Young's Modulus of Elasticity for a beam of multiple materials

[jsanders2008]

I am trying to calculate the maximum deflection of the wall of a walk-in air-handling unit. The wall is comprised of two steel sheets with foam insulation inbetween, and can be considered a rectangular beam, fixed at two ends, with a uniform load.

As I understand it, the equation for calculating deflection in a beam fixed on two ends with a uniform load is as follows:

d = 5 * F * L^3 / 384 * E * I,

where d is the deflection of the beam,
F is the force of the load,
L is the length of the beam,
E is the modulus of elasticity (Young's modulus) of the material, and
I is the second moment of area.

In this case, I believe the second moment of area would be as follows:

I = b * h^3 / 12,

where b is the width/depth of the beam, and
h is the thickness of the beam.

(The dimensions of the beam are L x b x h. See attached file for an illustration.)

Assuming the previous information is correct, my problem is that I don't know what to use for the modulus of elasticity (E), because there are three layers in the beam, which are made of two different materials.

I'm hoping that there is a way to calculate an "equivalent modulus of elasticity" based on the materials and thicknesses of each layer in the beam. Does anyone know if this is possible, and if so, how to do it?

I appreciate all help. Thanks.

 

[RPstress]

This is a sandwich plate; you can probably safely treat it as a sandwich beam. I recommend accessing

http://www.hexcel.com

and

http://www.diabgroup.com

and downloading their technical sandwich analysis handbooks.

http://www.hexcel.com/NR/rdonlyres/...B3-7EFF685966D2/0/7586_HexWeb_Sand_Design.pdf

http://www.diabgroup.com/europe/literature/e_pdf_files/man_pdf/sandwich_hb.pdf

http://www.diabgroup.com/americas/u_literature/u_lit_man.html

has more info.

The Hexcel one is marginally more accessible. It's based on honeycomb core rather than foam, but it pretty much all still applies for the basics.

You're equations are about right, except for I. You really need that to be for sandwich. If the two faces are equal thickness t separated by distance d (the foam thickness) then
I = t * w * (d+t)^2 / 4.

Strictly, you should use
I = (t^3 * w /12) + (t * w * (h+t)^2 / 4).

Also strictly, you should allow for the foam core shear deflection as well. See the handbooks referenced above for more.

In addition, I'd pay some attention to the loading. For instance, is it distributed, or one or more point loads.

 

[jsanders2008]

Thanks for your help, RPstress. I really appreciate it.

I think I'm going to base my calculation off of the first example problem from the hexcel article (page 12), modifying the deflection coefficients to make it a uniform load distribution, which is the case for me.

 

[jsanders2008]

Correction:

When I use P = q / b (page 11) in the equation for deflection on page 13, all units cancel out, leaving me with a dimensionless answer for deflection, which is obviously not correct.

What am I doing wrong?

 

[jsanders2008]

Correction #2:

I checked the diabgroup document and I believe I've figured out what I did wrong; the equation on page 13 on the hexcel article needs another L in the numerator of both terms in order to account for the fact that P = q / b.

 

[RPstress]

Oops, my I's were the usual stressman's factor of two out! (Too small, so I was probably safe for statics...)

q in the Hexcel manual is a load (force) per unit length, aka a "running load", so it has units of force/length (e.g., lb/in). Dividing it by the width gives a pressure (e.g., psi).