## Partial Steel Beam Reinforcement Anchor Force

## Partial Steel Beam Reinforcement Anchor Force

(OP)

Hello,

I'm working on a steel beam reinforcement consisting of a new W-shaped beam welded below an existing W-shaped girder, which looks like this:

I'm trying to determine the anchorage force and extension required for partial reinforcement. According to my reference below from the Canadian Steel Handbook, the formula provided consist of the area of the reinforcement times the distance from the centroid of the reinforcement to the centroid of the entire combined section, which is the same variable (Q) used in shear flow calculations. My question is, would this formula still apply to my W-shaped reinforcement? Or is it limited to cover plates?

I'm concerned that there's an implicit assumption that the plate has uniform stress if assumed to be thin, and with the W-shaped reinforcement, there is a considerable stress distribution across the depth of the section. Any thoughts on this? Thanks.

I'm working on a steel beam reinforcement consisting of a new W-shaped beam welded below an existing W-shaped girder, which looks like this:

I'm trying to determine the anchorage force and extension required for partial reinforcement. According to my reference below from the Canadian Steel Handbook, the formula provided consist of the area of the reinforcement times the distance from the centroid of the reinforcement to the centroid of the entire combined section, which is the same variable (Q) used in shear flow calculations. My question is, would this formula still apply to my W-shaped reinforcement? Or is it limited to cover plates?

I'm concerned that there's an implicit assumption that the plate has uniform stress if assumed to be thin, and with the W-shaped reinforcement, there is a considerable stress distribution across the depth of the section. Any thoughts on this? Thanks.

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

Thanks for the reply.

I agree shear flow is not limited to cover plates and yes it's VQ/I, but the anchorage force formula provided in my reference is MQ/I.

Do you think there is still no difference and the anchorage force formula is also applicable to W-shaped reinforcements considering that the cover plate is a relatively flat element, and the W-shaped reinforcement has depth in it?

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

I feel that your instincts with that are sound. My understanding is that the final deformation state in the reinforcement member may be viewed as having two components:

1) Transverse load inducing a reinforcement member centroidal curvature matching that of the composite member. This generates a vertical force demand at the connection which VQ/IT does not cover and;

2) Axial load inducing a reinforcement centroidal stretching that has the effect of offsetting the centroidal curvature to larger radii, thus producing true composite action. This leads to a shear slip tendency which creates a horizontal force demand in the welds and for which we do the VQ/It.

The weld demand described in #1 is insignificant for reinforcing members with small moments of inertia. With increasing moments of inertia, however, the effect becomes more pronounced (it may still be insignificant relative to VQ/I though). For this reason, and for general stability, I feel that it is a good practice to install partial height stiffeners as shown below along with some concentrated, local welding at that location. That, in addition to the MQ/I stuff.

## RE: Partial Steel Beam Reinforcement Anchor Force

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

The cut-off point is important, however. The added W section requires a certain distance to become fully effective. Axial stress at the end is zero.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

If you deny the existence of the force shown in blue below, then how would you put the differential element back into equilibrium with respect to vertical force? For convenience, I've pretended that all of the flexural tension resides in the lower flange. I don't believe that compromises the argument but, if I'm wrong, I'll be grateful to hear about it.

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

There needs to be shear in the reinforcement piece regardless of whether or not the original beam is adequate for shear on its own. And, since the shear in the reinforcement has to get in and out of that piece

somehow, then I feel that the flange bending & weld tension issues exist regardless of the shear capacity of the original beam.Do you know how to quantify the length of this "transition zone" such that its capacity can be evaluated? I don't other than to maybe ballpark it as 2X the depth of the reinforcing piece or something like that. Given that uncertainty, just throwing in a stiffener set to make sure that the issue is resolved convincingly seems entirely reasonable to me.

I typically throw in a stiffener to serve as a stabilizer as shown below, independent of the issue that we're discussing here and simply as a matter of good practice. The fact that the stiffener can also act to relieve cross flange bending is effectively just a negligible cost bonus.

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

I am still in the process of thinking about it. I believe your model is wrong. Consider the reinforcement beam by itself. Apply an eccentric force to both ends of the beam, which is the horizontal weld shear. That is tantamount to a beam with an equal and opposite moment at each end. Shear throughout the span is zero.

That is where I am at the 'moment' (if you'll pardon the expression).

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

I've considered it:

1) The MQ/I welds at the ends of the reinforcing certainly feel like a concentrated moment in the way that you've suggested. However, the welds between the ends are normally introducing additional shear as well which implies a shear force that is varying along the length of the member. And that returns us to my differential element FBD.

2) In any normal built up beam, I feel that this logic chain applies:

a) One requires the new, low flange force to vary along the length of the beam.

b) [a] dictates that the reinforcing member horizontal shear increases along the length of the beam.

c) [b] dictates that the reinforcing member vertical shear increases along the length of the beam since vertical and horizontal shear are everywhere and always complimentary.

## RE: Partial Steel Beam Reinforcement Anchor Force

Regarding the del_V in the free body diagram, and assuming you're talking about a location away from the end of the reinforcement (away from the disturbed region), the balancing force is the external load that causes the del_V, or rather the portion of the external load that is supported by the reinforcement since del_V is defined over the reinforcement depth only. The welds need to transfer this if you don't assume direct bearing by the main beam on the reinforcement. I would design the welds for this.

The welds alternatively need to 'hang' a load if applied at the bottom flange.

In either case, the del_V load is presumably modest or you'd need stiffeners everywhere.

At the end of the reinforcement, del_V isn't only due to external load as already identified in the discussion above. I wonder though whether a single stiffener at the end of the reinforcement is enough. I think that a pair at say 2/3*D_reinf would be better but not sure if absolutely required. Or maybe the spacing should also relate to the flange width.

## RE: Partial Steel Beam Reinforcement Anchor Force

It's needed if you want to be able to accommodate uniform loads and any moment diagram that isn't linear varying.

As I mentioned previously, I see it as being of potential significance where:

1) The reinforcing possesses a significant moment of inertia in its own right and;

2) Even then probably only at the ends where the reinforcement shear must migrate into the original member.

It's often a moot point between the ends because many beams of this sort will be loaded above the reinforcing member, putting the joint in compression as you mentioned. And, where loading would be at the bottom of the beam, it's hard to imagine designers forgetting to design the welds for direct tension.

Whether one wants additional stiffeners or not, I can't see much logic in omitting the stiffeners at the ends. Those stiffeners have awesomely direct load transfer from the reinforcing web straight into the existing beam web, sans flange bending. Stiffeners located anywhere else would involve some reliance on flange bending around the stiffeners and non-uniform stresses in the welds.

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

I disagree. Shear and moment come as a compatible set that can't really be separated. Varying reinforcement shear absolutely will be accompanied by varying global shear AND varying moment. To say that one is the "cause" and the other is not is spurious in my opinion.

I agree, the transition surely is a disturbed region. What do you propose for dealing with that? My thinking has been:

1) Do the stiffener as proposed so that, at worst, you might rupture some welds at the end of the reinforcement but not rip the thing clean off.

2) Design the MQ/I welds to

alsoperform the "hanger" job at the end of the beam so reduce the odds of rupturing any welds. Luckily, this will naturally be a location of concentrated welding.3) As you said, hope that flange flexibility helps to relive any stresses arising from compatibility. This would tend to be more true further from the stiffener I think.

## RE: Partial Steel Beam Reinforcement Anchor Force

1) No matter how you set out the reinforcement, the horizontal shear forces induced by the welds will create a force set that generates no vertical shear in the reinforcement at all.

2) Because of #1, the shear carried in the reinforcement will be that which you would expect if the original beam and reinforcement beam were acting

non-compositely: V_reinf = V_overall x I_reinf / ( I_reinf + I_original ).I find these results surprising and am curious to know if others see this differently.

## RE: Partial Steel Beam Reinforcement Anchor Force

I agree with #1, but not with #2. Gravity load is required to counter the upward curvature caused by the shear induced by the weld. Additional load is required to bring the reinforcing beam down to the compatible deflection.

I have not really thought it through thoroughly, but I suspect the shear to be taken by the stiffeners would be approximately equal to V

_{1}* h_{r}.t_{r}/(h.t + h_{r}.t_{r}) where V_{1}is the shear at the end of the reinforcing beam and the product of h*t represents web area of the upper beam and the subscript 'r' refers to the reinforcing beam.EDIT: A more accurate method would be to calculate the deflection of the composite beam, then the reinforcing beam, keeping in mind the correction for curvature due to weld.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

Right, but the shear proposed below would be be that associated with

exactlythat gravity load. Part of it would nullify the upwards curvature due to the weld shear and the remainder would push the reinforcing down to the appropriate elevation in the deflected, composite beam. If all of the proposed load went towards downwards curvature, starting from level, then it would push the reinforcing down too far and composite behavior would not be captured.V_reinf = V_overall x I_reinf / ( I_reinf + I_original )

## RE: Partial Steel Beam Reinforcement Anchor Force

V_reinf = V_overall x I_reinf / ( I_reinf + I_original )

I_composite is not mentioned, but it determines the actual deflection of the beam.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

I_comp does determine the deflection of the beam but

notthe total shear in each of the two parts. That's why I said it was surprising.## RE: Partial Steel Beam Reinforcement Anchor Force

That's the case, but I was responding to your statement quoted below, where [b] might follow from [a] but not necessarily. 'Cause' and 'effect' are ok in this context IMO because the external load causes the shear to change, at least in Australian vocabulary where we call shear force/bending moment/etc 'design action effects', ie the effects caused by design actions (loads).

"a) One requires the new, low flange force to vary along the length of the beam.b) [a] dictates that the reinforcing member horizontal shear increases along the length of the beam."

On to the more interesting discussion above, I admit I've lost track of whether it's in relation to the B-region or the D-region in places. I'm also going to hedge by saying I too haven't come to a satisfactory conclusion in my own mind but think the stiffener force will be related to several stiffnesses that would be complex to assess. At the termination of the reinforcement, the overall bending moment is essentially the same on the unreinforced and reinforced side, but the Bernoulli curvature M/EI should be quite different. Assuming there's enough flexibility or ductility to avoid failure, the stiffness of the connection will determine how quickly the moment is split between the main beam and the reinforcement (over what length unequal curvature occurs between main beam and reinforcement), and therefore the magnitude of the forces involved. Coping the top flange of the reinforcement near the stiffener is something I'm toying with to introduce flexibility and move the force couple further apart.

## RE: Partial Steel Beam Reinforcement Anchor Force

Edit: Or maybe chop out most of the reinf top flange width at the end, do away with the stiffeners, and just have web-to-web welds for a length aka the reinf section is an inverted tee at the end.

## RE: Partial Steel Beam Reinforcement Anchor Force

That would be a pretty great detail from an engineer's perspective. I'd also thought of partial coping as an attractive solution. Of course, as is probably the case with most engineers, I'd back off in practice to just the stiffener for fear of being considered -- and possibly truly being -- ridiculous.

## RE: Partial Steel Beam Reinforcement Anchor Force

Right up there with fear of major collapse as a motive for engineer behaviour. I'm not being sarcastic, even if it sounds like it,

Which leads me to this detail. Less welding overall (replaced by gas cutting), no additional overhead welding, doesn't look ridiculous. The reason for going down this track is I'm still not sure whether the concentrated force that transverse stiffeners enable doesn't also have a downside. This structure matches the increase in section properties to the St Venant principle so there isn't any tendency for weld forces to deviate from weld capacity. Maybe St Venant means we don't need to cut out the lower left corner of the reinforcement.

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

For uniform load on composite beam,

Mr = T.h/2 = V.Q.L.h/8Ic @ midspan varying to zero at supports (parabolic shape)

To balance Mr, we need Wr acting down on lower beam, which it gets from the upper beam bearing on it.

Wr*L/8 = V.Q.L.h/8Ic

Wr = V.Q.h/Ic

Vr = Wr/2 = V.Q.h/2Ic---------------------------------------------------(1)

Vr is reaction each end carried by stiffeners into upper beam.

Wr is total (virtual) load acting on reinf. beam

V is applied shear on composite beam @ each end of reinf. beam

Q is statical moment of reinf. beam.

h is height of reinf. beam.

Ic is moment of inertia of composite section.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

1) This statement is no longer true in the sense that I'd intended it. Where the reinforcement would be set out asymmetrically, or the load asymmetrical, the reinforcement end moments may be unequal and thus produce a shear in the reinforcement which complicates / neuters my efforts at simplification.

2) Recently, here on eng-tips, I've seen a number of folks advocate the use of W-shapes as reinforcement rather than WT-shapes. The logic being that top side flange welding is easier than underside T-stem welding. But the issues raised here complicate that somewhat.

3) Even with traditional, WT-shape reinforcing, I still see a problem. It seems reasonable to me that the MQ/I concentrated weld should also resist the reinforcing end moments which is something that I've never seen discussed in print. To varying extent, this issue would afflict all forms of reinforcement that possess significant, independent Ix's. So flange plates, flange channels, rods, and small angles would be relatively insensitive to this phenomenon. But stacked wide flanges, WT-shapes, large angles, and channels welded to girder webs should consider it.

## RE: Partial Steel Beam Reinforcement Anchor Force

The anchorage force is the sum of horizontal shear from the end of the reinforcement beam to the section under consideration. It is zero at each end and maximum at the point of zero shear. It can be shown to be parabolic if the load is uniform (see below).

If the weld shear were the only effect at play, the reinforcement beam would arch upward, but the uniform load acting downward on the original beam prevents upward arching. In effect, the reinf. beam uses part of the applied load to counteract upward arching.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

It really isn't an approximation. It can't be off by a significant margin. Consider the performance of a built up beam with a single point load. The percentage of P taken by the reinforcement beam is dependent on Q, h and Ic. The shear and moment diagrams are illustrated below.

If it works for a single load, then, by the principle of superposition, it works for any combination of point loads.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

You can use the added moment to determine the length of the added member (which you should have done anyway) and use that moment to calculate the added q and use that q to determine reactions. I've always extended the add on section for a depth beyond the point of cutoff, but there is no need for that.Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

With regard to the parabolic approximation that I mentioned, your last couple of posts have been addressing something entirely different from what I'd intended. My intention is shown below.

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

Tension from weld to reinf. beam = T = V*Q/Ic *L/2 *Edit: i2 should read 1/2 = V.Q.L/4Ic

In the above, T is the total tension applied to the reinf. beam by the weld each side of midspan.

It is the area under the shear diagram of half the reinf. beam. Its units are force.

In the above sketch, Vr is the calculated shear at the end of the reinf. beam. Its units are force.

KootK is not mistaken; shear stress varies linearly over the reinf. beam. It has a value of Vr at each end and tapers linearly to 0 at midspan.

In the above sketch, the red shading indicates the linear variation of shear stress. The value at each end is Vr in units of force.

The suggestion of a value of MQ/I at each end is incorrect. The tension at any point is the area under the shear diagram (summation of shear force) between the end of the beam and the point under consideration. It could also be called the anchorage force and is parabolic in shape when the applied load is uniform.

As noted, the moment curve is similar, varying only by a factor of h/2.

I hope this clears up any misunderstanding.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

Again, I am not concerned with the variation of stress

along the length of the beam. Rather, I am concerned with the variation of stress over theheightof the reinforcing member which is parabolic and not linear under all conditions of load. In your calculations, are you not calculating the shear in the reinforcement at its ends as the area under the relevant portion of the diagram below? If so, that would be integration over a parabolic function rather than a linear one, would it not? Or are you attempting to calculate Vr as the sum of the area under the VQ/I diagram over the shear span (incorrect in my opinion)?I strongly disagree for the case of partial reinforcement. What is it you think MQ/I does if not this?

That is true but, in the case of partial reinforcement, MQ/I is precisely the "missing" portion of VQ/I that would have been present if the reinforcing extended the full length of the beam. These two statements are therefore equivalent:

1) The total anchorage force is the summation of VQ/I over the shear span of the ORIGINAL member imagining that the partial reinforcing had run the full length and;

2) The total anchorage force is the summation of MQ/I + VQ/I taken over the shear span of the REINFORCING member.

## RE: Partial Steel Beam Reinforcement Anchor Force

Because I'm a glutton for punishment, and because I want to do my share of the heavy lifting, I went ahead and attempted the integration. My result is shown below. As I predicted, it's slightly more than BAret's [0.14 * V]. It's also a cubic function which makes sense given that it's the integration of a parabolic function.

Below, I've also run the case where the roles of the main member and the reinforcing are reversed. The value that it yields is [1.00 - 0.156 = 0.844] which, again, jives with one's intuition.

## RE: Partial Steel Beam Reinforcement Anchor Force

_{x}Q/Ic is the horizontal shear per unit length in the pair of welds between upper and lower beams. (V_{x}Q/Ic)*L/2*1/2 is the total anchorage force at midspan, assuming uniform load. Anchorage force at the end of the lower beam is zero. It increases at a decreasing rate toward midspan.Integrating the triangle shaded red (or the area under the curve) is not relevant to the question at hand.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

If that is the case, then I'm afraid that I don't understand your methodology. Can you post some kind of sketch to show how the horizontal shear can be used in this way to arrive at the correct value of the vertical shear in the reinforcing?

I disagree strongly with that assertion. As far as determining the shear in the reinforcing member goes, I consider the integration of the VQ/I function over the height of the reinforcing to be pretty much the gold standard as it's classic textbook stuff rather than our own handiwork.

Below, I've compared my latest equation on the left with your equation on the right, as I understand it. I've also reversed the variables so as to determine the shear carried by each member of the assembly. My values add up to 1.00 as one would expect given that all of the global shear must me carried somewhere. Your values add up to 0.563 which would seem to leave 43.7% of the global shear unaccounted for. Are you able to reconcile that somehow?

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

Edit: I botched the description of the horizontal shear flow, it's not the area of the parabolic curve but rather the value of tau at the interface which is what you are concerned with for the attachment determination. Not sure why you are trying to get the area of the tau curve as that is not what the shear flow is, the shear flow is tau at the specific elevation of the fasteners x the length between fasteners. The anchorage force is the sum of the dx shear flow again at the fastener interface where integral V*Q/I*b dx = Q/I*b integral V dx where integral V dx is equal to moment, M(removed the image will redo the sketch and add it back)

Now with a reinforcing section of significant depth you have shear lag to contend with which should probably be taken into account when looking at the anchorage length.

edit2: Image showing the shear lag

My Personal Open Source Structural Applications:

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## RE: Partial Steel Beam Reinforcement Anchor Force

Because much of this thread is concerned with finding the vertical shear in the reinforcement and figuring out how that shear gets into, and more importantly

out of, the reinforcing member when it is partial length. As discussed above, this requirement is inadditionthe the horizontal shear requirement (VQ/I).BAretired and I actually discussed many of these same concepts with respect to the nature [MQ/I] in this previous, very informative thread. Folks following along may find that of interest. My perspective is/was summarized reasonably well in the posts surrounding the sketch below, reproduced from the previous thread.

## RE: Partial Steel Beam Reinforcement Anchor Force

1) MQ/I or;

2) VQ/I(partial) + MQ/I or;

3) VQ/I(full) or;

4) VQ/I(partial) + MQ/I + Shear Lag or;

5) VQ/I(full) + Shear Lag.

6) The old school practice of welding for [As x Fy] of the reinforcing at the ends, either in isolation or in combination with any of #1 through #4.

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

I'm afraid not. I do very much like the idea of baby stepping through things in order to isolate the fundamental nature of our differences though.

## RE: Partial Steel Beam Reinforcement Anchor Force

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## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

assumes rectangular main and reinforcing section and uniform tau across the width.

edit:missed a B in the Q formulaMy Personal Open Source Structural Applications:

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## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

Are we headed towards an "agree to disagree" here? I previously pointed out what I believe to a number of numerical inconsistencies with your proposed reinforcing beam reaction estimates. Are you not interested in addressing any of that? It was my hope that, in exploring those things, we could reconcile our differences. I do agree with everything shown in your latest sketch other than the conclusion shown in the last line of it.

## RE: Partial Steel Beam Reinforcement Anchor Force

3bh/2. This accounts for the fact that the maximum shear is 1.5 times the average shear.If you agree with the second last line, i.e. the moment in the reinf. beam, then it is a mystery to me why you don't agree with the last line.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

For others, it may not be so rare, so it is worthwhile getting it right.## RE: Partial Steel Beam Reinforcement Anchor Force

BA I believe this statement is incorrect, VQ/I is the horizontal shear stress at a specific slice in the overall section. You need to integrate once more to capture the full parabolic tau curve to yield total horizontal shear which also equals the total vertical shear.

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## RE: Partial Steel Beam Reinforcement Anchor Force

where MQ/I address the red hot spots in this diagram:

My Personal Open Source Structural Applications:

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## RE: Partial Steel Beam Reinforcement Anchor Force

I was thinking the exact opposite. As far as I know, you've not explained the physical reasoning to support the logical step represented by going from your second to last line to the last line. Maybe you did explain that somewhere above and I just missed it somehow, I don't know. Would you humor me and, perhaps for the second time:

1) Explain the physical reasoning behind that step in words to the best of your ability.

2) Post a free body diagram that has the value shown clouded below shown on it someplace, in equilibrium with the rest of the forces in play? I've attempted this for you below but am sure that I've misunderstood as:

a) the free body diagram neglects some of the forces that are in play and;

b) the model would produce spurious numerical results as I mentioned previously. In the case with the stacked rectangles, it would predict a value other than 50% of the vertical shear in each piece with the joint located at mid-height (4"). Clearly that's not right.

c) it's not in vertical equilibrium without the transverse load applied to the top that we've long been discussing.

## RE: Partial Steel Beam Reinforcement Anchor Force

That's right. Your FEM output helps to illustrate this stuff nicely and I'm grateful for that. The items highlighted in yellow below indicate the values that I'm interested in determining, sketched on my understanding of what a complete FBD would be. It's been an exciting thread for me personally as I previously held a misconception about the distribution of vertical shear and didn't even realize that the end moments were a thing.

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

Anyway, VQ/Ib (#/in

^{2}or N/mm^{2}) is the horizontal shear stress through any section of the beam. In particular, it is the horizontal shear stress at every section other than the weld line. On the weld line, the horizontal shear per unit of length (#/" or N/mm) is VQ/I.VQ/I causes a net compression and tension in the original and reinforcement beam respectively. It is applied at the edge of each beam and, in both cases, causes upward arching, assuming no other forces acting.

The moment from the weld force varies linearly from zero at the ends to a maximum at the load point. The reinf. beam reactions are consistent with the moment diagram, which you apparently agree with. I have shown the anchorage force and the resulting moment in my diagram above, also the reaction consistent with that moment.

RI consider it elementary statics and wonder what all the fuss is about._{left}= M_{max}/a; R_{right}= M_{max}/b.I think I just did, but can expand on it if needed.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

1) The value of the concentrated moment is zero. It does not exist.

2) The upper beam is lifting under the weld shear. It does contribute dead weight to the reinf. beam, but all the reinf. beam needs to balance the weld shear moment is the point load.

3) The hanger force is what I am calling the reinforcement beam reaction.

4) The arrows on the sketch representing horizontal weld force gives the impression that it is acting at the ends of the beam, but it is zero at both ends and maximum at the point load P.BA

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

If I've understood correctly, the moment diagram you've drawn is due to the distributed horizontal shear force (shear flow) applied to the top edge of the reinforcement section. This is eccentric to the reinforcement centroid so could be considered a concentric distributed axial force and distributed torque. The moment diagram is the integration of the torque.

In that case, there is no shear force associated with the bending moment and no vertical reaction at the ends. It is the distributed case of a beam with equal/opposite end moments.

## RE: Partial Steel Beam Reinforcement Anchor Force

Any chance that strikes a chord with you as a possible point of reconciliation?

## RE: Partial Steel Beam Reinforcement Anchor Force

Since the NA of the composite section may be in the section above, would that not cause the centre of tension to move up towards the upper member, and not towards the outer edge? or, am I misunderstanding something here?

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

The same but this time with fixed restraint on the corner nodes:

Edit:had the load direction reversedMy Personal Open Source Structural Applications:

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## RE: Partial Steel Beam Reinforcement Anchor Force

- This image shows vertical stress?

- Span and reinf length?

- Loading?

- The two cross sections?

- Is there a stiffener at the reinf termination?

- Pin-pin or pin-roller?

- What are the forces at the red hotspot and the blue balloon near the reinf termination point? Looks like a tension hanger force compression and balloon (a couple aka concentrated moment).

Thanks.

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

- Span and reinf length? 10ft overall span with reinforcement from 3ft to 7ft (modeled as two shells but ultimately share mesh interface)

- Loading? 1kip/ft vertical load applied at the top of the main shell

- The two cross sections?b=6" h=12" main shell, b=6" h=6" reinf. shell

- Is there a stiffener at the reinf termination?nope

- Pin-pin or pin-roller?pin roller, supports located at mid depth of the main shell left side pin xyz right side pin yz roller x

- What are the forces at the red hotspot and the blue balloon near the reinf termination point? Looks like a tension hanger force compression and balloon (a couple aka concentrated moment).dumped the file after the screencaps, only takes a minute or two to put together so can pull this info sometime later.

And in this image, is the red hot spot actually a hot spot or just M*y/I bending stress based on the shallower section properties? What is the red stress magnitude and what is the blue stress at the top face directly above?actually not 100% on this will recheck along with the above

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## RE: Partial Steel Beam Reinforcement Anchor Force

Any chance of a vertical direct stress plot? I am on a phone without a real computer for a while.

## RE: Partial Steel Beam Reinforcement Anchor Force

Not as I'm envisioning it dik. What follows should clear that aspect up.

Here's my modified version of what I suspect BA's method is:

1) Choose to work with a single point load for two reasons:

a) Since the proportion of the vertical shear going to the reinforcement will be agnostic to the loading, pick a load that simplifies things knowing that the results will translate fully to other situations. Moreover, all loads can be envisioned as a collection of point loads if desired.

b) A single concentrated load is a fine choice to work with because it results in no transvers load on the reinforcement member along the shear spans. And that simplifies the free body diagram that will come next.

2) Use the free body diagram below which includes all of the loads present on A full length reinforcement half span to work out the reinforcement end shear and end reaction.

## RE: Partial Steel Beam Reinforcement Anchor Force

Thanks...

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

The following approximate procedure is not exact, but is thought to be conservative. Anything more exact is likely not worth the calculation effort.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

So all along I've been thinking to myself "easy enough for a rectangle but what about wide flange reinforcing? Meh, we'll leave that for later".

Later is now. I did the algebra and this works out fairly simply for

anyreinforcing cross section using only properties that a designer will have on hand after the basic reinforcing design is complete:1) Ic = composite moment of inertia.

2) Q = the usual value.

3) Sr = the section modulus of the reinforcing alone.

4) Ar = the area of the reinforcing alone.

It reduces to the equation yellow below which would be multiplied by the overall beam shear to get the shear in the reinforcing and hanger load.

The equation should be valid for any shape: rectangle, wide flange, tee, flat plate...

EDIT: I think this will only work for vertically symmetric reinforcement as shown. So not tee's just yet...

## RE: Partial Steel Beam Reinforcement Anchor Force

The suggested approximate formula yields V

_{ratio}of 2/8 = 0.25 (very conservative)If: d = 8", h = 4", b = 1"

then, by the KootK formula, V

_{ratio}= 0.5The suggested approximate formula yields V

_{ratio}of 4/8 = 0.50 (perfect)BA

## RE: Partial Steel Beam Reinforcement Anchor Force

1) I_c = composite moment of inertia.

2) Q = first moment of area about the weld line.

3) S_rb = the section modulus of the reinforcing alone referenced to the bottom of the section.

4) A_r = the area of the reinforcing alone.

5) y_cbc = distance from centroid of composite section to the bottom of the reinforcing.

5) y_cbr = distance from centroid of reinforcing alone to bottom of reinforcing.

6) y_ctr = distance from centroid of reinforcing alone to top of reinforcing.

## RE: Partial Steel Beam Reinforcement Anchor Force

Ir = I of reinforcement about own centroid

Qr = Q of reinforcement when considered as part of composite section (the usual Q value)

Cr = distance from centroid of reinforcement to interface with the main section

Ic = I of the composite section

Explanation:

The trapezoidal longitudinal stress on the reinforcement is separated into an axial (force) component acting at the centroid of the reinforcement, and a moment component.

The axial force component = Qr/Ic * Mtotal.

The moment component = Ir/Ic * Mtotal.

Draw a free body diagram of the reinforcement similar to KootK, including the shear flow (= the change in axial force over the length of the FBD) and the equal/opposite reinforcement shear force at each end.

Sum moments to zero and rearrange to find Vr/Vtotal.

Edit: this is the same as KootK's equation (but I think neater and provides more insight). KootK's Srb terms are a convoluted way of writing Ir. This would be because KootK approached from a different direction (stress vs force/moment).

## RE: Partial Steel Beam Reinforcement Anchor Force

Edit: N1 = (M1,total).Q/Ic, not calculated from just the reinf moment.

## RE: Partial Steel Beam Reinforcement Anchor Force

It's beautiful steveh49. I've compared it against my stuff both numerically and algebraically and it checks out 100%. The algebra took a little doing. I feel like I'm in the 11th grade all over again this week. I'll noodle on your latest question over the weekend. For now, what does the term "St. Venant length" mean to you? I'm familiar with St. Venant's principal but not St.Venant's length. Is it a particular value or multiple of the member depth? Or just a concept, that of being far enough away from disturbances that the net effect is equivalent to a statically equivalent setup without the disturbances.

## RE: Partial Steel Beam Reinforcement Anchor Force

You'll probably kick yourself at how close you were to the simplified form. In your 8Apr 20:04 post, replace Q/Ar with y_crc (distance between composite section centroid and reinforcement centroid) then rearrange the Srb terms to Srb(y_cbc - y_crc) = Srb.y_cbr = Ir.

What I call the St Venant length is from the St Venant principle, approximately the beam depth. But no fixed number which is why I left it at St V length. There will probably be different opinions. Celt83's stress plots give an indication for the rectangular cross section.

## RE: Partial Steel Beam Reinforcement Anchor Force

Firstly, KootK alerted me to the fact that there was a reaction of the reinforcing beam which had to be taken by a hanger from the original beam. I didn't see it for a long time, but he was right.

Secondly, even after I did see it, I calculated only part of the reaction, namely the reaction from the axial force component, neglecting the moment component. Eventually, I could see that my answer was wrong because the reaction of the two beams did not add up to the reaction of the composite beam.

KootK came up with the correct answer, but steveh49 provided a simplification which I am embarrassed to admit, I missed entirely, namely:

Vreinf / Vtotal = (Ir + Qr * Cr) / IcIr = I of reinforcement about own centroid

Qr = Q of reinforcement when considered as part of composite section (the usual Q value)

Cr = distance from centroid of reinforcement to interface with the main section

Ic = I of the composite section

It is beautiful in its simplicity.

So, thank you KootK and Steveh49.

BA

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik

## RE: Partial Steel Beam Reinforcement Anchor Force

## RE: Partial Steel Beam Reinforcement Anchor Force

I really appreciate the effort that went into this discussion and into deriving a simplified formula for everyone.

I suppose my next question would be the effect of preloads prior to reinforcing...but I would leave that for another time as I have decided to shore and unload the girder before reinforcing. Thanks again guys much appreciated.

## RE: Partial Steel Beam Reinforcement Anchor Force

-Dik