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Fig. 1 Bracket

Fig. 2

For clarity, here's the set-up:

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Fig. 1 Bracket

Fig. 2

For clarity, here's the set-up:

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JBA

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In this case, there isn't much deflection (~0.003"), but if there were significant deflection, then that means that the bracket is exerting a reactive moment on the beam. How would you analyze a situation like that? Where on the bracket is the moment affecting?

I only left it distributed because a uniform distributed load should be similar to a point load at the center.PS If it is a static load at the center of the beam then don't show a series of distributed loads in your illustration.

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JBA

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Makes sense, but then how do you determine what those forces are? If the center load is F, then the shear force experienced by the bottom of the bracket when there is no deflection is F/2. However, when significant deflection occurs and the bottom is fixed so as to not deflect, the beam's end will deflect upwards and push against the top of the plate. As you said, now we have tension on the back plate and shear on the top plate. Are those forces also F/2? When does the magnitude of M factor in to the equation of stress in this scenario?

Edit:

I was thinking of treating this scenario as a cantilever beam with a fixed end. If the end of a simply supported beam deflects 1", then the force a bracket would have to exert on that end to prevent the deflection is the same force that causes a 1" deflection on the cantilever beam. Would you agree with this reasoning?

But then, even if my reasoning makes sense, when modeling the cantilever beam, how long should this beam be? If I use the same length as the original beam, then I get a very small force; but if I model the cantilever beam to span the length of the bracket, I get a much higher force -which seems more reasonable to me compared to the previous case, but I still can't tell whether it's a reliable assumption.

Edit:

I was thinking of treating this scenario as a cantilever beam with a fixed end. If the end of a simply supported beam deflects 1", then the force a bracket would have to exert on that end to prevent the deflection is the same force that causes a 1" deflection on the cantilever beam. Would you agree with this reasoning?

But then, even if my reasoning makes sense, when modeling the cantilever beam, how long should this beam be? If I use the same length as the original beam, then I get a very small force; but if I model the cantilever beam to span the length of the bracket, I get a much higher force -which seems more reasonable to me compared to the previous case, but I still can't tell whether it's a reliable assumption.

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JBA

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At the same time, I really do not know how accurate this representation would be (it would on the conservative side but probably too much so).

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JBA

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Edit: Amazing what you can find in books

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This is what I wanted to know. I know the value, according to those tables. What I'm not certain is how to apply them considering that the connection between the wall and the beam is treated as a point with reaction forces and moments. But, in the given case, if we zoom into that "point", it becomes a bracket. So the moment we calculate is acting on the bottom plate of the bracket? How would you use the height of the bracket to determine the separating force?...the reaction moment at each end of the beam = WL/8 whichthe moment that your end plate will see about its lower edgeandusing the end plate height gives you the separating force of the beam on the top of the plate.

See figure 1 for clarification of the problem I'm posing. In figure 2, if the bottom plate is fixed so it may not deflect and the beam is bending, the red circle indicates a point of stress due to contact between the bending of the beam and the top plate's resistance to that bending. My impression is that the usual reaction forces we solve in Figure 1 are the sum of forces and moments acting on the bracket -however, to truly design a bracket, we would have to expand all the forces represented by that value in the location that they're applied.

If R1, according to the table, is W/2 and M is WL/8, R1 is the sum of vertical forces acting on the bracket while M is the sum of moments. But if we expand R1 and M into its components, what does its distribution look like? We can confidently claim that the greatest component for R1 acts on the bottom plate of the bracket and therefore any other component is irrelevant, but what about the moment? I'd say the moment is shared somewhere between the bottom plate (as shown) and the red circle. Is what I'm trying to say make sense?

Fig. 1

Fig. 2

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JBA

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For a detailed analysis the actual stresses the on the bracket the bracket must be broken down into it component parts and the load transfers from the fasteners from the beam to those elements and from the back plate to the wall because they are what transfers the moments and forces from the beam to the bracket and from the bracket to the wall.

So until you diagram all of those elements you cannot understand how the beam forces and moments will affect the bracket. Stop confusing yourself by using diagrams containing both primary and reaction forces and moments. Look at the problem in the sense of how the beam load and moment flows from the beam to the to bracket fasteners, then from the fasteners to the bracket elements and from those elements to the wall fasteners and ultimately to the wall.

Once you do this you will see that there are a large array of stresses acting on the bracket elements during this transfer.

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Addendum to the above:

If you are trying to match the simplified analysis of the FEA then the first sentence above applies depending upon how you are restraining the top of the back plate of the bracket and at which point you are modeling the connection of that top plate to the beam. Looking at the figure you show you are not treating the bracket as an integral part of the beam with top restraint but with only a vertical restraint along the extension of the bottom plate of the bracket and since you have it integral with the beam also along that segment of the beam. As a result the FEA analysis actually results in a bending moment about the end of the extension of the bottom plate of the bracket. A better representation would be vertical and horizontal restraints at the back edge of the bottom of the bracket at the wall and a horizontal restraint at the top back edge of the bracket. Modeled in this manner then the analysis will give you the separating force at that top restraint and therefore the tension force on the top plate of the bracket from the beam.

If you are trying to match the simplified analysis of the FEA then the first sentence above applies depending upon how you are restraining the top of the back plate of the bracket and at which point you are modeling the connection of that top plate to the beam. Looking at the figure you show you are not treating the bracket as an integral part of the beam with top restraint but with only a vertical restraint along the extension of the bottom plate of the bracket and since you have it integral with the beam also along that segment of the beam. As a result the FEA analysis actually results in a bending moment about the end of the extension of the bottom plate of the bracket. A better representation would be vertical and horizontal restraints at the back edge of the bottom of the bracket at the wall and a horizontal restraint at the top back edge of the bracket. Modeled in this manner then the analysis will give you the separating force at that top restraint and therefore the tension force on the top plate of the bracket from the beam.

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