevent the development of the design yield stress, particularly when the flexural design is done to Mp capacity.

2) Buckling is only prevented, rigorously, if the member retains it's elastic stiffness at the critical cross section.

 

[KootK]

As I'd mentioned previously, I'm a whole lot less concerned about potential flexural hinge development for pin ended columns with moments applied at their ends. The situations that would bother me more are ones similar to what I've shown below. A wind column with a mid-height girt would be a practical example.

And yes, I do understand the implications of the interaction check being a linear equation. Namely, that the presence of some axial implies the presence of an applied moment less than full Mp.

 

[Settingsun]

I'll also reboot: the 1.18 factor (and the 0.85 factor IMO) aren't related to strain hardening. They're a slightly unconservative approximation to the plastic interaction diagram. Set axial and minor bending to zero and it leaves you comparing 85% of moment to the design plastic capacity. (I'm assuming that Mrx = design plastic moment capacity in CSA for compact sections).

There's not necessarily margin in the U factor. The term that adjusts for shape of moment diagram is left out for simplicity and this isn't always on the conservative side.

KOOTK FUDGE CODE: IF [Axial Load > 10% Axial Capacity] THEN [P/A + U x M / Sx < 1.00] END IF

For small moment, you'll still have partial yielding and therefore loss of stiffness, due to residual stresses.

Because the subject is plastic capacity, buckling would be inelastic. I think the margin comes from buckling not being bifurcation but actually an axial-moment interaction issue exceeding the capacity. The increase in moment due to axial load is accounted for so there would need to be additional load beyond the ULS load to kick over the edge. The capacity factor keeps you away from the worst of stiffness loss (refer to Enable's image 23 May 21 18:15) - you'll go a whole career without seeing steel that is under the minimum yield so the 10~15% capacity reduction actually keeps you under full plasticity. The 'column curve' also takes some account for inelasticity. Strain hardening gives reserve for compact sections.


[EDIT: My apologies. I thought I could slip the below in before you returned.]
The code procedure equates the actions due to the design loads to a failure criterion under the assumed design model. I agree there isn't any margin beyond the load and capacity factors, and any number of unquantifiables. On the other side, there are sometimes unconservative simplifications that the code writers can live with.

 

[KootK]

The term that adjusts for shape of moment diagram is left out for simplicity and this isn't always on the conservative side.

I don't believe that term is left out. It's the w1 term, right?

For no reason that general interest, the first three clips below are from Chajes as:

1) The exact solution for a central point load.

2) The exact solution for a uniform load.

3) A description of the method's approximate applicability to broad range of cases with some modification.

For small moment, you'll still have partial yielding and therefore loss of stiffness, due to residual stresses.

Sure, the same is true of columns that are not beam columns and is baked into the cake of the SSRC column curves.

I think the margin comes from buckling not being bifurcation but actually an axial-moment interaction issue exceeding the capacity.

I believe that is also baked into the SSRC column curves that form the basis of general column design. It's not like pure columns of practical proportions typically get to close to their Euler buckling capacity.

1.18 factor (and the 0.85 factor IMO) aren't related to strain hardening. They're a slightly unconservative approximation to the plastic interaction diagram.

I see it the same. The bottom two clips are from AISC's Plastic Design in Steel The second one shows the assumed stress condition under combined axial and bending at the plastic condition. In the absence of things like strain hardening, I believe that still describes the approach of a plastic hinge as one approaches the capacity.

 

[KootK]

The increase in moment due to axial load is accounted for so there would need to be additional load beyond the ULS load to kick over the edge.

In response to that, I stand by the statement below. If ULS loads are producing a worse than ULS stress condition, it's still potentially a problem.

As a result, it seems to me that even though ULS loads are considered at design, they may still be producing cross sectional stresses that plastify the section. At the least, I so far see no guarantee that this is not the case. The setup doesn't need any "additional" load to push the stresses into the plastic range in my opinion. And really, that's the fundamental issue that concern me.

...you'll go a whole career without seeing steel that is under the minimum yield so the 10~15% capacity reduction actually keeps you under full plasticity.

Again, while that may be the practical reality, I consider it to be already incorporated into the reliability models that underpin LRFD design methods and, therefore, not something that designers should be relying upon for extra margin.

The capacity factor keeps you away from the worst of stiffness loss (refer to Enable's image 23 May 21 18:15)

Assuming that you're referring to the sketch below, I agree that would help. But, again, it's non-rigorous help that kind of goes into this bucket of possible stuff that probably does help but nobody seems to come right out and say this was done to prevent flexural hinging in the middle of beam columns. And that is what has been my concern / interest.

 

[Settingsun]

I don’t see the case of a beam-column with midspan load plastifying at midspan as particularly different than a simply-supported beam designed to plastic moment capacity (putting aside consequences), or a column designed using the effective length method (accounting for midspan bending moment in a way that hides the bending moment). In both cases, you hit plastic capacity within the span so notionally a mechanism is formed. The load that caused the mechanism is still carried though (or infinitesimally less load actually). It takes additional load to cause a problem.

A more succinct argument is that we have the secant modulus as the effective stiffness for the design load, not the tangent modulus (zero) which would apply to additional load (of which there is none as the ULS load must be an overestimate for the design method to work).

When I said that steel always achieves the specified yield, I wasn’t saying that we can take the capacity factor that’s meant to account for understrength steel and use it for other purposes. Because steel fails to achieve strength so rarely, the capacity factor doesn’t need to account for this. The probability of understrength steel is less than the acceptable probability of capacity-deficit failure (as opposed to overload failure). It’s OK for the capacity factor to be used to paper over certain issues, such as unconservatism in the 0.85 factor at high moment, or unconservatism in the U factor, or ignoring the small additional deflection when the stiffness reduces after first yield.

even though ULS loads are considered at design, they may still be producing cross sectional stresses that plastify the section. At the least, I so far see no guarantee that this is not the case. The setup doesn't need any "additional" load to push the stresses into the plastic range in my opinion. And really, that's the fundamental issue that concern me.

I can’t point you to anything that guarantees sections don’t hit plasticity in the method. Quite the opposite: the method equates the loads to plastic capacity, and gives tiny factors (that are sometimes unconservative) to allow us to get as close as possible to that plastic limit. Look at what the 0.85 factor achieves in the image below: two-fifths of SFA yet they still push that tiny bit harder. Likewise, the U factor could be conservatively approximated with another (1+0.23(Cf/Ce)) factor in the numerator but that’s apparently too much conservatism. (That’s the missing factor for equal end moments to be used with w1=1.0.) As a person who occasionally sets foot inside buildings, I also wonder why this is the case but can convince myself intellectually that it’s OK.

 

[KootK]

In both cases, you hit plastic capacity within the span so notionally a mechanism is formed. The load that caused the mechanism is still carried though (or infinitesimally less load actually). It takes additional load to cause a problem.

I don't believe that it does take additional load to cause a problem for the case of buckling. All it takes is a perturbation which may or may not be a load. Telling you stuff that I know you know already, one way to see buckling unfolding is like this:

1) Your stiffness with respect to some degree of freedom goes to zero at or below ULS loading.

2) Some perturbation sets the thing in motion. This can be a load, a nominal asymmetry in geometry or material properties, a light breeze etc...

3) The buckling motion itself creates additional moment at the critical location in excess of the amplified ULS load value.

A more succinct argument is that we have the secant modulus as the effective stiffness for the design load, not the tangent modulus (zero)

4) I assume that you're referring to the Shanley column model of things. In that context, I believe that the secant modulus allow us to approach the reduced modulus but not zero tangent modulus. A hinge is still a hinge after all and, therefore, a stability problem.

5) I currently view the secant modulus business in combination with strain hardening as the most likely candidate for a meaningful and deliberate attempt to prevent hinging in beam columns. I'm now convinced that there's nothing within the interaction equation to do that job given that:

a) Cf/Cr does nothing in this respect in light of our understanding of the 0.85/1.18 factor.

b) The U & 0.85 factors are often, but not always, conservative. And the impact is often relatively small.

Because steel fails to achieve strength so rarely, the capacity factor doesn’t need to account for this.

I'm at a loss with this. Perhaps it's some regional difference in perspective when it comes to structural liability. See the two tables below taken from this paper. Based on that, and in the North American context, it seems pretty clear to me that we do in fact consider material understrength and do account for it with a value less than 1.0.

the method equates the loads to plastic capacity, and gives tiny factors (that are sometimes unconservative) to allow us to get as close as possible to that plastic limit.

Agreed. I always have to remind myself that organizations like AISC and CISC have a marketing function and constantly strive to be competitive relative to other materials. Heck, I can hardly poke my head in on LinkedIn these days without being spammed to death about how mass timber is the best thing since aerobic life forms.

 

[KootK]

...or unconservatism in the U factor

I'm curious, in what situations do you feel that the U-factor might be unconservative?

 

[Settingsun]

I don’t get steel mill certificates every day so my experience is anecdotal rather than statistical, but here are some I can easily lay my hands on from a major project that required submission of them:

350 MPa specified - circular hollow section
408 424 417 432 422 405 415 443 365 375 374 456 415 405 400 430 445 420 445

355 MPa specified - circular hollow section (substituted for 350 MPa)
499 506 (I wonder if these were a dual grade product, like S355/S420 or even S460)

500 MPa specified - reinforcing bar
548 556 568

The 365 MPa is the closest to minimum specified that I can recall ever seeing. I personally don’t expect to see a large number of certificates failing the specified yield, which is what I would expect based on the Galambos numbers. If average is 1.05*Fy and the coefficient of variation is 10%, around 33% of samples fail in a normal distribution (?), and a not-negligible proportion would fail by more than the capacity reduction factor. Is that your expectation?


For the U factor, I haven't derived it myself but McGregor/Wight say there should be a (1+0.23(Cf/Cr)) factor in the numerator for uniform moment. Leaving it out would be unconservative for any value of compressive force.

 

[Settingsun]

1) Your stiffness with respect to some degree of freedom goes to zero at or below ULS loading.

2) Some perturbation sets the thing in motion. This can be a load, a nominal asymmetry in geometry or material properties, a light breeze etc...

3) The buckling motion itself creates additional moment at the critical location in excess of the amplified ULS load value.

To the extent that our predictions of structural behaviour are accurate:

#1 would be a design or construction error, or the low probability of failure that limit state design accepts.

#2 is accounted for, at least in the Australian code. There is either a perturbation in the analysis (nominal loading or minimum eccentricity), or the column is pin-ended with no transverse load which is covered by the column curve.

#3 is accounted for by the perturbation and second-order analysis/moment magnification, combined with the column curve on the capacity side (hidden additional moment). Additional moment in excess of the ULS value requires loading in excess of the ULS load. No argument that things might go pear-shaped at that point.

The image below shows where I see ULS loading landing, although sometime above I was thinking of an elastic/perfectly-plastic design model when referring to infinitesimal changes in load. I added the red lines to represent an overestimate of lower-bound capacity due to small code unconservatism; and the resulting unconservative design capacity. The horizontal red line should be compared to the black curve b (the actual capacity).

From the same article:

Based on the Galambos numbers for average yield stress and coefficient of variation, are North American steel makers having to deal with this extremely difficult circumstance? Do design engineers approve steel that's 10% under strength?

I take the last paragraph as at least contemplating an increased capacity 'reduction' factor because steel strength doesn't need to be reduced. The Australian code commentary lists material strength as one of five factors that the reduction factor accounts for:

- Understrength member/connection
- Strength in structure vs isolated member in lab test
- Design inaccuracy / inadequate understanding of structure behaviour
- Required ductility & reliability
- Accidental eccentricities.

The reduction is 10%. I can't imagine 10% will cover it if all five factors are negative. I suspect the first two beneficially offset the last three.

 

[KootK]

For the U factor, I haven't derived it myself but McGregor/Wight say there should be a (1+0.23(Cf/Cr)) factor in the numerator for uniform moment. Leaving it out would be unconservative for any value of compressive force.

Yeah, I figured that uniform moment would turn out to be the culprit. Fortunately, that condition is pretty rare, particularly in steel where you usually encounter less beam/column continuity than you do in concrete. I've included the Wight stuff below for the benefit of any interested parties. It appears pretty spot on for a lot of practical conditions obviously. Not especially conservative, mind you, just pretty accurate.

If average is 1.05*Fy and the coefficient of variation is 10%, around 33% of samples fail in a normal distribution (?), and a not-negligible proportion would fail by more than the capacity reduction factor. Is that your expectation?

I haven't thought too hard about it but, yeah, something like that. My argument is not that material strengths don't come in reliably high. They do. Rather, my argument is that I don't consider it to be within a particular designer's purview to make use of that fact, disregarding what our standards have to say about it. My impression of how the things work in the steel production world is something like this:

1) The design community establishes material requirements and safety factors based on research etc.

2) The steel production industry reacts to [1] by adjusting their metallurgy so that they are comfortable with the risks involved in their product possibly coming in lower than spec. This surely often means that a supplier's internal specifications exceed those requested of the design community. There may also be other reasons to adjust metallurgy in a manner that would produce over-spec Fy's that have nothing to do with trying to meet Fy.

3) If designers raise their expectations of Fy based on mill certificates etc, producers will probably just return to #2 and create an even more demanding internal spec. In this respect, it kind of becomes a tail chasing exercise.

are North American steel makers having to deal with this extremely difficult circumstance?

I, for one, don't know and don't much care. Like I said, I simply do not consider it my purview to mess with the established, code reliability models in routine design.

Do design engineers approve steel that's 10% under strength?

This designer does as long as it's not a systemic problem where everything is coming low all of the time. In my opinion, this is one of the things that material safety factors are for.

#3 is accounted for by the perturbation and second-order analysis/moment magnification, combined with the column curve on the capacity side (hidden additional moment). Additional moment in excess of the ULS value requires loading in excess of the ULS load. No argument that things might go pear-shaped at that point.

I still disagree. As you know, North American column curves are based on SSRC curves and the concept of maximum strength as applied to imperfect columns and represented conceptually by path [G] in the diagram below frame the Guide to Stability Design Criteria for Metal Structures. That maximum strength represents a condition of neutral equilibrium between external and internal moments, just as it does in good old Euler buckling. Whether you want to call Pmax or a point on the curve 1% to the right of Pmax the "failure load" is immaterial for practical design purposes: Pmax represent the upper limit of usable column stability.

 

[Settingsun]

Why doesn't the column curve account for this?

 

[KootK]

Why doesn't the column curve account for this?

I'm struggling to address that because, unfortunately, I'm baffled by your perspective on this. That said, I know that we were raised on different versions of what "buckling" means and that surely factors in. So, instead of answering your challenge, I'll issue one instead. My perspective:

1) The green dot below represents the maximum usable axial capacity of a column as reported on SSRC column curves.

2) While the green dot represents neutral stability, the factored load that would correspond to that is, for all practical purposes the "capacity". Nobody's dead yet but there's no more to be had.

What of that do you disagree with such that you feel more load is required to induce failure beyond that which would take us to the green dot? Me no understando.

 

[KootK]

That said, I know that we were raised on different versions of what "buckling" means and that surely factors in.

Speaking of that, I think that I've rediscovered the source of some of my dogma in that arena (Chajes). This doesn't change anything that we discussed previously of course.

 

[Settingsun]

The green dot is the nominal capacity calculated from code column curves. It’s supposed to be a lower characteristic value in limit states design, so ~95% of columns will do better.

We then reduce this by a further ~10% by applying the capacity reduction factor, to account for the various issues I quoted from the Australian code commentary. The capacity has high reliability.

The ULS design loads are specified so that few structures will experience them, and these loads are set to be no more than (green dot minus 10%). Even fewer structures will get to the green dot load level and, of those that do, very few will actually fail at the green dot level.

Nobody’s dead yet but there’s no more to be had.

That’s my position too, but in my view that means you got away with it by the skin of your teeth. There needs to be something (possibly infinitesimal) to push the system over the edge. But it also means that you’re in the perfect storm of severe overload of a statically weak structure, which engineering design doesn’t promise to save you from.

I don’t know about the US/CSA codes, but the Australian codes are pretty strict in the stability stakes. I’ve posted a scatter plot of beam tests vs the Australian code capacity (I don’t have one for columns). It’s the lower characteristic value as expected, before applying the capacity reduction factor.

I’ve also posted our column curves. We have five depending on residual stress level (method of fabrication). In 2011, the US column curve was similar to the -0.5 curve, so we have three weaker curves that are used where warranted.

 

[KootK]

EDIT: My apologies. I thought I could slip the below in before you returned

And my apologies for not seeing your edits until just now. That combined with you later additions, has me believing that we've reached substantial agreement. Thanks for contributing to this one even though it's not your native code. I learned a lot after you jumped in.