Equivalent stiffness of frame with bracing 4

[SiggiN]

Hi!

Suppose you have a frame like the enclosed picture

If I want to calculate the equivalent stiffness of the columns and bracing is it correct use the following stifnness':

k_col = E*I/L1^3

k_brace = E*A/L2*cos^2(45deg)

k_eq = 4*k_col + 8*k_brace --> Not sure about summing these stiffness' ?

( in 3D I have 4 columns and 2 brace mebmbers pr. column, one as shown and the other out of the picture plane)

Once k_eq is known the system can be modeled as a cantiviler beam with a mass on top?

Thank you!

Regards
Siggi

 

[r13]

I am not sure the equations you were using, but I suggest to place a 1 kip lateral load and obtain the corresponding lateral deflection, then k_eq = 1/Δ.

 

[BAretired]

k_col = E*I/L1^3

k_brace = E*A/L2*cos^2(45deg)

k_eq = 4*k_col + 8*k_brace --> Not sure about summing these stiffness' ?

( in 3D I have 4 columns and 2 brace mebmbers pr. column, one as shown and the other out of the picture plane)

Once k_eq is known the system can be modeled as a cantiviler beam with a mass on top?

L1 and L2 are not defined. I assume L1 is the height of column and L2 is the height of brace connection.

By your formula, if braces are removed, k_eq = 4EI/L1³...should be 3EI/L1³ for one column or 12EIL1³ for four columns.

By your formula, if braces are infinitely stiff, k_eq = ∞

So your formula is not correct at the two extreme values. Try retired13's method. That should be correct.

BA

 

[KootK]

k_eq = 4*k_col + 8*k_brace --> Not sure about summing these stiffness' ?

You instincts on this are good as this is fundamentally incorrect. As the stiffnesses of additional members are added to the system, the combined stiffness should be getting smaller rather than larger. The math is like circuits in parallel: 1/k_eq = 1/k_col + 1/k_brace.

k_brace = E*A/L2*cos^2(45deg)

While I too like retired13's unit load method, I don't know if you have access to frame analysis software. Once useful trick you can use with this is to recognize that the brace axial stiffness won't contribute anything meaningful to the combined system stiffness as the combined stiffness will be dominated by the bending of the column member. So, if you consider the column bending stiffness correctly, you can just ignore the brace stiffness altogether.

k_col = E*I/L1^3

I don't believe that is the correct formula. Among other things, I don't thin that it accounts for the flexibility of the column back span.

My recommendation, if doing this by hand, would be to use the model shown below. If you can find an equation for the tip deflection of a point loaded propped cantilever, you should be able to rearrange that to get the stiffness parameter you seek directly.

 

[r13]

You don't need computer to perform the unit load analysis, I bet BA can teach you the classic method by hand, as it is his strong suit

 

[BAretired]

As the stiffnesses of additional members are added to the system, [highlight yellow]the combined stiffness should be getting smaller rather than larger[/highlight]. The math is like circuits in parallel: 1/k_eq = 1/k_col + 1/k_brace.

I don't think so, but if the brace is pin connected to the column, its stiffness will have only a minor effect on overall stiffness. If it is moment connected, it could have a substantial effect.

Agree with retired13, hand calcs will suffice.

BA

 

[KootK]

I don't think so, but if the brace is pin connected to the column, its stiffness will have only a minor effect on overall stiffness.

It's difficult to tell with any certainty but, based on OP's math, I was assuming that he was attempting a formulation like the one shown below. And, in that case, it is most definitely flexibities that need to be summed and not stiffnesses. Hence my comment about the need to sum reciprocals of stiffness. Obviously I agree that adding the brace to the unbraced cantilever would have the effect of increasing the stiffness of the system.

If it is moment connected, it could have a substantial effect.

Sure but, then, is there really any doubt that OP means to treat the connection as a pinned joint as the overwhelming majority of structural engineers would in a similar situation?

Agree with retired13, hand calcs will suffice.

In that case, I'm not sure that I see the point of the recommendation. OP is effectively already doing a version of that... just, I think, incorrectly.

 

[r13]

I hope OP does pick up the fundamental here.

 

[BAretired]

@KootK,

I agree that usual engineering practice would be to assume a propped cantilever and ignore the change in length of the brace.

BA

 

[SiggiN]

Hi guys!

Thank you all for your contributions!

I realized I was a bit quick when writing the equetions i do apologize!

In this case the brace is welded to the columns, so it never occured to me to model the problem using propped cantilever.

I'm still learning usual/best "engineering practice" (relativly recent graduate).

I might dirgress but, I have always been under the assumption that welded conections act like fixed connections, to be able to support moment. Wrt. to safety/conservatism will a pinned conenctin always be on the safe side?

Thank you all!

Regards
Siggi

 

[Blackstar123]

kootK, as always, you explain complex concepts in so easy terms. Still, I'm confused about two things when transferring the problem into an equivalent 2 span beam, why are you ignoring:
1. Fixity at the column support
2. Rigidity at beam-column joint.

 

[KootK]

I might digress but, I have always been under the assumption that welded connections act like fixed connections, to be able to support moment. Wrt. to safety/conservatism will a pinned conenctin always be on the safe side?

That's a nuanced thing:

1) It is correct that pretty much all welded connections are capable of transferring moment between the connected members.

2) A welded connection will not cease to draw moment just because the designer would wish for it to be pinned.

3) In a system that is trussed, the stiffness of the truss mechanism relative to the any parallel moment frame mechanisms will often, but not always, be such that incidental, welded moment connections will be prevented from developing significant stresses as a result of the transfer of bending moments between members.

4) Neglecting the contribution of welded moment connections will almost always be conservative with respect to system stiffness and estimates of deflection. This is absolutely the case for the problem that you've tabled here.

5) Neglecting the contribution of welded moment connections will almost always, to some degree, be unconservative with respect to the design of the connections themselves.

6) Neglecting the contribution of welded moment connections is kind of a crap shoot with respect to its impact on member design and requires additional judgement. For this problem, I expect that considering the joints to be moment connected would benefit the design of the column but punish the design of the brace which would now see end moments in addition to axial force. Of course, on the other hand, if the ends of the brace are considered fixed, that would reduce the effective buckling length and increase brace axial capacity in a way that might benefit the design of the brace member in an overall sense. The types of members selected an their mid-span bracing conditions will impact this as well.

 

[KootK]

Still, I'm confused about two things when transferring the problem into an equivalent 2 span beam, why are you ignoring:
1. Fixity at the column support
2. Rigidity at beam-column joint.

1) As I noted in my last post, ignoring joint fixity and rigidity will be conservative with respect to deflection control here and, as I understand things, deflection control is the motivation for this exercise.

2) At the same time that I expect my solution to be a conservative solution for deflection, I also don't expect it to be an overly conservative solution. In my experience, this approach will produce a practical design without much waste.

3) I deem my solution to be an expedient one that would save designer effort. Attempting to include the effects of joint rigidity complicates the analysis, forces one to have to consider realistic %-rigidities at the connections and whether or not the details of the constructed connections support the assumption of joint rigidity for force transfer. In my experience, something like this may:

a) have a single angle side welded to the post for a brace.

b) have a bolted base connection or a welded connection to a member that itself is meaningfully flexible.

All this stuff complicates a properly executed design in a way that, in my opinion, is unjustified given the scale and nature of the problem.

4) In my time as a structural engineer, I've known a lot of other structural engineers and have been privy to their design assumptions on various things. That informs my opinion that the overwhelming majority of structural engineers would treat these connections as pins analytically. I place great value upon this since most of what I know about structural engineering has been handed down to me from my mentors and colleagues. When in Rome...