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Material properties analysis of rectangular beam section

Material properties analysis of rectangular beam section

Material properties analysis of rectangular beam section

(OP)
Hello all,

I'm wondering if anyone has insight into a materials engineering problem. I'm performing a non-destructive and a destructive bending test, destructive compression test, electrical resistance, and hardness test on two different wood beams that each have a different sized rectangular cross section, different material, and different length. Beam 1 is a new composite wood material, and beam 2 is a uniform unknown wood species (might be oak). The dimension are listed below.

Beam 1: 5.5" x 5.5" x 60" (new composite wood material)

Beam 2: 9" x 9" x 72" (wood species unknown)

They are different wood species, and I'm wondering how to compare the engineering properties obtained from these different strength tests that I'm going to perform. I'll be obtaining the modulus of elasticity and modulus of rupture from the bending test. The max compressive strength at failure, and the hardness using a janka ball test. In general, I'm wondering how the strength properties compare, that is : Is the new composite wood material (Beam 1) more resistant to deflection than the Beam 2, is it generally stronger under a bending/compressive load environment? The problem I'm running into is that it's difficult to compare the test results of two different cross sectional areas.

Any advice?

Thanks,

RE: Material properties analysis of rectangular beam section

You can calculate the properties of interest to make a comparison.

S = bd2/6 (section modulus)
I = bd3/12 (moment of inertia)
In the above, b is width and d is depth.

If you do a bend test, the maximum fiber stress in the beam is M/S where M is the bending moment. If you use a single load at midspan, M = PL/4 and Δmax = PL3/48EI. So you can relate maximum fiber stress and modulus of elasticity for the two specimens. To find the strength of each sample, you would need to carry the test to destruction, so make sure you do that last.

BA

RE: Material properties analysis of rectangular beam section

I would think you'd want to figure that out before you made any tests, or even planned to make them.

RE: Material properties analysis of rectangular beam section

(OP)
Thanks for your help BA, makes sense. I have a couple of questions about this.

It would be an easier conceptual comparison if the two beams had the same cross sectional area, but they don't. I have a question about the modulus of elasticity first though:

How do I make a direct comparison of the modulus of elasticity of Beam 1 and Beam 2, which is a result of the bending test, since the cross sectional areas are different? I mean, lets say that I get E1 to be 1,000,000 psi and E2 to be 2,000,000 psi. One question I have is, if I were to have Beam 1 with a larger cross section, say instead of 5.5 x 5.5 the cross section is 10.5 x 10.5, would the modulus of elasticity increase? Is the modulus of elasticity a function of the geometry of the cross section? Or is it an inherent material property that won't change with changing cross section?

From what I understand, the modulus of elasticity is the ratio of the change in stress with respect to the change in strain. So I assume that since the cross section increased, then the moment of inertia increased, so the strain should decrease with the same load applied, which would affect the modulus. Correct?

Now, lets think about this a little bit differently: Lets say that the cross sectional dimensions of Beam 1 and Beam 2 are the same, and the lengths of the beams are the same. So the only difference is the material that the beams are made out of. In this case, it seems to me to be an easier comparison of the engineering properties obtained. For example in this case, if I got E1 to be 1,000,000 psi, and E2 to be 2,000,000 psi it's clear that the Beam 2 would be stiffer than Beam 1. In this case, since the cross sections are the same sizes, I can state that the material in Beam 1 is generally more ductile than the material in Beam 2. That is, the material is Beam 1 is less resistant to deformation. How can I make the same type of analysis/conclusions, with the above example where the beams have differing cross sectional areas and lengths?

Here is another question I have: If I carry beam 1 to failure in the bending test and I determine the modulus of elasticity in the elastic region, and the modulus of rupture. Is it possible to estimate what the modulus of rupture, or the modulus of elasticity would be if the beam was a larger cross section. So, if I determine that the modulus of elasticity of Beam 1 which has a cross sectional dimensions of 5.5" x 5.5" to be 1,000,000 psi and the modulus of rupture is 11,000 psi. Can I estimate what the modulus of rupture and modulus of elasticity would be if the beam was 10" x 10" instead of 5.5" x 5.5"?

Thank you,

RE: Material properties analysis of rectangular beam section

The modulus of elasticity is a material property, independent of the size or shape of the beam. If you look up formulas for deflection in beams, they include the modulus of elasticity as one variable, the moment of inertia as another variable, and you can calculate the effects of changing beam dimension.
I suppose you could deduce the modulus of elasticity from load tests on a full-size beam without taking samples.
I'm not a wood expert, but I would assume that the modulus might vary by direction in wood.

RE: Material properties analysis of rectangular beam section

(OP)
O.k.

So you're saying that if I conducted a bending test on two separate beams, where beam 1 and beam 2 are the same material and the only difference is the cross sectional size, then the resulting stress strain curve would be the same slope in the elastic region for both members? Of course, they would never be exactly the same, but theoretically they should be? Irregardless of the cross sectional area difference?

Thanks,

RE: Material properties analysis of rectangular beam section

Beam 1 5.5 x 5.5 beam, span 60"
S = 5.5(5.5)2/6 = 27.73 in3
I = 5.5(5.5)3/12 = 76.25 in4

Beam 2 9 x 9 beam, 72" span
S = 9(9)2/6 = 121.5 in3
I = 9(9)3/12 = 547 in4

Strength Comparison
Suppose that Beam 1 fails when the maximum fiber stress is 2,000 psi.
Since f = M/S, M = f.S = 2,000(27.73) = 55,460 "# at failure
PL/4 = M so P = 55,460*4/60 = 3,697#

If Beam 2 also fails at 2,000 psi, then M at failure = 2,000(121.5 = 243,000"#.
and P = M*4/L = 243,000*4/72 = 13,500#

If Beam 2 fails at a higher or lower value, it would display proportionately higher or lower strength than Beam 1.

Deflection Comparison

Assume Beam 1 deflects 0.25" when P = 2,500#
Then PL3/48EI = 0.25
E = PL3/48*0.25I = 2500(60)3/(12*76.25) = 590,000 psi

If Beam 2 has the same E value as Beam 1, then:
Δ = PL3/48EI = P(72)3/48*590,000*547 = 2.41e-5(P)
Or P/Δ = 48EI/L3 = 48*590,000*547/723 = 41,500#/"

If E for Beam 2 is higher than E for Beam1, the mid-span load per inch of deflection would be proportionately higher.

Please note: the above results were calculated while typing and may be incorrect. They need to be checked.




BA

RE: Material properties analysis of rectangular beam section

The properties of wood are quite variable even for the same species and grade. If the material were purely elastic, the formulas given in my earlier posts are correct but wood is not a perfectly elastic material. The presence of knots may affect strength values.

Under load, wood tends to creep so that long term deflection is greater than instantaneous deflection. As noted by JStephen above, wood properties are not the same in all directions because wood is not an isotropic material. The direction we are primarily considering here is parallel to the grain.

BA

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