Plastic Analysis

[parrot77]

I have a question about a specific plastic analysis case, some of you guys may be of help. I have a beam fixed at one end and a cantilever on the other. Most common continuous systems will require 3 plastic hinges and the collapse mechanism will be independent of the load on the adjacent spans. But in this case the work done by span load Q (see attached sketch) will have to overcome the negative dispacement from the cantilever load q assuming only two plastic hinges form. Can anyone tell me how to evaluate the 'negative work done by the cantilever load q?

 

[BAretired]

The work done by the load on the cantilever is -qC*[θ]C/2 where C is the length of the cantilever.

BA

 

[abusementpark]

I forget the exact expression for work done by a distributed load in that scenario. However, isn't it also possible that a partial collapse mechanism occurs where the cantilever end failures first over the support?

Here is a link to a free program that can handle plastic analysis. I used it a lot to check my work in grad school - probably the only time I will ever use plastic design.

http://www.mastan2.com/

 

[rb1957]

i also think you'd have another plastic hinge at the prop (the next moment peak)

 

[BAretired]

Come on guys, this is a simple problem. Why make it an issue?

BA

 

[parrot77]

Thanks for your replies guys. BA looking at it again last night, you are correct, and yes you are right.. on reflection it is quite simple. C/2@ is the average displacement by the load qC hence work done = - qC^2@/2

Thanks again

 

[rb1957]

BA,

but this isn't a simple cantilever, more like a simple beam,

and the OP noticed that the overhang, the way he'd initially considered it, was doing -ve work. so he (IMHO) rightly asked for our opinions.

seems that the 3rd hinge would develop at the prop, getting rid of th eoverhand -ve work question, and the solution depends only on the point load. no?

 

[parrot77]

rb, for this collapse mechanism I am assuming that the point load is the dominant one, and that the udl on the cantilever is not sufficient to develope a plastic hinge at the support. Thus the mechanism described needs to overcome the cantilever moment in order to form the plastic hinges under the point load and at the fixed end. The cantilever can be assessed as a seperate mechanism by simply comparing the cantilever moment to the plastic moment capacity. ie for this failure mode only one hinge needs to be formed at the support. I think BA is correct

 

[rb1957]

i don't think it'll work like a cantilever, due to the prop.

i think max moment will be under the load, and that's where the hinge will form.

 

[dhengr]

I second BA’s ‘come on you guys.’

You must first look at the cantilever to the right, since it influences the moments in the back span. However, you can not tolerate a plastic moment at the canti. moment reaction, that would be a hinge, but it’s a failure mechanism, thus verboten. Draw two moment diagrams for the whole beam: one with just the canti. loaded; and the other with just the back span loaded with the concentrated load; and superimpose the two moment diagrams.
Otherwise, the whole concept is that a plastic hinge will form at the max. moment, the fixed reaction and rotation will occur, and start redistributing load to the next max. moment point, under the concentrated load. When a plastic hinge forms there, you have a mechanism, and the game is over.