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# Partial Steel Beam Reinforcement Anchor Force7

## 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.

### RE: Partial Steel Beam Reinforcement Anchor Force

Shear flow is VQ/I. It is not limited to cover plates.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

(OP)

#### Quote (BAretired)

Shear flow is VQ/I. It is not limited to cover plates.

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

The anchorage force is equal to the total axial force in the reinforcing member, MQ/I. See below.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (Baffled_Engineer)

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?

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

Does this help? Depends on your loading to determine the cut-off point, and I'd extend the section past the cut-off point. Modify the shear values to determine the intermittent weld. Check to see you are happy with the numbers.

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

I see no need for the stiffeners recommended by KootK. They accomplish nothing. Stress in the weld is parallel to the neutral axis of the section. Curvature of the added W section is a natural result of the eccentric axial load applied. There is no tendency for cross flange bending, unless perhaps from the heat of the weld, but I don't think that is what we are talking about.

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

BART: Agree, but it is nice to see him using partial depth stiffeners...

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

#### Quote (BAretired)

There is no tendency for cross flange bending...

#### Quote (dik)

BART: Agree...

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

Upon further consideration, I believe KootK has a valid point. The lower beam carries a portion of the total shear. Since it does not extend to the support, it needs stiffeners to carry its reaction into the upper beam. In the absence of stiffeners, the reaction would need to be carried by flange bending.

BA

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

I concurr... thanks, Koot...

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

There is a transition zone at both ends of the reinforcement beam where the section is not fully effective. There may be a tendency to bend the flanges within the transition zone, but, if the original beam is adequate to resist the maximum shear between support and reinforcement (which it must be or the reinforcement would continue to the end), then the addition of stiffeners at the ends of the lower beam does not contribute to the strength of the built up beam.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BAretired)

There may be a tendency to bend the flanges within the transition zone, but, if the original beam is adequate to resist the maximum shear between support and reinforcement (which it must be or the reinforcement would continue to the end), then the addition of stiffeners at the ends of the lower beam does not contribute to the strength of the built up beam.

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

To clarify for OP, if he's still around, this is really what I'd be aiming for.

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (KootK)

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.

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

#### Quote (BAretired)

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.

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

The flange force will vary under constant shear. Varying shear isn't needed.

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

#### Quote (steve49h)

Varying shear isn't needed.

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

#### Quote (steve49h)

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

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.

#### Quote (steve49h)

I think that a pair at say 2/3*D_reinf would be better but not sure if absolutely required.

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

At long last, I now agree that KootK is correct. Under uniform load, horizontal shear from the flange welds causes the lower beam to curve upward (tension on top). But the beam curves downward, so some of the applied load is acting directly on the lower beam (through the upper beam). Reactions at each end must be transferred to the upper beam by stiffener plates.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

I agree with the end stiffeners and was going to point out that they should be at the end of the reinforcement rather than where first drawn but you got in with the current detail first. I also thought my local code had a requirement not to rely on mixed bearing/weld connections but that doesn't seem to be the case. I would then only have the end stiffener; I think that a little flange flexibility will avoid excessive compatibility stresses. It is still the case that any varying reinforcement shear is caused by varying overall shear and not varying moment; and that the disturbed zone is a different beast than the interior which I thought wasn't coming out in the discussion.

### RE: Partial Steel Beam Reinforcement Anchor Force

(OP)
The stiffeners at the ends makes sense to me as they provide stability and load transfer of the vertical shear component in the reinforcing beam back into the existing girder. I would have to design for those stiffeners in addition to the anchorage forces from the horizontal shear flow. Thanks guys I appreciate the help.

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (steveh49)

It is still the case that any varying reinforcement shear is caused by varying overall shear and not varying moment;

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.

#### Quote (steveh49)

...and that the disturbed zone is a different beast than the interior which I thought wasn't coming out in the discussion.

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 also perform 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

I feel that an interesting follow up to this is the question of how much shear winds up in the reinforcement and needs to be dealt with at the stiffener? One robust way to do that is to simply integrate VQ/I over the depth of the reinforcement. That said, while I struggle to find a way to prove this rigorously in the general case, I believe that the following is true, at least for simply supported beams:

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

#### Quote (KootK)

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.

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 V1 * hr.tr/(h.t + hr.tr) where V1 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

#### Quote (BAretired)

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.

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

In the proposed formula:
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

#### Quote (BAretired)

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

I_comp does determine the deflection of the beam but not the total shear in each of the two parts. That's why I said it was surprising.

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (KootK)

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.

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

I'm in the happy territory of knowing this is over-thinking but also under-thinking at the same time.

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

#### Quote (steve49h)

I'm in the happy territory of knowing this is over-thinking but also under-thinking at the same time.

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

#### Quote:

fear of being considered -- and possibly truly being -- ridiculous

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

Along similar lines, there must be a similar mechanism at play even in non-reinforced members under certain conditions of support detailing. After all, a non-built up member isn't so different from a built up member save the rather excellent connection at the interface.

### RE: Partial Steel Beam Reinforcement Anchor Force

I'm beginning to think that we've been missing something important here. For the general case of partial reinforcement, I feel that a free body diagram of the reinforcing piece ought to include end moments as shown in my first sketch below. My second sketch shows a possible detailing arrangement to suit such a model.

### RE: Partial Steel Beam Reinforcement Anchor Force

And then, of course, the next logical step would be something like this which starts to look like what any bridge engineer or PEMB designer would surely suggest for a variable cross section member...

### RE: Partial Steel Beam Reinforcement Anchor Force

Koot, that moment is what I've been trying to address. The stress in the reinforcement can be thought of as a moment and axial force superimposed. As per BA, the flange welds introduce the axial force in a way that opposes the required moment so increase the concentrated moment. I don't know whether the couple will be too close with consequent weld-popping forces unless we make it be further apart or the stress development more gradual. And trying to do it without the second stiffener as a challenge.

### RE: Partial Steel Beam Reinforcement Anchor Force

Ahhh... Now I see the genius of your first detail. The back end stiffener does the moment job. Given that MQ/I and these end moments both dissipate towards the supports, I feel like it's generally a good idea to just run the reinforcement as near to the supports as spatial constraints will allow.

### RE: Partial Steel Beam Reinforcement Anchor Force

Tension from weld to reinf. beam = T = V*Q/Ic *L/2 *i2 = V.Q.L/4Ic

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

Following is a comparison between two proposed methods of calculating the stiffener reactions, assuming uniform load on the composite beam.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

Thanks BART... time for another SMath program... save me a lot of work... I owe you a beer...

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

You're welcome, dik, but I don't suppose this issue comes up very often. I can't remember ever strengthening a beam with anything other than a plate. Prior to seeing this thread, and given this problem, it would never have occurred to me to provide for a reaction at the end of the reinforcing beam.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

Thanks for the mathing BAret. I've done some modest verification on that and agree with your latest post, noting that the expression below is a linear approximation of what is actually a parabolic function. In some cases it would produce erroneous results but, in the particular case to which you've applied it, would produce a reasonably accurate, slight underestimate of the shear in the lower piece.

### RE: Partial Steel Beam Reinforcement Anchor Force

I've always just added a plate on the end to stabilise the section (usually T)... i Don't recall the last time I did a W Section composite... the topic was great... Kudos to the Baffled guy...

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

@Koot... what is the parabolic function, if I may ask?

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

I haven't bothered to work out the details dik. I just know that it takes a parabolic form from mechanics of materials. My first step in checking BA's stuff was to reverse the roles of the main member and reinforcement to see if the total equaled 100% of the total shear. It was off by a significant margin. This is why. BA's approximation is more applicable to some cases than others. Which is fine, of course, as that's usually the way of things with approximation.

### RE: Partial Steel Beam Reinforcement Anchor Force

Some observations related to the revelation (to me) that partial reinforcement has end moments.

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.

#### Quote (KootK)

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) 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

#### Quote (dik)

@Koot... what is the parabolic function, if I may ask?

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

#### Quote (KootK)

I haven't bothered to work out the details dik. I just know that it takes a parabolic form from mechanics of materials. My first step in checking BA's stuff was to reverse the roles of the main member and reinforcement to see if the total equaled 100% of the total shear. It was off by a significant margin. This is why. BA's approximation is more applicable to some cases than others. Which is fine, of course, as that's usually the way of things with approximation.

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

Thanks BART... clear as mud, with the required reactions equal to the shear values shown by the green SFD portion. UDL would be similar approach, but would have slightly different diagrams. The BMD doesn't even come into play... thanks gentlemen... 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

#### Quote (BAret)

It really isn't an approximation.

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

@BAret: I believe that your recent diagrams need the modifications shown below. The first is just MQ/I; the second represents the "disturbed region" business that steveh49 and I were discussing earlier in the thread.

### RE: Partial Steel Beam Reinforcement Anchor Force

First, there was a typo in one of my earlier posts (2 Apr 21 20:00)

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

Yup... that's what I understood from posting of 4:45... CAM, again...

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

#### Quote (BAret)

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.

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 the height of 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)?

#### Quote (BAret)

The suggestion of a value of MQ/I at each end is incorrect.

I strongly disagree for the case of partial reinforcement. What is it you think MQ/I does if not this?

#### Quote (BAret)

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.

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

#### Quote (dik)

@Koot... what is the parabolic function, if I may ask?

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

KootK, I did not use that triangle in my analysis. VxQ/Ic is the horizontal shear per unit length in the pair of welds between upper and lower beams. (VxQ/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

#### Quote (KootK, BA)

Again, I am not concerned with the variation of stress along the length of the beam. You should be. Rather, I am concerned with the variation of stress over the height of the reinforcing member which is parabolic and not linear under all conditions of load. You should not be concerned about this as it is not relevant.

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? NO!!!

If so, that would be integration over a parabolic function rather than a linear one, would it not? It would be if that was what I was doing. 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)? Yes, indeed, that is what I am doing (correct in my opinion).

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BAret)

KootK, I did not use that triangle in my analysis. VxQ/Ic is the horizontal shear per unit length in the pair of welds between upper and lower beams. (VxQ/Ic)*L/2*1/2 is the total anchorage force at midspan, assuming uniform load.

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?

#### Quote (BAret)

Integrating the triangle shaded red (or the area under the curve) is not relevant to the question at hand.

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

Here's another version assuming the reinforcement to be the same depth as the main member. Clearly, the shear carried in the reinforcing is 50% in this case, right? Symmetry?

### 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

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### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (Celt83)

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.

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 in addition the 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

I certainly agree with the shear lag business but consider that to be separate, and in addition to, the MQ/I business and the associated end moments in the reinforcing. The term "anchorage" is a bit problematic in this context as one might take that to mean any of the following:

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

If the reinforcing beam runs full length of the original beam, under the action of a single point load, can we agree that the sketch below is accurate?

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BAret)

If the reinforcing beam runs full length of the original beam, under the action of a single point load, can we agree that the sketch below is accurate?

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

#### Quote (KootK)

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.
Did not pick up on this, making more sense now.

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### RE: Partial Steel Beam Reinforcement Anchor Force

No worries, I'm glad to have your input on this as with all things first principles.

### RE: Partial Steel Beam Reinforcement Anchor Force

Ok I looked at tau based on the anchorage value since we are talking about the vertical shear in the end of the reinforcement, result looks pretty similar to yours.
assumes rectangular main and reinforcing section and uniform tau across the width.

edit: missed a B in the Q formula

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### RE: Partial Steel Beam Reinforcement Anchor Force

For a single concentrated load, the shear force and shear flow is constant and doesn't depend on the length of the beam or reinforcement. BAretired's method of integrating this over the reinforcement length to calculate the reinforcement hanging reaction would mean the reaction grows without limit as the length increases.

### RE: Partial Steel Beam Reinforcement Anchor Force

With respect to the question of what percentage of the total vertical shear is carried in the reinforcement, I see that as being completely determined by cross section geometry alone with no dependence whatsoever on the magnitude of the load nor it's distribution.

### RE: Partial Steel Beam Reinforcement Anchor Force

I have shown below the condition for full length reinf. beam with a single point load. I believe it is theoretically correct. Moment for the composite section is Pab/L, neglecting beam weight.

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BAretired)

I believe it is theoretically correct.

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

KootK, I am fully aware that you do not agree. I don't know how I can persuade you that your notion of integrating the curve is wrong. I am also aware that the vertical curve varies parabolically from top to bottom, but that does not enter the picture because the weld line shear already takes into account the parabolic variation with the term Q/Ic. For a rectangular cross section, such as you have shown, and I repeat below, Q/Ic has a maximum value of 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

I do not want to "agree to disagree". I would prefer to agree and I am perfectly willing to hear your arguments, but I'm afraid we are dragging the discussion out rather long for a situation so rare that I, for one, cannot remember ever having had to contend with in over fifty years of engineering practice. For others, it may not be so rare, so it is worthwhile getting it right.

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BARetired)

I am also aware that the vertical curve varies parabolically from top to bottom, but that does not enter the picture because the weld line shear already takes into account the parabolic variation with the term Q/Ic. For a rectangular cross section, such as you have shown, and I repeat below, Q/Ic has a maximum value of 3bh/2. This accounts for the fact that the maximum shear is 1.5 times the average shear.

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

Do you wish to continue or not BA? Call it and I shall respect your wishes. As you probably know, I rarely ever run out of steam on the deep dives. And I have several, targeted ideas for how we might move things forward if you're amenable to that. During the course of this discussion, I've already asked you several pointed questions that you've not made any explicit attempts to address. If you'd consent to attempting to answer my questions, I suspect that there's a fine chance that we could reach a consensus and that one or both of us could learn something valuable.

#### Quote (BAretired)

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.

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

#### Quote (celt83)

KootK correct me if I'm wrong but I believe what your after is quantifying the force generating these stress concentrations...

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

@celt83: Tell me that doesn't make for a kick-ass diagram with my ideas superimposed on your FEM stresses?

### RE: Partial Steel Beam Reinforcement Anchor Force

Sorry I didn't respond earlier. One of the nurses at the health clinic called and advised that I could move my 'jab' from 7:30 tonight to right away, so I took the opportunity to do it. I have now had both jabs.

Anyway, VQ/Ib (#/in2 or N/mm2) 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. Rleft = Mmax/a; Rright = Mmax/b. I consider it elementary statics and wonder what all the fuss is about.

#### Quote (KootK)

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.

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

Was that the Jabberwocky?

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

BAretired, referring to your posts 7/4/21 16:04 & 23:28:

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

@BAret: the center of tension in the reinforcement is not located at h/2 but, rather, closer to the outer edge of the member. That, owing to the fact that there's a flexural tension stress gradient across the height of the reinforcing. The stress diagram is a trapezoid, not a rectangle. I believe that this would increase your predicted reinforcing end shear, perhaps bringing it in line with my formulation (I haven't verified the algebra yet).

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

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (is not located at h/2 but, rather, closer to the outer edge of the member.)

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?

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

FEM backup for what I believe BA is referring to, cross section restrained at each end at the top outside corners with a varying axial load, VQ/I, applied at the top surface. Yields horizontal reactions only:

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

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### RE: Partial Steel Beam Reinforcement Anchor Force

Celt83, could you share some details of this analysis:

- This image shows vertical stress?
- Span and reinf length?
- 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

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?

### RE: Partial Steel Beam Reinforcement Anchor Force

- This image shows vertical stress? Correct Vertical Shear Stress
- Span and reinf length? 10ft overall span with reinforcement from 3ft to 7ft (modeled as two shells but ultimately share mesh interface)
- 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

Thanks. I thought the first image was vertical direct stress. Will have to think some more...

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

#### Quote (dik)

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?

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...

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

It seems to me that we can calculate flexural stresses directly from the moment on the composite section, which is readily found. The problem we are having is in deciding how much reaction the reinforcing beam takes.

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

@BAretired: I'm pretty sure that I've got this sorted out now. I executed what I believe to be the corrected version of your method as I proposed in my last post. It now produces the exact same result as my original method of integrating the shear stress function over height of the reinforcement. Unless someone can poke a hole in what I've done, I consider this "case closed".

### RE: Partial Steel Beam Reinforcement Anchor Force

Interestingly, my differential element study from the top would produce the same result for a concentrated load when the location of the delT force is made accurate. I'd recognized that discrepancy at the time and it was actually what prevented me from attempting to estimate the vertical shear in the reinforcing then.

#### Quote (KootK Long Ago)

For convenience, I've pretended that all of the flexural tension resides in the lower flange.

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (BAretired)

Anything more exact is likely not worth the calculation effort.

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 any reinforcing 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

You get a little purple star just for the shear effort!! No pun intended.

The suggested approximate formula yields Vratio of 2/8 = 0.25 (very conservative)

If: d = 8", h = 4", b = 1"
then, by the KootK formula, Vratio = 0.5

The suggested approximate formula yields Vratio of 4/8 = 0.50 (perfect)

BA

### RE: Partial Steel Beam Reinforcement Anchor Force

This should be the truly general version of the formula that would apply to all shapes, symmetrical or not, including tees. For sport, I again ran it against the rectangle example. It would be more meaningful to run it against a wide flange reinforced with a tee. However, I may never get around to doing the integration on that for verification.

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

2
Vreinf / Vtotal = (Ir + Qr * Cr) / Ic

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

And now that's settled, I want to move back to the question of reinforcement that terminates within the span rather than running full length. There's only one question left in my mind, which was raised earlier. Hopefully the image below is self-explanatory, and the North American crew solves it while I sleep.

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

### RE: Partial Steel Beam Reinforcement Anchor Force

#### Quote (steveh49)

Edit: this is the same as KootK's equation (but I think neater and provides more insight).

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

KootK, it's very tidy how simple the Vratio equation turns out. I was happy enough to integrate the shear stress since the question doesn't come up often.

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

Well, I really learned something in this thread.

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) / Ic

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

It is beautiful in its simplicity.

So, thank you KootK and Steveh49.

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

For Plate reinforcing... have to modify my programs for T,W and L... same methodology...

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

@BA: thanks for reporting back to let us know that we've reached a substantial consensus. A degree of closure is a boon for both this thread's active participants and for posterity I think. Being pressed to defend my ideas has also been a boon for me personally. I've been chipping away at the various aspects of elastic, composite design theory on and off for the better part of five years now (mostly with wood). The insights that I've gained here have me dangerously close to complete-ish understanding I suspect.

### RE: Partial Steel Beam Reinforcement Anchor Force

(OP)
Thanks for the help guys.

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

Your question was great... I learned a bunch.

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

-Dik

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