Where did this equation come from? Bent sheet metal Z 1

[pattheengineer]

The part in question is a piece of sheet metal bent into a Z with two 90 degree angles. The top part is against a vertical wall and is attached via bolts, the it sticks out from the wall and then bends downward at 90 degrees with a mass hanging off of it.

Someone sent me an analysis of the part with an equation and no explanation where the equation came from and was wondering if anyone would have any idea.

S[sub]vm[sub][/sub][/sub]=P/A + 6M/bt[sup]2[/sup]

S[sub]vm[/sub] is the maximum stress
P is the force from the mass hanging off the Z bend
A is the cross sectional area
M is the moment created by the mass
b is the width of the Z bend
t is the thickness of the sheet metal

Also does anyone know of a good reference for bent sheet metal analysis?

 

[Sparweb]

This is the kind of thing your college profs were trying to prepare you for.
The first term is simple direct stress, the second term is stress due to bending. Since bending stresses produce both tensile and compressive stresses, somewhere in the Z part there is a direct tension stress being added to a flexural tension stress (and maybe a corresponding pair of compressive stresses). The analyst that provided you the equation is using a short-hand form that has already been worked out from first principles.


STF

 

[Andries]

Hi,

You are not a student are you? Anyhow, most strength of materials books will provide you with the required background, but try any book on aircraft structures (Bruhn, Peery (1950), etc.

Andries

 

[rb1957]

personally, I wouldn't've used this expression for the structure you're describing. where's the axial load ?

I'd look at the vertical (loaded) web in shear, and the angles as caps loaded by a couple, and pay particular attention to how it's attached to the support.

First, draw a free body.

another day in paradise, or is paradise one day closer ?

 

[pattheengineer]

Not a student, I was reviewing a young engineers report and I wanted to make sure he didn't plop out an equation that I did not know about for bent sheet metal forms.

I realized this morning what I was looking at was actually the equation for bending in a straight member. I was so use to seeing the S = Mc/I that the 6M/bt[sup]2[/sup] went right over my head.

Anyway what he was doing was trying to show a finite element model was valid for the maximum stress in the bend by using a straight member equation. Problem really was his FEA model showed less stress than the straight member hand calculation did.

The equation that he used would have been ok if he had used a correction factor for a bent member. Not sure what is wrong with his FEA model yet, that one is for him to figure out.

 

[rb1957]

bending stress = 6M/bt2 means he's reacting the bending only over the vertical web. surely the upper and lower flanges will contribute greatly to the I of the section ?

and why have a P/A axial component ? the loading is transverse (ie down on a horizontal beam).

did he draw a free body ?

can you add a sketch of the beam and load, just as we're on the same page ?

I suspect that the flanges of the Zed are not attached to "the rest of the world", that there are some fasteners in the web that react the applied load. so at the base of the beam the flanges are ineffective. it actually makes for an interesting and complicated detailed analysis (as the flange load develops over the length of the beam, and then "shear lags" into the web, into the base fasteners. shear buckling of the web is something that should be checked (or show the shear stresses to be small).

another day in paradise, or is paradise one day closer ?

 

[hokie66]

As the OP described it, bending about the strong axis is not relevant. His 'zee' section is on its side, with one flange attached to a wall. So it is essentially a plate bending exercise, where the plate is the web.

 

[rb1957]

is that right ? (I could read his description that way) ... the Zed is aligned with the web horizontal and the two stiffeners vertical, one up and one down, and the down one is loaded and the up one reacts the load ?

if so, then this is much more complicated than first thought. it's easy to easy the loaded leg acting in bending (6M/bt^2) but how does this load get across the web to the other leg which wants to react it ? what would happen is a lot of torsion ... which you can see if you draw a free body diagram. this is something not considered in your equation.

another day in paradise, or is paradise one day closer ?

 

[stressebookllc]

First of all Svm seems to indicate Von Mises stress, which is not what the stress being calculated actually is.

Based on the calculation (which is not uncommon for combined stresses of a lug for example that sees transverse and axial loading), the critical section must be a section that experiences both axial or tension load, as well as a pure bending load and in a lot of real world situations also a shear load. The axial and bending parts must be added for the Rt, and the shear stress for Rs.

In this case however, if a section on the flat portions is considered, then the loading is pure bending. The axial stress is purely in the vertical part of the Z.

Now, since it is pure bending, cozzone plastic bending margin can be used.

In addition, he has to check the fasteners for prying effect.

http://www.stressebook.com

Stressing Stresslessly!

 

[stressebookllc]

Thinking back to the moment continuity part, yes the stresses due to the bending moment and axial load do come together on the center vertical leg right before or after the turn or bend is made. It makes sense to write the margin that way.

http://www.stressebook.com

Stressing Stresslessly!