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# Steel fracture in bending3

## Steel fracture in bending

(OP)
I am involved in a security gate/vehicle restraint design in which I need to provide a steel wide-flange cross beam across two bollards to stop a moving vehicle (15,000 pounds traveling at 30 mph).

I based my analysis on plastic theory/conservation of energy and, on paper, it performs as expected. However, I am unsure as to how to address the fact that, after the plastic hinge forms and the cross beam continues to deflect through the plastic deformation, there must be some limit at which the steel stresses must reach the fracture point. It can't deform infinitely, at some point the beam will rupture and fall apart.

Tensile elongation limits in carbon steel are one thing, but what method can be used to calculate fracture limits in bending? I am unable to find any specific methodologies.

Your input was a huge benefit to me in previous posts, I hope in can be again. Thanks.

### RE: Steel fracture in bending

Isn't the "fracture limits in bending" the Fut (Ultimate).

### RE: Steel fracture in bending

(OP)
Boo1,
Yes, F ultimate is the "stress value" at which it will fail and/or fracture. My problem is determining that value in relationship to bending. I'm only finding such values in relation to axial tensile elongation...such as some set percentage of elongation, such as 15% to 20%, before failure occurs.

After impact, I want the cross beam to continue absorbing energy after the plastic hinge forms. I won't get any stress increases, but I can continue to use the strain increase...the flat part of the stress/strain curve after upper yield limit is reached...to remove energy. The way I see it is that strain hardening will occur at some point...the stress/strain curve will begin to rise again...and brittle failure will occur.

Is it reasonable for me to assume that the rotation of the cross beam section at the plastic hinge is somehow directly related to an axial tensile elongation? If so, how can I equate the two?

Perhaps I'm thinking too hard, I'm merely concerned that the deflection limit I have chosen for the cross beam is actually greater than the deflection for which the cross beam is capable prior to it failing. My method of failure is something like this..

1. Vehicle impacts the cross beam at the center of the gap between the bollards.
2. The cross beam ends extend approximately 18" past the bollards.
3. The plastic hinge forms and deflects back through the bollard gap.
4. The cross beam ends contact the bollards and begin to slide inward, toward the impact point, as the vehicle travels through the gap.
5. An equilateral triangle forms between the plastic hinge and the bollards.
6. At some point, the vehicle/hinge will deflect far enough that the cross beam ends will "slip" past the bollard face...indicating failure of the system to restrain the vehicle. (Admittedly, I have hooks on the beam ends that will catch the bollard before that happens)
7. The vehicle/hinge can travel about 4 foot before the beam ends, theoretically, slip past. That travel distance absorbs a great deal of energy.

The burning question is "Does the beam fail, ultimately, prior to that 4 foot deflection?"

This, of course, is just one impact scenario, but I think the answer will control in every one.

Thanks.

### RE: Steel fracture in bending

At that theoretical 4 foot deflection , what is the strain in the outermost fiber in steel cross section?You need to fix up a limiting strain value first and then give the corresponding material properties("E" and "I") and calculate your deflection  value!!

### RE: Steel fracture in bending

(OP)
The limiting strain value IS my question, though.

Assuming E=29000 ksi, and utilizing plastic theory, Ix is negligible once I form the plastic hinge. I do get to remove some energy elastically, but it's minimal compared to the plastic amount. Zx is my controlling property. But how far, realistically, can I stretch plastic theory before fracture?

I should state that I don't care about the final state of the vehicle, or gate for that matter...both can be destroyed as long as the vehicle stops within 3'-20'.

### RE: Steel fracture in bending

The percentage elongation is an indication of the rupture strain of the steel.  You mentioned 15%, or 0.15 in/in rupture strain.  A36 steel has a rupture strain in the order of 35%.  For a member in flexure, the extreme fiber at rupture has a strain equal to the rupture strain.  The area under the stress-strain curve up to rupture is the strain energy the beam can provide up to failure.

### RE: Steel fracture in bending

Ntollinger,

I have tried to sort this out myself, with no more success than you seem to have had.

Reading through the early chapters of "The Steel Skeleton" by J. Baker and others, I suspect that the only way (or at least the most reliable) you can be certain of the full post-hinge behaviour is by testing a prototype to failure.

Depending on the nature of the facility you are trying to protect, that could possibly be seen as a trivial cost, particularly if your results could be used for other facilitites.

Fortunately for me, the dimensions of the beam that I was proposing were similar to the angles used in tests by Baker etc.  I just ended up assuming that I would get the same hinge rotation (30 degrees) that Baker etc found in their tests.

### RE: Steel fracture in bending

There are some indeterminate issues that make it very difficult to produce a reliable calculation :

1) The length of the plastic hinge that forms depends on the shape of the part of the vehicle that contacts the beam.  If the vehicle is a round shape then the plastic hinge may be quite long, with a correspondingly low elongation per unit length of beam flange.

2) In real life it is quite likely that a buckled compression flange will occur at an early stage of the collision deformation, after which little energy will be absorbed.

An effective approach to absorbing collision energy is to arrange for a structure which forms new successive plastic hinges as the deformation increases during the collision.

### RE: Steel fracture in bending

The limiting strain value is the average elongation at failure.  For instance, if you have material with an elongation of 20 percent, then the limit of extreme fiber strain is 20 percent.  This can provide a lot of deflection in a reasonably long span.  Let's assume you have a 15-foot span.  You should be able to get 3 additional feet of "stretch" out of the extreme fiber, so that just prior to fracture, your new length for the extreme fiber is 18 feet.  Now, this is "theoretical".  In practice you will have some variation in properties along the length and the stress will not be evenly distributed, so I would use a limiting value somewhat below the average elongation, say 50 or 60 percent.

### RE: Steel fracture in bending

Preventing buckling of the compression flange remains a problem in reality and the analysis. Is a flanged member the most appropriate section? Would a circular section or cable not perform the same function with straightforward analysis?

### RE: Steel fracture in bending

For a look at commercially available systems that have an even higher requirement than yours, check out:

www.deltascientific.com

I have no affiliation with them, but found them by doing a google web search for pop up crash barriers.

### RE: Steel fracture in bending

(OP)
letrab,

Your point is well taken, the flanged member is a requirement of the manufacturer I'm working for. My above discussions are simplified descriptions of the system. In reality, the gate is a cantilevered one much like the SC3000 shown in butelja's website reference to www.deltascientific.com. My client will be mounting a decorative picket fence to the cross-beam which will then slide back out of the way for the gate's operation...thus a cable won't do what I need...and a circular section, while it is a possibility, may become a fabrication issue for the fence attachment. Plastic width-thickness ratios are met, do you have any recommendations for other criteria? Any additional methods of analysis are greatly appreciated.

My discussions are more for the mode of failure when the support posts for the cantilever buckle out of the way, which they do quite quickly, leaving the bollards behind the cross-beam to deal with the remaining impact energy. I still have to include the analysis for the bollards as an impact point as well...of course the fracture analysis for a bending steel pipe bollard is just another facet of this steel fracture in bending discussion.

All of you are providing excellent input and I really appreciate it. My finding this forum was truly a stroke of luck.

### RE: Steel fracture in bending

Ntollinger,

You should have a look at the Association of American Railroads (AAR) Manual of Standards and Recommended Practices Section C Part II Appendix C - Plastic Design Factors.  This documents a widely used (in the past 50 yrs of railcar structural design) method of evaluating the ultimate bending capacity of a steel beam using an Mc/I approach but with a very high (sometimes above Fu) fictitious extreme fiber stress.  This is the best method I know of squeezing the very most bending strength of a chosen steel section, given, of course, that local buckling is not a concern by using low width/thickness ratios. No mention of deflections though, for what that's worth...

### RE: Steel fracture in bending

(OP)
trainguy,

I like the sound of that. Now begins the flurry of activity in trying to find a copy of said Section C Part II Appendix C.

Would you or anyone else know of any websites that may have this section in a downloadable form (.pdf, text file)? Ordering manuals always seems to take so long...

I'm at nathant@qproq.com if e-mailing such links are easier.

Thanks much

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