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Non-ME
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Formula for calculating force to bend a metal beam?

11/03/2008 5:04 PM

How do I calculate how much force is required to deflect a tiny beam of spring steel .020" when force is applied to the center of the beam. The beam is .550" long and .100" wide and about .007" thick.

Unfortunately I don't know a lot more about the spring steel characteristics.

Online references would be great.

Thanks,

Dave
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ba/ael
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#1

Re: Formula for calculating force to bend a metal beam?

11/03/2008 6:26 PM

Δ = deflection

L = span

E = Modulus of Elasticity (I assumed 30,000,000 psi)

I = Moment of Inertia

b = width

d = depth

Δ = PL3/(48EI)

I = bd3/12 for a rectangular cross section

I think you will need about 1/2 pound to deflect the beam 0.02". This should produce a maximum flexural stress of about 840 psi at midspan (not very much).

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#2
In reply to #1

Re: Formula for calculating force to bend a metal beam?

11/03/2008 7:29 PM

Thank you Bruce. That's about the force I was looking for. Where did you come up with 30,000,000 psi? Any chance you can list the units used in this equations?

- Dave
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ba/ael
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#3
In reply to #2

Re: Formula for calculating force to bend a metal beam?

11/03/2008 8:37 PM

The modulus of elasticity doesn't vary much among steels. I looked up the properties of spring steel and it listed E as 210,000 MPa (or about 30,450,000 psi).

The units fall into place naturally. They can be metric or Imperial. Since you provided the width and depth of the beam in inches, I worked in Imperial.

So b = 0.1", d = 0.07", L = 0.55", I = 2.858e-9 in4, E = 30,000,000 psi (pounds per square inch). Deflection Δ is in inches and P, the force applied at midspan is in pounds.

If all of these terms had been entered in metric units, i.e. Newtons and millimeters or combinations thereof, P would have been given in Newtons.

The expression Δ=PL3/(48EI) may be rewritten as P=Δ*(48EI)/L3 to solve for P where P is the force required to cause a deflection of Δ.

So P=0.02(48*30e6*2.858e-9)/0.553 = 0.495# (say 1/2 pound).

The units of P must be pounds because all the terms had consistent units and the unit of force was pound.

It is always wise to do a dimensional analysis of an expression to make sure that the units work out to what you expect. Using the letters F and L for force and length, the expression for P is dimensionally L*(F/L2*L4)/L3 which is F*L5/L5 which is F. So P will be in force units which is what we expect.

The expression for Moment of Inertia, bd3/12, dimensionally is L*L3 or L4. So "I" will be in in4 (or mm4 if you work in metric units).

Hope this helps in your understanding.

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#4
In reply to #3

Re: Formula for calculating force to bend a metal beam?

11/04/2008 10:21 AM

Excellent Bruce.

In case anyone cares, there was one typo (that didn't make it into the equations): d = .007" (not .07").

Thanks again,

Dave
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ba/ael
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#5
In reply to #4

Re: Formula for calculating force to bend a metal beam?

11/04/2008 1:43 PM

Actually, it did make it into the equations because the moment of inertia, I = bd3/12 includes both b and d. And the expression for P includes I. When you think about it, the beam dimensions must have something to do with deflection.

As it turns out, I used a depth of beam of 0.007" when checking deflections but I used a depth of 0.07" when calculating bending stress and incorrectly reported a maximum stress of 840 psi (my bad). The maximum stress turns out to be 84,100 psi which is a substantial stress.

To convert to psi, multiply by 145. The yield point of this steel is very high (1100 MPa or about 160,000 psi) so a stress of 84,000 psi represents approximately half of its yield value.

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omw7
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#6

Re: Formula for calculating force to bend a metal beam?

11/05/2008 5:42 AM

This is not a linear calculation and the beam theory that you are applying is inadequate.

Deflection is 20/550=1/27.

Depth to span is 7/550= 1/79

Both of these would indicate that beam theory is an approximation.

Further questions would be: what are the support conditions? Are they fixed and does catenary action play a part (at 1/27 deflection this is the case). Does the way the spring is attached give rise to spring support conditions which increases the difficulty. How accurate does the deflection have to be?

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ba/ael
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#7
In reply to #6

Re: Formula for calculating force to bend a metal beam?

11/05/2008 11:58 AM

You are quite right. The deflection is approximate as most structural calculations are. It is tantamount to finding that a simple beam having a span of 27'-6" deflects a foot at midspan. That is a bit much for small deflection theory.

Boundary conditions are assumed to be hinge at one end and roller at the other, i.e. a simple beam. If the ends are restrained, either horizontally or rotationally, the deflection will be considerably less. Alternatively, it will take a larger force to produce a stipulated deflection.

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