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# Flexural Strength of a Steel Plate 2

## Flexural Strength of a Steel Plate

(OP)
I have a 1/2" cantilevered steel plate welded to a steel beam. The plate is experiencing a 13 kip load as shown in the image below:

My question is, can I use AISC 14th F11. RECTANGULAR BARS AND ROUNDS and apply those equations to calculate flexural strength? My understanding was that bars are different than plates, even though they both have rectangular sections.

Can anyone provide structural theory that can justify it? I have seen some explanations on here talking about the Z/S ratio.

At work, I am being told to just check (phi)Mn = Fy*S, and I also want to know whether it is reasonable to assume that the plate can get to the plastic moment without buckling or other modes of failure?

### RE: Flexural Strength of a Steel Plate

I'd run it first using F11, but I'd set your unbraced length at 16". See how much of a hit you'd take to the nominal bending strength.

The more I look at it, the more it looks like a stiffened seated connection (but rotated 90 degrees). If someone in your office has a Salmon and Johnson steel design textbook, they have a procedure for designing that stiffener.

### RE: Flexural Strength of a Steel Plate

(OP)
Thanks, winelandv. This is actually a "stop plate" that is welded to a crane runway beam, as a means to stop a bridge crane at one end of the runway while it is in motion.

The force is the longitudinal force from the crane. I don't think the plate needs to be 8" tall, but as of now, if I perform the following check per AISC F11:

(Lb)*(d)/(t^2) <= 0.08*(E)/Fy?

Lb = 2*8" (cantilever) = 16"
d = 8"
t = 3/4" (I know I said 1/2" in original post, I was mistaken)
E = 29,000 ksi
Fy = 36 ksi (A36 steel)

The problem is I am getting 227.5 on the left vs 64.4 on the right

So I fail this check for "bars" bent about the major axis...meaning I can't apply F11 to this?

I wonder why AISC does not have any direct specifications for plates in the Design Chapters?

### RE: Flexural Strength of a Steel Plate

You can still apply F11. You need to turn the page to F11.2.(b). It gives you another check to see if your ratio is below 1.9*E/Fy. It also includes Mn for your plate being in this region.

Finally, F11.2.(c) has the Mn equation for when your ratio is larger than the 1.9*E/Fy.

### RE: Flexural Strength of a Steel Plate

#### Quote (gmoney731)

My understanding was that bars are different than plates, even though they both have rectangular sections.
There is no structural difference between bars and plates. The only distinction between them is that bars are fabricated from flat stock that is 8" wide or less, while plate stock is wider. But for design, the steel member doesn't know whether it was fabricated from bar stock or plate stock. So for practical purposes on drawings, most engineers just call both items "plates" on their drawings and don't dictate to the contractor/fabricator which type of stock must be used.

### RE: Flexural Strength of a Steel Plate

13 kip = 57 kN (according to a quick google search; correct me if I am wrong)

This is a significant load. With given dimensions (approx 200mm x 200mm x 25mm in SI units), it may or may not resist the load without buckling or locally yielding. Furthermore, if the crane has any significant speed (say, a few meters per second) and a significant mass (probable) when impacting, the load should be calculated with a dynamic analysis. How did you determine that load? Make sure that the load is conservatively estimated.

An option is to model the plate as a membrane (thin slab, taking only in-plane forces), apply the load and rigid connection, and check if the von Mises stress exceeds yield. Then, a buckling check (eigenvalue buckling analysis) can be done as a check for stability.

If you want to model it by hand, you may use linear thin slab elements (triangles, 2 or 4 elements), hand-calculate the element stiffnesses and nodal load, apply boundary conditions (zero displacement at nodes along clamped edge) and solve the system for nodal displacements. Then, use the interpolation inside the element to compute in-plane forces and stresses. Thus will take some time, and is just a more error-prone version of the computer analysis, which you no doubt have access to.

Using the beam simplification will incur significant modelling error, since it amounts to idealizing the plate as a cantilever, but it may be used to double-check the FEA results.

EDIT: to clarify, the other modes of failure include at least shear buckling, compression buckling, local yielding and crushing. Furthermore, plastic sectional capacity is unlikely to be achieved, and thus the elastic section modulus would be the "conservative" (if other modelling errors are ignored) choice for estimating strength of the plate as if it were a beam.

### RE: Flexural Strength of a Steel Plate

Another note on the design: it would be better to place the stop plate in such a way that the load is distributed along the entire left edge, converting the 13 kip point load into a line load. This would reduce stresses in the plate, reduce the buckling load and reduce the extent of/risk of local yielding and crushing of the plate edge.

### RE: Flexural Strength of a Steel Plate

For design procedure....

Your extended plate looks just like an extended shear tab connection. You just don't have the bolt holes and your eccentricity is a little larger.

I'm 99% sure there are design examples of this that go through all the limit states in the AISC Design Examples.

### RE: Flexural Strength of a Steel Plate

Gmoney731:
What’s the dia. of your wheels? Aren’t those wheel stops usually contoured to match the wheel shape/dia.? You don’t want the wheel hitting on a sharp top corner of that stop pl., that could damage the wheel over time. And, at the same time the pl. should be high enough so the wheel can not climb it on contact. You do not want the wheel flanges disengaging the rail. I’d have the front edge of that 8" high pl. be vert. for 1" to 1.5" high, then contoured so the wheel hit it over some edge length/height at about 4" high. An inch or so either side of that 4" high elev. the wheel and the pl. contours actually match, then there is some slight clearance above and below this area. Then on that leading edge, and on the sides of the bot. edge, I’d grind 1/4 or 5/16" x 3" long bevels; fill those bevels with weld, then put a fillet weld over the bevel and all around the pl., to the rail. Don’t leave any notches, undercuts or craters in the welds or pl. in th front edge region. Otherwise, your problem is primarily a shear and potential pl. buckling problem. That front weld is designed for (13.2k)(4" + 1" ht., whl. contact ht.)/(8" base length - 1" ea. end or abt. 6" base lever arm) = 11k of upward force on the front of that weld. Then the reinforcing fillet all around takes the 13.2k shear to the rail.

### RE: Flexural Strength of a Steel Plate

We normally weld a bit of beam off cut or similar above the beam, often the crane support beams are not stock length so we us the offcuts. as dhengr has indicated, a single flat would cause issues if the bump was engaged with the wheel.

### RE: Flexural Strength of a Steel Plate

I would check it for:

Combined flexure (Mn from F11) and shear using the AISC Manual Part 9, Equation 9-1.

### RE: Flexural Strength of a Steel Plate

Look at Shakya & Vinnakota, AISC Engineering Journal Q3 2008. They worked out an efficient approach to triangular gusset design by looking at them as a series of columns that get shorter and shorter. Then they integrated across the height of the triangle to sum the capacities.

### RE: Flexural Strength of a Steel Plate

Not saying I would do it this way, but what about a steel strut-and-tie model?

I imagine this approach would be lower-bound (same as for concrete), but maybe I'm wrong.

### RE: Flexural Strength of a Steel Plate

That is similar to the approach I mentioned except they add all the adjacent struts up with integration.

### RE: Flexural Strength of a Steel Plate

gusmurr,

That approach (strut-and-tie) is always an upper-bound method (not necessarily conservative), since it involves approximating a kinematic mechanism (failure mode), and the accuracy of the solution depends on how close the solution is to the actual collapse mechanism. A simple way to put it: assuming some plastic failure mechanism will invariably overestimate the stiffness of the member (a plate in this case), which will underestimate the displacement field and therefore underestimate the internal forces involved in an average sense across the plate domain.

### RE: Flexural Strength of a Steel Plate

#### Quote (winelandv)

The more I look at it, the more it looks like a stiffened seated connection (but rotated 90 degrees).

That's the treatment that I would go with whether it's:

1) Salmon & Johnson.
2) Vinnakota which I've not yet tried.
3) Strut and tie which I've done on many occasions.

All of those work for me. I don't favor the F11 approach because, in my mind, F11 is for beams having a span to depth ratio that implies Bernoulli flexure. At 8" x 8", this is definitely not Bernoulli flexure.

Also like the stiffened seat approach, I would toss in a perpendicular plate as shown below if that would be spatially acceptable. This will stabilize your system at the point of load application which is structural stability gold. Yeah, there's still a buckling concern in the plate parallel to load but at least that's a pin-pin system. Without the perpendicular plate, your stability relies on the parallel plate cantilevering up from it's fillet welds which is a bit icky in my opinion.

### RE: Flexural Strength of a Steel Plate

#### Quote (gusmurr)

Not saying I would do it this way, but what about a steel strut-and-tie model?

I've done that a number of times and support the approach. Yeah, it's lower bound. However, if one uses reasonable proportions for the strut and tie and ignores most of the bounded plate area, as you have, it's hard to imagine going too far wrong with that.

### RE: Flexural Strength of a Steel Plate

With regard to the unstiffened plate stability, I've also been wondering how legitimate it is to use paired fillet weld in bending as shown below. I'm sure that it has some bending capacity but it strikes me as a little sketchy just the same. This paper, which I sadly do not have, seems to suggest that the right combination of fillet welds could "develop" the plate in this way for seismic applications. That's obviously encouraging.

### RE: Flexural Strength of a Steel Plate

#### Quote (JoelTXCive)

Your extended plate looks just like an extended shear tab connection. You just don't have the bolt holes and your eccentricity is a little larger.

I had that same thought. And that method may well be appropriate. There were, however, two stability differences between the shear tab and OP's condition that gave me pause:

1) The shear tab is uniformly loaded in shear which is a more stable condition than is OP's front loading.

2) Do extended shear tabs rotationally stabilize their supported beams? Or do supported beams rotationally stabilize the shear tabs which are designed to go plastic in flexure? I don't actually know the answer that but, if it's the latter, then that would negatively impact the stability of OP's situation.

### RE: Flexural Strength of a Steel Plate

To provide a flavor for the Vinnakota design aid...

### RE: Flexural Strength of a Steel Plate

centondollar, KootK,

Thanks for the responses. It seems you both have different opinions on whether that approach would be lower- or upper-bound. I agree with centondollar that traditional yield line analysis is always upper-bound. And we know that a concrete S&T model is always lower-bound. But a steel strut-and-tie model? In my mind it should be lower-bound but I can't really explain why.

KootK, do you have any references or could point me in the right direction regarding steel S&T models?

Thanks

### RE: Flexural Strength of a Steel Plate

KootK,

#### Quote (KootK)

With regard to the unstiffened plate stability, I've also been wondering how legitimate it is to use paired fillet weld in bending as shown below. I'm sure that it has some bending capacity but it strikes me as a little sketchy just the same.

In Padeye design, it is standard to use those fillets to resist any out of plane bending that the padeye sees. Granted, the padeye assembly has a tension force, but I don't see why those fillets couldn't do the same for a compressive load. Unlike a padeye, we're loading the top at one end (instead of the middle), so the question would be, how much of the plate (and by extension, the fillet welds) could we assume to engage for stability, and then is that stiff enough. My hunch would be that you would need to thicken the plate to get acceptable stiffness to stabilize the plate.

At any rate, I think our OP should slap the perpendicular plate on this assembly, do the stiffened seat, and move on to the next challenge.

### RE: Flexural Strength of a Steel Plate

#### Quote (gusmurr)

KootK, do you have any references or could point me in the right direction regarding steel S&T models?

Every steel truss that's ever been designed is a lower bound, steel, strut and tie model.

### RE: Flexural Strength of a Steel Plate

#### Quote (windlandv)

In Padeye design, it is standard to use those fillets to resist any out of plane bending that the padeye sees.

That helps alleviate my concern. I'll take it, thanks.

### RE: Flexural Strength of a Steel Plate

"And we know that a concrete S&T model is always lower-bound. But a steel strut-and-tie model? In my mind it should be lower-bound but I can't really explain why."
Concrete strut-and-tie models are upper bound solutions in practice, since they are based on assuming some stress distribution (which is analogous to assuming locations of plastic hinges and the failure mechanism) which may or may not be decided using elastic solutions as reference, and then solving the model. If the stress field is reasonably well known beforehand, and the struts and ties are located accordingly, the error may be acceptably small.

STM is different from the normal member design philosophy, in which an equilibrium configuration is found (by e.g. beam theory) and the section is designed by assuming plastic cross-sectional behavior; then, the cross-sectional resistance is always larger than the internal forces generated by the (equilibrated) external forces. In STM, the equilibrium configuration is not straightforward, and thus the orientation and placement of struts, ties and nodes is also not straightforward, and thus the error is not readily measurable.

In my opinion, it will be easier to perform an elastic membrane analysis than to find an accurate STM model for this case.

### RE: Flexural Strength of a Steel Plate

KootK:
"Every steel truss that's ever been designed is a lower bound, steel, strut and tie model. "

Steel trusses are designed by applying elastic theory and (if done by hand) a dimension reduction model that reduces members into bars. There is no guesswork involved, unlike in the STM, in which the structural resistance is idealized by assuming discrete bar-type resistance for a structural member that in reality is exposed to a more complicated stress field. Real structures (trusses and rods not included) are not collections of struts and ties, and the attempt to model them as such is not necessarily a conservative solution.

### RE: Flexural Strength of a Steel Plate

#### Quote (KootK)

Do extended shear tabs rotationally stabilize their supported beams? Or do supported beams rotationally stabilize the shear tabs which are designed to go plastic in flexure?

The section that details the buckling check for shear tabs in the Steel Construction Manual says, "This check assumes that beam is supported near the end of the plate...", so they are assuming that the beam helps prevent rotation. This means that the cantilever steel plate does not meet the fundamental assumption used for the shear tab buckling check.

However, the way that the buckling check for shear tabs works is that you use the double-coped beam procedure. This procedure has you calculate a Cb value that is at least greater than 1.84, and then use it in the section F11 check. This supports the idea that the F11 check would be valid to use, but I would be using a Cb value of 1.0 and would also likely double the unbraced length since the load is being applied above the shear center.

I would also still include the combined shear and flexure yielding check that is recommended for shear tabs (Equation 10-5):
(Vu/ΦVn)² + (Mu/ΦMn)² ≤ 1.0

Structural Engineering Software: www.structuralcentral.com

### RE: Flexural Strength of a Steel Plate

#### Quote (centondollar)

Concrete strut-and-tie models are upper bound solutions in practice...

I disagree with you utterly on that, as does every reference on strut and tie design that I've ever encountered, including the fourth paragraph of this one: Link. However, from recent experience on this thread, I know that you hold your opinion on this matter very strongly even though no one here or in the world at large seems to share that opinion. And that's okay since there's something to be said for being a rebel and trusting your own instincts.

Still, I've no interest in wasting my own time in trying persuade the unpersuadable. So, by and large, I'd like to just leave you to your opinion if you'll allow it. I will, however, attempt to answer gusmurr's question with respect to just what it is that makes yield line upper bound and STM lower bound. If you have something new and interesting (rather than old and repetitive) to share in response to that, I'll be grateful to hear about it.

### RE: Flexural Strength of a Steel Plate

The engineer is mostly in control of his or her own audacity and fearlessness when it comes to STM.

As I´ve mentioned previously, equilibrium is by definition not satisfied in a strut-and-tie model, since the continuous domain (for which some elastic solution is often available) is discretized (often very roughly) into bars. Concrete plates and beams are not collections of trusses, and modelling them as trusses will not necessarily capture a sufficient amount of the equilibrium solution (e.g., elastic 2D solution), even though such models may be useful! In short, the STM is not foolproof, unlike e.g., ordinary RC beam design in which equilibrium is used to derive design forces and the plastic section design ensures capacity against such equilibrium.

Choose the truss model poorly, and the STM design will fall short and fail.

PS. There is no tool for an ordinary engineer to estimate the ductility of a corbel, deep beam, pile cap or other complicated structure with such accuracy that a method warrants the word "safe".

### RE: Flexural Strength of a Steel Plate

#### Quote (gusmurr)

But a steel strut-and-tie model? In my mind it should be lower-bound but I can't really explain why.

I see it like this.

PLATE DESIGN WITH THE YIELD LINE METHOD

When you employ the yield line method in practice to plate elements, you're usually using the "work method" which is based on virtual work and energy balance. If you conduct a close examination of the individual panels between your work lines, you will find that they are almost never in perfect equilibrium. This is primarily what makes the method an upper bound method. Two of the available methods for dealing with that imperfect equilibrium include:

1) Most commonly, if you're reasonably close to being in equilibrium, you just add 10% to your design moments and move on. You know, "engineering".

2) Less commonly, you can utilize an alternate method known as the "equilibrium method". As the name implies, this method does enforce equilibrium. It does that by introducing nodal loads on your panels to get the job done. This method is computationally inefficient so, when it is used, it's usually used as a way of guiding the practitioner in the modification of her work method model towards one closer to the true, equilibrium solution.

MEMBER DESIGN WITH THE STRUT AND TIE METHOD

When you design with the strut and tie method, you model your system as a determinate truss within the confines of the real, physical structure. By definition, a properly analyzed, determinate truss is in equilibrium with the external loads applied to it. So you have a complete load path from the start that does satisfy equilibrium and the "design" is reduced to simply checking the capacity of the parts and pieces. This is why the strut and tie method is a lower bound method.

Selecting a strut and tie method that satisfies equilibrium is comically simple. Truly, any determinate truss arrangement will do from an ultimate limit state perspective so long as the system possess enough ductility to transition from its elastic state of stress to the plastic stress field assumed by the strut and tie model without being torn apart. That said, good strut and tie models do closely align with elastic stress fields. The reasons for that are primarily:

1) Such a model will require less ductility to transition successfully from the elastic state of stress to the plastic one and;

2) Such a model will tend to deflect less.

### RE: Flexural Strength of a Steel Plate

#### Quote (ProgrammingPE)

The section that details the buckling check for shear tabs in the Steel Construction Manual says, "This check assumes that beam is supported near the end of the plate...", so they are assuming that the beam helps prevent rotation. This means that the cantilever steel plate does not meet the fundamental assumption used for the shear tab buckling check.

Thanks for that. I'd feared that might be the case. It's a bit disconcerting in that often, when I see the extended shear tab connection used, the separate rotational restrain mechanism is not present.

#### Quote (ProgrammingPE)

However, the way that the buckling check for shear tabs works is that you use the double-coped beam procedure. This procedure has you calculate a Cb value that is at least greater than 1.84, and then use it in the section F11 check. This supports the idea that the F11 check would be valid to use, but I would be using a Cb value of 1.0 and would also likely double the unbraced length since the load is being applied above the shear center.

I see what you mean. It seems that you'd have to quadruple your unbraced length, however, given that the double coped procedure is predicated upon a pin-pin-ish buckling mode as shown below (from the Dowswell work) rather than a cantilevered model as we have here.

### RE: Flexural Strength of a Steel Plate

Interesting thoughts, KootK. Just a few notes:

The strut and tie method involves approximating a stress field with discrete truss elements. This means, in practice, that the load path is not "complete" - it is roughly approximated. Analogously, the displacement method (FEA) underestimates energy, overestimates stiffness and is thus an upper-bound method for which the solution converges (under certain conditions) when the discretization and/or element type is improved. Converging a STM model is, however, not as straightforward as converging a finite element solution.

Regarding selection of STM, it is not comically simple. The reason? Precisely what you point out: ductility is not an easily measurable quality (for RC, steel or any other material). Another reason for the difficulty in finding a proper STM model is the fact that a deep beam, corbel or pile cap is not actually a collection of struts and ties: those are only extremely idealized models that are accurate for certain materials (e.g., steel) under some restricted set of loadings.

If STM were as simple as you would like to believe, it would not be the bane of every standard writer and reinforced concrete design society, authority and researcher. It is more of an art than a science, for the reasons I have laid out in this post, and the literature will support this claim.

As a side note, the modified compression field theory (MCFT) was introduced some 30 years ago for predicting shear resistance in deep beams and thin wall slabs precisely due to the inadequacy of the STM model (currently applied in the Eurocodes) for predicting shear capacity; at large load levels (including transverse compression loading) for deep members, the STM formulas grossly overestimate the capacity. AASHTO has caught up to the development of RC shear design methods and now presents MCFT as the preferred design method for shear in RC bridge girders. The lesson: STM is not always "safe". Real structures are not collections of bars.

### RE: Flexural Strength of a Steel Plate

#### Quote (centondollar)

As a side note, the modified compression field theory (MCFT) was introduced some 30 years ago for predicting shear resistance in deep beams and thin wall slabs precisely due to the inadequacy of the STM model

I suspect that you've confused strut and tie modelling with the truss model of shear resistance. Those are two very different things. I believe MFCT addresses deficiencies with the truss model of shear resistance, not the modern strut and tie modelling approach. Read all about it here: Link.

Saying that the ancient truss model of beam shear resistance is true, modern STM is a bit like saying a "car" is a Ferrari.

### RE: Flexural Strength of a Steel Plate

#### Quote (centondollar)

It is more of an art than a science, for the reasons I have laid out in this post, and the literature will support this claim.

You've mentioned the literature supporting your positions repeatedly. Would you be so kind as to show us some of that literature so that we might digest and evaluate it for ourselves?

I have a special interest in strut and tie modelling and posses much of the seminal literature on the topic. In all of my travels you are the very first person that I've ever known to refute that STM is a lower bound method.

Are you able to point us to anything in print that would suggest that you are not entirely alone on planet earth with respect to your belief that STM is an upper bound method?

### RE: Flexural Strength of a Steel Plate

#### Quote (centondollar)

Converging a STM model is, however, not as straightforward as converging a finite element solution.

In all of the STM design examples that I've reviewed from ACI, FIB, Schlaich, numerous US DOT's, and others I have not once seen anyone attempt to "converge" an STM model. It seems to be you and you alone who feel that is necessary. I believe that an STM model requires no "converging" because:

1) STM models are in equilibrium with their external loads by definition.

2) Any approximations made in the development of STM models are well within the range of normal engineering approximation.

3) STM is a lower bound method.

Can you point to any example that we might review in which someone has attempted to "converge" and STM model? We pretty much always attempt to have our STM models mirror elastic stress distributions but that's done at the beginning of the design process and is not at all iterative.

I submit that no one is "converging" STM models because they don't require converging.

### RE: Flexural Strength of a Steel Plate

#### Quote (centondollar)

Regarding selection of STM, it is not comically simple. The reason? Precisely what you point out: ductility is not an easily measurable quality (for RC, steel or any other material).

Ductility is, in fact, very easy to measure. The trick is that you measure it in a laboratory setting. Then you use the results of that laboratory testing to inform the simplified design procedures used by engineers in practice so that they are not stuck with the onerous task of somehow trying to calculate how ductile a thing is on the fly. This process is basically what allows us to have "connections" in all of our favorite materials.

I presume that you are aware of the significant amount of testing that has be done to validate the strut and tie method of concrete design over the years? Many of the code provisions that govern strut and tie design exist, in part, to ensure that the members possess enough ductility for the STM method to be valid. This includes things like:

1) Limits on max/min strut angles.

2) Requirements for side face reinforcing in beams.

3) Requirements for transverse reinforcement to confine struts.

....

### RE: Flexural Strength of a Steel Plate

I've been putting off diving into strut & tie in any significant way because it has little utility in my work and time is scarce. But....I can't well sit idly by while people argue if it's lower bound vs upper bound, leaving me more discombobulated than when I opened the thread!

Just ordered:

Strut & Tie Model for Beginners
Structural Concrete: Srut-and-Tie Models for Unified Design

Any recommendations to add to the list?

### RE: Flexural Strength of a Steel Plate

With all this talk of STM and what have you - I found this lecture on Compatible Stress Field Method and the development of software for same to be completely fascinating. Perhaps you guys will enjoy. He also discusses many of the items in the (concrete related) discussion in this thread. And bond strength models

I'm a bit / a lot of a layman when it comes to STM stuff so I have no opinion on this other than "neat!". (I'm not affiliated with idea statica)

### RE: Flexural Strength of a Steel Plate

#### Quote (Enable)

Any recommendations to add to the list?

1) Paid. The Chen book that you purchased is actually my favorite text for a deep dive on the state of the art. Well done.

2) Free. In my opinion, the best beginner's guide on strut and tie is still the 1987 one by Schlaich that kinda got the ball rolling on modern STM: Link. Even if you didn't care about STM, this is worth a read merely as a view to the inner workings of the mind of a genius.

3) Free. The US Fedral Highway Administrations have a ton of high quality, free resources available. This is just one example of many: Link.

4) Free. PCI has a beautifully formatted little ditty on beams: Link

5) Paid. ACI has a nice, 2021 guide document out: Link

6) Paid. ACI SP-208 & SP-273 STM examples: Link

7) Paid. MPA's Concrete Center Guide to STM models (Euro): Link

### RE: Flexural Strength of a Steel Plate

To dial it back to steel for a moment, this is from Akbar Tamboli's Handbook of Steel Connections. This section was authored by Larry Muir and Bill Thornton who, as we know, are really just fringe element nobodys when it comes to steel connection design. These three paragraphs are basically the framework upon which steel connection design is built.

### RE: Flexural Strength of a Steel Plate

"In all of the STM design examples that I've reviewed from ACI, FIB, Schlaich, numerous US DOT's, and others I have not once seen anyone attempt to "converge" an STM model. It seems to be you and you alone who feel that is necessary. I believe that an STM model requires no "converging" because:"

The accuracy of STM depends on how well you estimate equilibrium with the truss (bar) model, and thus, any configuration will not suffice.

"1) STM models are in equilibrium with their external loads by definition."
This is not strictly speaking true. Imagining three bars in a deep beam loaded by mid-span point load does not mean that those bars exist and that equilibrium is achieved. The same goes for more complicated structures.

"2) Any approximations made in the development of STM models are well within the range of normal engineering approximation."
That depends entirely on how well the engineer mimics an elastic (or otherwise uniquely determined) stress field.

"3) STM is a lower bound method."
You write this, but it does not make it so.

"Can you point to any example that we might review in which someone has attempted to "converge" and STM model? We pretty much always attempt to have our STM models mirror elastic stress distributions but that's done at the beginning of the design process and is not at all iterative."
The strain energy of the STM should equal the strain energy of the unique (elastic or by other means derived) equilibrium solution.

"I submit that no one is "converging" STM models because they don't require converging."
The correct statement is that the STM, like many other things in engineering, is surrounded by misunderstanding.

### RE: Flexural Strength of a Steel Plate

"Ductility is, in fact, very easy to measure. The trick is that you measure it in a laboratory setting. Then you use the results of that laboratory testing to inform the simplified design procedures used by engineers in practice so that they are not stuck with the onerous task of somehow trying to calculate how ductile a thing is on the fly. This process is basically what allows us to have "connections" in all of our favorite materials."
The words "easy" and "experiment" do not go together, and experiments do not always provide results that can be extrapolated to more complicated situations.

"I presume that you are aware of the significant amount of testing that has be done to validate the strut and tie method of concrete design over the years? Many of the code provisions that govern strut and tie design exist, in part, to ensure that the members possess enough ductility for the STM method to be valid. This includes things like:"
I know that the STM (which, by the way, is at the core of the shear design model you posted earlier) cannot properly predict slender RC beam or plate behavior.

### RE: Flexural Strength of a Steel Plate

"To dial it back to steel for a moment, this is from Akbar Tamboli's Handbook of Steel Connections. This section was authored by Larry Muir and Bill Thornton who, as we know, are really just fringe element nobodys when it comes to steel connection design. These three paragraphs are basically the framework upon which steel connection design is built."
No gaps and tears, he writes. Satisfying equilibrium, he writes. Those are exactly requirements that many STM models (not all, mind you) do not fulfil.

I do not object to the statements you keep quoting (I too have learned this in university), but to the way you interpret them. Equilibrium (from e.g., elastic solutions) cannot be satisfied for a complicated geometry (the stress field for a simple 2D membrane in-plane stress problem can be quite complicated) and loading if one uses a strut-and-tie model with bars so few and regularly spaced that it looks like an ordinary steel truss. Capturing the equilibrium for any non-trivial problem is hard, and - if done in the scientific way - requires comparison of strain energy of the known unique (e.g., elastic) solution and the strain energy of the proposed STM.

### RE: Flexural Strength of a Steel Plate

#### Quote (gusmurr)

I imagine this approach would be lower-bound (same as for concrete), but maybe I'm wrong.

Is it really lower-bound to assume a pin restraint along the edge of your compression strut? That hypothetical strut is restrained by a spring along its edge, not a hard pin. If you deleted the pin and considered the strut a free cantilever (i.e. effective length k=2), then yeah, lower bound.

### RE: Flexural Strength of a Steel Plate

Tomfh, I agree with that completely. The pin was a pretty crude assumption. Ignoring that, it should be lower bound.

### RE: Flexural Strength of a Steel Plate

#### Quote (gusmurr)

The pin was a pretty crude assumption.

I didn't even notice the pin. That's what I get for Eng-Tipping on my phone. This is the STM model that I favor and mistakenly thought that you had originally proposed. The vertical is fixed for moment out of page (k=2) and pinned for moment in the plane of the page. I was too lazy to attempt an isometric.

### RE: Flexural Strength of a Steel Plate

#### Quote (COD)

"1) STM models are in equilibrium with their external loads by definition."
This is not strictly speaking true. Imagining three bars in a deep beam loaded by mid-span point load does not mean that those bars exist and that equilibrium is achieved.

It doesn't mean that those bars do exist and are in equilibrium but, rather, that they could exist if they need to and would be in equilibrium in that that. And that is what matters.

#### Quote (COD)

"2) Any approximations made in the development of STM models are well within the range of normal engineering approximation."
That depends entirely on how well the engineer mimics an elastic (or otherwise uniquely determined) stress field.

So do it well then. That's the engineering part of the exercise. As many test validated STM models have proven over the years, skilled engineers have little difficulty in developing STM models that perform well in this regard.

#### Quote (COD)

"3) STM is a lower bound method."
You write this, but it does not make it so.

My saying it doesn't make it so.
Everyone on planet earth who has ever written about STM (other than you) saying it... makes it highly, highly likely.

#### Quote (COD)

"Can you point to any example that we might review in which someone has attempted to "converge" and STM model? We pretty much always attempt to have our STM models mirror elastic stress distributions but that's done at the beginning of the design process and is not at all iterative."
The strain energy of the STM should equal the strain energy of the unique (elastic or by other means derived) equilibrium solution.

That is false as a lower bound method will never, by definition, have it's strain energy perfectly match that of the true, elastic solution.

I note that you have conveniently disregarded my request for any printed example of anyone ever having attempted convergence in STM modeling.

#### Quote (COD)

"I submit that no one is "converging" STM models because they don't require converging."
The correct statement is that the STM, like many other things in engineering, is surrounded by misunderstanding.

It seems to me that, when it comes to STM, the misunderstanding is predominantly yours. There exists a veritable army of researchers, authors, and practitioners who clearly feel that their understanding of the strut and tie method justifies their skillful use of it.

### RE: Flexural Strength of a Steel Plate

#### Quote (COD)

"Ductility is, in fact, very easy to measure. The trick is that you measure it in a laboratory setting. Then you use the results of that laboratory testing to inform the simplified design procedures used by engineers in practice so that they are not stuck with the onerous task of somehow trying to calculate how ductile a thing is on the fly. This process is basically what allows us to have "connections" in all of our favorite materials."
The words "easy" and "experiment" do not go together, and experiments do not always provide results that can be extrapolated to more complicated situations.

So what, then? Are we just going to stop trusting industry research to inform design practice altogether? How are we to design anything at all if that is to be the case? Rationally, industry testing should be the gold standard when it comes to informing design practice.

#### Quote (COD)

"I presume that you are aware of the significant amount of testing that has be done to validate the strut and tie method of concrete design over the years? Many of the code provisions that govern strut and tie design exist, in part, to ensure that the members possess enough ductility for the STM method to be valid. This includes things like:"
I know that the STM (which, by the way, is at the core of the shear design model you posted earlier) cannot properly predict slender RC beam or plate behavior.

STM informs the shear truss model. However, the shear truss model is far cry from being as sophisticated and complete a design method as is STM. As such, it is invalid for you to claim that, because we're moving away from the shear truss model, STM is somehow flawed. In that, it is your logic that is flawed.

### RE: Flexural Strength of a Steel Plate

Thanks for the recommendations KootK much appreciated.

### RE: Flexural Strength of a Steel Plate

#### Quote (COD)

No gaps and tears, he writes. Satisfying equilibrium, he writes. Those are exactly requirements that many STM models (not all, mind you) do not fulfil.

I submit that those requirement would indeed be met by virtually all of the rather pedestrian STM's that I've seen you object to over the last couple of months. In my opinion, you are spuriously deterring people here from attempting to make use of strut and tie design procedures in situations where they are widely deemed to be appropriate.

#### Quote (COD)

I do not object to the statements you keep quoting (I too have learned this in university), but to the way you interpret them. Equilibrium (from e.g., elastic solutions) cannot be satisfied for a complicated geometry (the stress field for a simple 2D membrane in-plane stress problem can be quite complicated) and loading if one uses a strut-and-tie model with bars so few and regularly spaced that it looks like an ordinary steel truss.

In this thread, you have objected to the use of a two member truss inside a square plate made of one of the most ductile materials used in construction. A two member truss! You consider that to be too "complicated" of a geometry for a skilled engineer to prosecute successfully? What could possibly be simpler than this situation?

### RE: Flexural Strength of a Steel Plate

I have definitely used the 'ignore parts of the plate to treat it as a truss with simple buckling assumptions' method for things like this before.

There could be arguments about ways in which similar methods could be non-conservative because there are considerations like internal force transfer that they may not account for, or that there are inherent stress/strain compatibility assumptions that may not be valid. It doesn't seem like a thing here. However, this isn't the same as being an upper bound solution.

Strut and tie, or this type of ad-hoc lite version of strut and tie seems to be pretty definitively not upper bound. An upper bound method will have potential solutions for capacity that are at the critical load or above it. It shows an upper bound for the potential capacity solution. The critical load is the minimum possible solution. No valid solution or model would have a lower capacity than the collapse load. It seems pretty trivial to prove that strut and tie is not an upper bound theorem and that you can make valid models that are below the failure load. This is also definitely the intent of the system.

Not being an upper bound methodology doesn't mean that there aren't ways to get non-conservative results from a lower bound methodology. There could be limits or assumptions that are invalid in certain situations, it could just be non-conservative for various reasons, and all sorts of other fun stuff. It's just not an upper bound theory.

### RE: Flexural Strength of a Steel Plate

Can someone please attach or summarise the F11 buckling method? Does it explicitly cover a situation like this, with a slender cantilever plate?

### RE: Flexural Strength of a Steel Plate

Here you go Tomfh. The section is meant to apply, generically, to beam-like members made from solid, rectangular sections.

Thanks!

### RE: Flexural Strength of a Steel Plate

you could simply trying a second solution
Box shape will always be stronger than flat plate
or latterly stiffened plates

### RE: Flexural Strength of a Steel Plate

I was looking at a plate last week by hand and instinctively used 1.5S for Z. I then looked it up and saw that it is 1.6S in the Spec. What not just say 1.5 for rectangular and 1.7 for rounds? This seems like an odd place to be stingy with paper and spec provisions. It also makes explaining things to a freshly-minted engineer more difficult. First I try to show them the logic for the derivation of a Spec provision, and then I can't explain what they put in the Spec, even for something as simple as S and Z of a rectangular bar.

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