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Stress/Deflection of Plate

Stress/Deflection of Plate

Stress/Deflection of Plate

(OP)
I'm working on some stress analysis for work. I'm designing a plate, part of which is meant to deflect under a load. Due to this sheet metal plate being somewhat non-straight forward geometry, I'm unsure if I'm attacking it correctly.

I'm putting symmetrical cuts into the plate to allow for easier deflection. Since its not one solid thickness, I didnt know if I could use a cantilever equation. I'm trying to make sure I have an adequate safety factor, but I seem to get numbers I dont trust when I do hand calculations. These numbers also dont match the FEA numbers I get (not that I trust those, either....)

I've included pictures, one of the CAD model showing what the actual plate looks like. The other, a cross section view at the cutout (hopefully this is the correct way to analyze).

Any help or suggestions would be greatly appreciated. I'd like to do well on this project!

http://img210.imageshack.us/i/beam1.png/

http://img121.imageshack.us/i/beam2.png/

I've been using the following equations:

Deflection at B = (Deflection at A)(Height of A/Height of B)

And for stress:

Stress = (3*(Deflection at B)*Modulus of Elasticity*T2)/(2*B^2)

And calculating the safety factor by taking the Yield Stength divided by the Stress.


I have seen this equation, though, for cantilever beams, to calculate deflection:

http://www.advancepipeliner.com/Reso...n_Formulae.pdf

However, for the equation of deflection at a specific point:

deflection = (Force*(B^2)*(3A-B))/(6*E*I)

I'm not sure how to calculate the "I" value. Why would there be inertia, and how do I solve it? When I look up equations, they seem to deal with rotation, and also require I plug in a width value for the plate. I didnt think width would be a factor for this?

Sorry for the long post, and any help is appreciated!

RE: Stress/Deflection of Plate

"Deflection at B = (Deflection at A)(Height of A/Height of B)"

Assuming B is at or above the step, OK, if the ratio of T2 to T1 is large.

"And for stress:

Stress = (3*(Deflection at B)*Modulus of Elasticity*T2)/(2*B^2)"

No. That is not true for beams of varying cross section.

IF B is ///exactly/// at the transition of T2 to T1 then the maximum stress in T1 at the wall is Stress = (3*(Deflection at B)*Modulus of Elasticity*T1)/(2*B^2)

"And calculating the safety factor by taking the Yield Stength divided by the Stress."

Well we're already wildly unsafe as the situation has not been correctly analysed. Did you do elastic beam theory at uni? If so revisit your notes. If not then it is interesting, it is not all that difficult, but it isn't trivial.


 

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies  http://eng-tips.com/market.cfm?

RE: Stress/Deflection of Plate

you say you've got a plate ... how wide ? how are the sides supported ?  it looks like you've done a section thru the applied load (at the middle of the plate ?).  

if your plate is wide, then you can model the plate as a cantilever (if you've got a fixed reaction, two rows of fasteners), and you need to decide on an effective width (probably something less than a 45deg angle projected from the load).

From roark, deflection of a cantilever with a point load is P*L^3/(3*EI) for a beam with a constant I.  For a beam with changing I, read up beam analysis in a strength of materials text ... it's too long to explain here, but it ain't hard math (just a lot of it).

RE: Stress/Deflection of Plate

(OP)
Seems I is mass moment of area. Would I use (1/12)bh^3? With b being the width (I didnt give this, assuming the plate is ~200 mm wide, and the height of the cut is "B", would the I value be I=(1/12)(200)(B)^3?

As far as plate being wide, yes Im going to assume >200mm

RE: Stress/Deflection of Plate

i probably wouldn't use the entire width of the plate, 'cause the sides would react some of the load.  maybe use the distance between the load and cantilevered edge, ie L.

but notice my two cavets ... is the edge fixed ? and how are you accounting for the different thickness ? the formula is for constant thickness, if you're worried about bending stress in the plate, use the thinner t (near the cantilever edge).

btw, your FE is probably giving you way high stresses and displacements.  

RE: Stress/Deflection of Plate

(OP)
I'm still hung up on the I calculation.

It says, I=(1/12)b*h^3

Is the "b", the width (or say 200mm, or less if we dont take the whole thing), and the h is the thickness (say .5mm?, what I designated T1).

Then, in the deflection calculation, the L would be the height of the cutout I drew in the cross section, labeled "B"?

RE: Stress/Deflection of Plate

if this "plate" is 0.5mm thick, it ain't a plate ... formulaes provided are good for deflection < 1/2 the thickness.  if you apply a plate formula you get (other than the wrong answer) a very large displacement and corresponding stress.  the thin membrane deflects and reacts the applied load with in-plane tension.  you Might assume that a strip the width of the plate (200mm i think)is bending in an arc and the applied load is reacted by a component of tension, something like T = P/2/sin(theta) and you determine theta from a right triangle with sides R, R-d, and W/2 (=100mm).

but this is for school, isn't it ?

RE: Stress/Deflection of Plate

(OP)
It's not for school, for a project at work.

Well, the plate would be something like 10 mm thick, it would just have that cut/groove put in to make it bend under less force than a 10mm thick plate would.

RE: Stress/Deflection of Plate

You need to get an engineer to help you, right there in person.
 

Mike Halloran
Pembroke Pines, FL, USA

RE: Stress/Deflection of Plate

"deflection = (Force*(B^2)*(3A-B))/(6*E*I)

I'm not sure how to calculate the "I" value. Why would there be inertia, and how do I solve it? When I look up equations, they seem to deal with rotation, and also require I plug in a width value for the plate. I didnt think width would be a factor for this?"

I agree with Mike, if you don't understand the difference between mass moment of inertia (lbm*in^2) and inertia for stiffness (in^4) and why width is important. A friendly suggestion would be to get a Mechanical Engineer to help you out.  Are you a designer or engineer?

Tobalcane
"If you avoid failure, you also avoid success."  

RE: Stress/Deflection of Plate

I would also like to add that the stress you are calculating is just one piece of the analysis.  You should continue to calculate the principal stresses and ultimately the Von Missies stress so you can use Sy and Su correctly.  So be carful.   

Tobalcane
"If you avoid failure, you also avoid success."  

RE: Stress/Deflection of Plate

(OP)
I am an engineer, I just havent dealt with this since school.

If I make a plate .5mm thick, 3 mm high, and 10 mm wide in CAD, run it through FEA, I get a deflection of 5.21 µm and a stress of 27.24 MPa.

Using the equations I've mentioned, I get 5.57 µm and a stress of 32.03 MPa.

So, these seem accurate. Its the step that throws me off.

The bending should only occur (in a marginal scale) at the "thin" part. The part above that, at 10 mm thick, should not flex.

RE: Stress/Deflection of Plate

Sorry I ment Von Mises

Tobalcane
"If you avoid failure, you also avoid success."  

RE: Stress/Deflection of Plate

kevin,

i'm sure we don't mean to sound like "pissy little sh!ts" but you're asking a whole bunch of really basic questions and you're using some very complicated pieces of software (FEA) that will mislead you quicker than you can imagine.  

And the job you're trying to do is really quite tricky ... if you're trying to control displacement with an undercut, why not use a thinner plate to start with ?

you're making little test models that you can check with hand calcs whihc is good, but it'll only get you so far; you're using complicated tools 'cause you're trying to solve a complicated problem, and it's hard to hand calc a complicated problem.

i imagine that you've tried your full size plate in the FE and gotten a ridiculous answer.  this is alomt certainly because the FEM doesn't apply membrane forces to react the applied load.

RE: Stress/Deflection of Plate

(OP)
rb, yes it is a bit silly. Its frustrating applying some of these basic equations when I'm not quite sure if they apply in the geometry I'm using. The reason I ask them is for some validation that I may use them in the manner here (due to the "step" in thickness). Some of the responses have made me feel a quite foolish for asking, perhaps I should have better thought out my questions.

Yes, the FEA maybe making things worse but I'm hoping its not so far off I cant use it for a reference. Perhaps it is.

I wouldve liked to use one solid, thinner plate, but I need the majority of the plate to be thicker (~10mm) so I can tap it, to hold some alignment optics. I would like to use something like a bolt at the top of this plate, so that I can get very little movement at the bottom (where the optic would be) to "tune" it with relative precision.

RE: Stress/Deflection of Plate

Kevin:
Yes, you probably should feel a bit foolish for that design, and most certainly you should understand your problem better, and think through your question better before asking.  Don't be too embarrassed though, there are plenty of people here trying to do problems that they don't have the foggiest idea about what they are doing, applying formulas they've picked out of the air and don't understand, and if all else fails trying to apply FEA, assuming that'll solve all their problems.  The CAD drawing you show is not a particularly good design for what you are trying to do, but you also still leave so much design info unsaid that it's kinda tough to comment.  How much length do you have for the canti.?  What's the weight of your optics, I'll bet its base is actually fairly stiff to support its own weight.  Where are you aiming it and how do you plan to tune it, for what alignment?  Can you add a 5 or 10 pound wt. under it to help tune the bm. and how much deflection do you want the bm. to have?  How is this contraption fixed to the wall, there will be deflection there too.  Does vibration of the wall effect your equip.?  I'm sure there are another dozen questions, but they are further into the process.  A .5mm strip will not do what you want, machined into the edge of a 10mm pl.  You would want spring steel for that, and to design it as a canti. leaf spring or flat spring.

The equations you want are for a canti. bm. with self wt. on it, plus a point load at the optics, and a length L.  Any Mechanics of Materials book should show you those, but you must understand the subject well enough to apply them.  And, you shouldn't be puzzling about the moment of inertial of the canti. pl. ( I = b d^3/12 ) for a stress and deflection calc., and mass moment of inertia because this isn't a dynamics problem.  That puzzlement shows a fair lack of understanding of the basics of the problem, and prompted the "get an engineer to help you" reply.  Our basic beam formulas and theory are based on small deflection theory, and generally go to hell when you exceed yield or have large deflection to thickness ratios, thus the membrane or thin shell theory comment.

Not knowing any of the answers to the questions above, I'm shooting from the hip, but try this on for size.  My dimensions are intended to suggest general proportions, I haven't run any of the numbers which I would do long hand, not with FEA, and you want to work well within the elastic range so you don't cause a permanent deformation.  You want a welded tee section with the base drilled for bolts to the wall.  Say 100mm high x 200mm long & 5mm thick.  Your canti. bm. is 3 - 4mm thick (that's d above) x 200mm wide ( b above) & 300 or 400mm long ( L above).  Slot holes in this pl. to match your optics base, with slots running from the tip toward the wall, so you can move your load and change the deflection.  Make part of you added weight a base pl. tapped to match the optics base, but sliding with the optics, under the canti. pl.  Provide a means of adding weights out at the tip of the canti. so you can further change its curvature and delta, in the vert. plane perpendicular to the wall.  If you wish you could add more load to one corner tip of the canti. pl., that would cause it to twist and change the alignment of the optics in a vert. plane parallel to the wall.

Work well within the elastic limit of the canti. pl., an approx. stress calc.; you want a fair approx. of delta, so you can set up your equip., but then you are going to fine tune it anyway, then an exact delta calc. isn't that important.  Not even worth turning on the computer.  You gotta study a little and really start to understand your own problem.  The rest is your's to do.
 

RE: Stress/Deflection of Plate

KevinH673, these should be the equations you are looking for, calling W the width where the thickness is T1:
fB=FB2(B/3+(A-B)/2)/(EI) deflection at 'B' due to load F and end moment F(A-B)
E=modulus of elasticity
I=WT13/12 moment of inertia of section
α=FB(B/2+A-B)/(EI) slope at 'B' due to load F and end moment F(A-B)
fA=fB+α(A-B) total deflection at 'A'

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Stress/Deflection of Plate

prex, those equations account for two thicknesses ?

RE: Stress/Deflection of Plate

No, they account for only T1 thickness (the only one that appears in them). With a ratio of 0.5 (T1) to 10 (T2) as proposed by OP, the deflection of the thicker portion is clearly negligible (and for the same reason the full width of the thinner portion should be taken into account, no need for a 45° angle spread from load).

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Stress/Deflection of Plate

(OP)
denghr, thanks for suggestions but this needs to be air tight. I can not go in and add weights, as the box this will go in needs to be sealed. I'm trying to use hand calculations, not FEA, to design it. I am here for guidance on using the correct equations.

prex, thank you for the help. I have tried your equations under 1/2 lbf load (2.18 N). For a plate 59 mm high (point of F), 5 mm high "cut", .75" thickness, and 20 mm wide, my deflection seems to be 7.75 micrometers at the top, and 6.65 micrometers at the point "B". These seem a bit low? Also, wouldnt the deflection at the top part (point A) be much higher than point B?

Could I not take the load at A, and solve for the load at B using moments, F@B=F@B*(A/B), or is this not valid due to change in thickness?

Then, take the load@B, and apply it to a beam of thickness T1, width "W", Length B

Where deflection=(F@B*B^3)/(3*E*I)
where I = W*T1^3/12

I'm hesitant to analyze the whole thing in one piece using deflection equations is that the top, thicker part will not bend (or will bend a negligable amount), all bending will happen at the bottom, thinner part.

RE: Stress/Deflection of Plate

kevin,

i don't understand why you're using an undercut to control deflection ... why not a constant thickness ?  why have 10mm thick plates when you could probably get the desired result with 2mm thick ?  what material ??

you've said the plate is 200mm long.  is this the side of a box ?  how big are the other sides.  you've said a point load.  is that an approximation of the applied loads ?  no distributed load ?  how big ??

i think you're focusing on the wrong part of the problem ... the deflection at A = deflection at B+slope at B*(A/B).

RE: Stress/Deflection of Plate

(OP)
rb,

We need the thickness to be about 10 mm for threading in the optic holder. The reason the undercut is to such a small thickness is because we arent using a large amount of force to cause deflection, just a bolt (double threaded, so internal thread moves micrometers per turn). The part where the cut is (and respectively the optic) we only desire to move hundreds of micrometers. We dont want hardly any movement in the "forward" direction, merely to change the angle. We also need to make sure that we stay in the elastic range.

RE: Stress/Deflection of Plate

(OP)
I'm unsure on calculating the slope. I know the deflection at A by how many turns of the bolt I do. I am not sure on the translation to B, though. I dont expect any bending, or not much, between points A and B due to the thickness of it, I would think that would act almost as a straight line. The movement/bending should happen between the base, and point B. How would you go about accounting for both? I thought a translation of the Force from A to B would allow you to calculate the deflection.

How I would expect it to move
http://img188.imageshack.us/i/draw1t.png/

RE: Stress/Deflection of Plate

sounds like quite a challenge.  how big is the box ?  i ask 'cause we're focusing on deflection between the load and the cantilevered edge and there's a trade-off between deflection to the base (as a cantilever) and deflection with the sides (bending like a beam).

have you thought about using a hinge ?  

RE: Stress/Deflection of Plate

(OP)
I have considered a hinge when we were first looking at this concept here. The problem with it would be too much movement in the "forward and backward" directions. The optic needs to stay, as best as possible, in the same position, while only changing angle. In practice, it of course wont work perfectly, but hopefully we can minimize the movement more with bending. Thanks for the suggestion though.

RE: Stress/Deflection of Plate

So you're designing a flexure, basically etched/ sawcut/ EDMed from the soid.
In which case, I'd just, at least at first, assume that the part doesn't deflect where it's 10mm thick.
In which case the flexures become cantilever beams with a force or deflection applied at some distance beyond the end of the cantilever beam.
I think I'd treat that as a cantilever beam with a moment on the end.
To avoid cross- coupling, you might need to iteratively move the location of the flexures a little until the axis of the central hole (and the axis of the attached optics?) doesn't translate, but only rotates, as the force/deflection is applied to the projected tip of the cantilever.
It might be illuminative to calculate the curvature of the cantilever, in which case the geometry of the solution becomes analgous to a tube bending problem, just with an extra large radius and extra small angle.
A couple hours iterating the equations in Excel should give you a good feel for it.


 

Mike Halloran
Pembroke Pines, FL, USA

RE: Stress/Deflection of Plate

i think your grove will act just like a hinge ... it'll set up a slope at B ... what if you put the hinge in the mid-depth of the plate, you want rotation without displacement.

RE: Stress/Deflection of Plate

KevinH673, what you plan to do 'The movement/bending should happen between the base, and point B. How would you go about accounting for both?' is exactly what is in my equations and is also suggested by MikeHalloran:
FB3/3EI  is the deflection at 'B' due to F translated at 'B'
F(A-B)B2/2EI  is the deflection at 'B' due to the moment of F acting at 'A'
FB2/2EI is the rotation at 'B' due to F translated at 'B'
F(A-B)B/EI  is the rotation at 'B' due to the moment of F acting at 'A'
Multiplying the rotation by A-B you get the increase in deflection (as a straight line as you say) from 'B' to 'A'.
All this should sum up to:
FB(B2/3+A2-AB)/EI
Hope this is the same as in my preceding post blush

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Stress/Deflection of Plate

i get ... deflection @ A =F/EI*([A-B]^3/6 + A^2*B/2 - A^3/6 + ([A-B]^2/2 - A^2/2)*[A-B] ... maybe it's the same

RE: Stress/Deflection of Plate

yeah, it's the same as prex's.

but i really don't think you will get the results you're looking for, rotation without translation

RE: Stress/Deflection of Plate

(OP)
Hi guys. This project was put on the backburner for a little while, and I've been back to it. I took your advice and went to the textbooks. I'm really not trying to use this site as a crutch, but I do find it as a good source to clear things up.

It may prove the design doesn't work. It is not my design, but I'd atleast like to validate it if it doesn't.

prex, back to your equations. Why do you sum up all of these? I thought if you choose one way to analyze it (such as using the moment), why do you also calculate using force such as a cantilever beam equation?

If you guys have some reading you could direct me to, I'd appreciate it. Trying to learn (or re-learn I should say) some of this.

RE: Stress/Deflection of Plate

Kevin, as I understand what you are doing you are creating a displacement with a threaded connection.  This creates a known displacement.  I think you need to start with the known displacement, at A the free end, and determine resulting displacment at some other location, at B.  You are describing a displacement input problem rather than a force input problem.
Perhaps this example may serve you:
http://www.engineersedge.com/beam_bending/beam_bending10.htm

Knowing the end displacement, calculate the displacement at the unknown point load and the unknown point load located between the free end and the fixed support.  Then calculate the stress in the flexing length to evaluate plastic or elastic result.  I assume the threaded adjustment does not introduce a moment to complicate the deflection of the unloaded end of the beam/plate.

Ted

RE: Stress/Deflection of Plate

KevinH673, it's simply substitution of a system of forces with an equivalent one. The force at A is equivalent to a force at B plus the moment F(A-B).
I'm not sure of what you are trying to do exactly (perhaps because I had forgotten everything of this thread): you say you want to (in)validate a design, but against what? You need something to compare with to validate.

prex
http://www.xcalcs.com : Online engineering calculations
http://www.megamag.it : Magnetic brakes and launchers for fun rides
http://www.levitans.com : Air bearing pads

RE: Stress/Deflection of Plate

(OP)
Well, I'm trying to make sure I have all my "ducks in a row" so that I can predict how this thing will behave. I can control the deflection at A, as there will be a bolt pressing against it controlling it. My goal is to predict how point "B" will deflect relative to "A". I also have FEA to analyze, which I only am using to validate the hand calculation.  

RE: Stress/Deflection of Plate

(OP)
"The force at A is equivalent to a force at B plus the moment F(A-B)."

prex, wouldn't that mean the force at A is larger than the force at B? Shouldn't it be the opposite?

Also, this claims slope is the integral of M/EI with respect to x, so slope = Mx/EI, or F(A-B)*B/EI as has been posted above, and that d^2y/dx^2 = M/EI; however, another site I had read claims d^2y/dx^2 to be Mx/EI. Which is correct? I have the link to one site here:

http://www.roymech.co.uk/Useful_Tables/Beams/Beam_theory.html

and the image that shows the equation:
http://www.roymech.co.uk/images11/beam_21.gif

However, I will attach the picture of the equation that says d^2y/dx^2 = Mx/EI

Which is correct? I assume the former, since it has been mentioned before. I'm just confused by the discrepency.

RE: Stress/Deflection of Plate

i believe your posted equation says M is function of x, sort of like f(x).

d2y/dx2 = M/EI ... M is a function of x ... integrate for slope (dy/dx), and again for displacement (y)

RE: Stress/Deflection of Plate

(OP)
rb, ok this makes sense then. I thought the x was an actual value. Thanks.

RE: Stress/Deflection of Plate

if you have a point load.  why don't you make your "cantilevered" section much much narrower, and keep some thickness in your "undercut" area so you can truly apply a straight beam calculation?  With the accuracy of measuring devices you should be able to pick up any changes in geometry.

I think the current approach is untenable as far as getting consistent results, because with that thin a piece of material you have much more potential problems with non-homogeneous effects, alot of stresses from machining.

RE: Stress/Deflection of Plate

Kevin,

This problem screams for an FEA that incorporates geometric non-linearity.  Do you have access to such expertise?  If not, you should sub-contract it.  This would help validate the complex hand-calcs.

tg

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