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# Plastic hinges - rotation capacity before fracture?4

## Plastic hinges - rotation capacity before fracture?

(OP)
I am trying to find a reliable way of estimating the maximum rotation that can occur at a plastic hinge before the ductility gives up and the member falls apart.

What I need to do is to calculate the total amount of energy that can be absorbed while a member continues to bend up to eventual fracture, after a one-off accidental event.

For my purpose, it does not matter that the member will be permanently deformed, and so the common definition of 'failure' as the development of some degree of permanent deformation is not relevant.

Can anyone point me to one or more good references that deal with this aspect of plastic theory?

I have several good texts on plastic analysis and design, but none of them go into the question of what happens after the full yield moment has been reached and a mechanism created.  My interest is in the subsequent behaviour of that mechanism.

IN the absence of good references, I would welcome any comments on my home-made approach (based on entirely guessed assumptions ).

I have got to an estimate of rotation capacity by the following process:

1.  I Assume that when a plastic hinge is developed, two zones of fully yielded material are created (one in tension, the other in compression).

2.  Looking at the member in side elevation, I have taken the limits of the yielded zones as being two intersecting lines, making an 'X' shaped zone with the lines at +/- 45 degrees to the axis of the member (ie included angle of 90 degrees).

3.  I then say that total fracture will occur when the longitudinal strain within these zones reaches the  specified minimum elongation for the material.

4.  From there on, I can get an angular rotation at fracture by some fairly simple arithmetic.

Has anyone any better sugestions?  All responses will be welcomed, even if they all point out some stupid flaw in my thinking.

### RE: Plastic hinges - rotation capacity before fracture?

AUSTIM,

What you need is a Moment-curvature diagram for your column.  This will give you some idea of the demand/capacity and ductility.  One program that I know of for this application available at the right price (free) is at SC Solutions website: http://www.scsolutions.com/bridge.html

The program is called SEQMC - SEQAD Moment Curvature Analysis.

It is easy to use and understand but just know that any output will have a disclaimer on it noting it is a demo version not for professional use.

Hope this helps!

### RE: Plastic hinges - rotation capacity before fracture?

(OP)
Thanks Qshake.  I should have added that I am looking at steel sections, but that software also looks useful.

### RE: Plastic hinges - rotation capacity before fracture?

The steel member proportions and type of connection used will have an influence on the ductile range.  Steel sections have various flange thickness/outstand ratio, various web thickness/height ratios, and similarly for the connections.
Some steel codes of practice use classes of sections to classify sections according to these characteristics.  One of the reasons is that these characteristics determine how much hinge rotation will occur before the ductile (plastic) rotation progresses to buckling, particularly in compression zones.  Once this happens, then considerable rotation occurs with fairly low energy absorption, and the hinge has gone beyond the stage of usefulness, even if it has not actually fallen apart.
I do not know of an easy way to predict this change from ductile to buckling for a particular case.

### RE: Plastic hinges - rotation capacity before fracture?

(OP)
Hi, RiBeneke,

Thanks for your input.  Again, I realise that I should have described my problem in better detail at the start.

I am trying to develop a safe emergency exit ladder (about 25 metres high) up a narrow shaft within an underground gold mine in Tasmania.

The general plan is for a 'chain' of linked ladder sections, all suspended from the top.  Hinged 'seats' will be provided every 6 metres or so, so that any one who can't cope with the 25m climb in one hit may stop and rest on the way, and also so that anyone falling will not fall the full distance of 25 metres.

Thus the energy that I am seeking to absorb is merely the energy of one Tasmanian miner falling 6 metres or so.  My concern is that one man falling should not cause the total collapse of the ladder system, taking any other miners on the ladder at the time with it.

The sections in which I would like to absorb the energy are the stiles of the ladder, which will be fairly stubby rectangular flat sections, and not subject to serious buckling concerns.  Thus I am only interested in the ductile stage of post-yield behaviour.

The rest of this post is only slightly related to my initial posting (merely the ramblings of an ageing engineer ).  Feel free to ignore it all.

Back in 1952 I was introduced to the basics of Plastic Theory by Professor Sir John Baker, the leading light in the field in the UK at that time.  He told us of the development of the Morrison Shelter, and its place in the introduction of plastic design to practical use.

Early in World War II the war cabinet were concerned about the long-term effect on the civilian population of having to shelter from air raids every night in the only type of domestic shelter then available.  This was the Anderson shelter, which was constructed by excavating a trench in the garden, roofing it with curved corrugated steel (similar to Armco), and then placing the excavated material over the steel roof.  Drainage was generally non-existent, and the shelters were not viewed as pleasant places to sleep in by many of the younger members of the population.

Along came Prof. Baker with a plan for a steel frame (just a box with angle steel at all corners plus mesh to keep out debris) that would fit under most kitchen tables.  He had calculated that the top angles of this shelter would deflect by about 1 foot if the house above it were to collapse in an air attack, leaving the residents terrified but still alive.  We were shown one photograph of a post-event shelter displaying the exact 1 foot deflection, with space for 2 or 3 prone people remaining underneath the bent roof members.

Prof. Baker had a face-to-face meeting with Winston Churchill (who was in his bath at the time!), and the decision was made to produce a million such shelters.  (But they were named after the Home Secretary, not the inventor)

Back to my problem, which is fundamentally the same - given an estimated energy input, how do I determine the maximum hinge rotation that I can get out of a simple non-buckling section before it 'snaps'?

### RE: Plastic hinges - rotation capacity before fracture?

Before going on I would roughly estimate whether a reasonable proportioning of members would make the job, based on your procedure in the first post. As I suppose these hinged seats have to be lowered by people for passing through during escape, there are weight limits that could make your solution unrealistic.
Also I would ask myself if the limiting case could not be with 3 or 4 (under panic) people resting onto the seat one on top of the other.
As you correctly point out, what you are looking for is an energy absoption capacity, not so much a limit deformation. I think that you should stay very far from the limit elongation: I would say 2-3%, not more, otherwise you prediction may become unrealistic. By the way a limit of 1% in deformation (of welds only if I recall correctly, 0.5% in full metal) in calculations based on limit analysis (ASME III).
Just a few thoughts.

prex
motori@xcalcsREMOVE.com
http://www.xcalcs.com
Online tools for structural design

### RE: Plastic hinges - rotation capacity before fracture?

(OP)
Hi, prex.  Thaks for your input.

The 'seats' will be made of light aluminium grid, hinged and supported by cantilevered frames on each side of the ladder.  They will normally be in the lowered position, but can easily be raised as miners pass up (and between the side frames).

The use of the ladder will not be as a 'panic' escape.  There are safe refuges within the mine, and use of the ladderway will be under controlled procedures.  Current proposals will limit the traffic to a maximum of one miner within any one section between rest positions.

### RE: Plastic hinges - rotation capacity before fracture?

Hope everyone had a wonderful holiday...

Austim, so you're dealing with the other concrete are you?!  Well, I was wondering what was up with the fracturing but I was too quick to answer considering the rebar fracture.

I'll be back...

### RE: Plastic hinges - rotation capacity before fracture?

If you give me an e-mail address I will produce a .doc variant of some collections made on rotational capacity after my reading one text, some of which could (maybe) be useful to anwer your question.

### RE: Plastic hinges - rotation capacity before fracture?

(OP)
Thanks for the offer, ishvaaag.

all contributions will be welcome at tsewcau@yahoo.com.au

### RE: Plastic hinges - rotation capacity before fracture?

I suggest you make a mock-up of one section (a stile and seat frame) of your ladder and apply load to see how far it deflects in energy-absorbing fashion.  That way you will test the sections and the connections together.
I think it will yield the more useful information than could easily be done by calculation.
Remember that aluminium and steel also come in a variety of hardnesses that depend on alloy, production method and fabrication practice, and this also affects the ductility.

### RE: Plastic hinges - rotation capacity before fracture?

(OP)
Hi again,

Ishvaaag - many thanks for the *.doc.  There is a lot in there to digest; it all helps in one way or another.

Ribeneke - yes, I agree with your comments.  A test may well be the best way to prove my final design, but I still need some (pseudo-?) rational process to reach a decision as to what we should test.

I have made some progress on my own - looking at a 1948 edition of "The Steel Skeleton" by Baker, Horne and Heyman was useful in two ways - it has some good basic stuff.  It also has some great historic background to the design and performance of the domestic air raid shelters.  (I particularly like the story of the three people in Exeter who were blown 46 feet across the road, still in their shelter, into the front bedroom of the house opposite, and lived to tell the tale).  If nothing else, I now have the numeric data which Baker et al used regarding energy to be absorbed, span, and steel section sizes used, plus confirmatory evidence of actual deflection.

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