Portal frame vertical deflection

[takupfr]

Hi guys,

Quick one for you professionals: do you guys know a simple formula to quickly estimate the vertical deflection at the apex of a portal frame pinned at its bases?

My inputs would be:

w : UDL on rafter

L: column to column distance
h: top of column height
Alpha: angle that the rafter makes with the horizontal

Ir: second moment of inertia of rafter
Ic: second moment of inertia of column
E: elastic modulus

Hopefully you can help me with this one.

Regards,

Mathias.

 

[ANE91]

Only so many ways to skin this chicken

 

[takupfr]

Hi ANE91 and thank you for your answer.

I remember doing this in engineering school back in the days. The thing is i kinda forgot how... i am ashamed tbh.

Would you mind developping a little more to get to the end formula?

I just need something that can be used quickly without going through a full demonstration everytime.

 

[ANE91]

You don’t need a “full demonstration every time.” You code it once and thereafter only need to modify the inputs. It’s just Castigliano’s second theorem. I’m unaware of any approximate analysis methods for this scenario. Generally, the point of approximate analysis is to reduce an indeterminate problem to a determinate one, but a top-hinged moment frame is already determinate. If you don’t have the top hinge, then you can approximate the analysis by assuming the locations of points of inflection.

Direct stiffness method is more complicated. Virtual work is similar and could be faster.

 

[takupfr]

Hi, thank you again for taking time here.

I am sorry but i see the formulas for the reactions at supports but don’t see the formula to get the deflection at mid span.

I may be stupid.

 

[ANE91]

q_i is the deflection. Basically, you’re summing the axial shortening of columns and rafters and adding to that the flexural deflections of columns and rafters. We’re ignoring shear deformations because they are trivial. The full equation also has a torsional term, which we also don’t need here.

 

[ANE91]

I may be stupid.

...and don't denigrate yourself like this. It's unprofessional. Even the best engineers didn't know these things, at one point. Give yourself some grace.

 

[ANE91]

You know how deflection works, right? Visualize it. Quantify it.