Principle Stresses in Cantalever beams

KJohnston

Hi all,

I have a problem with cantalever beams, the beam is 1m long by 0.1m by 0.1m. A 10Kn force placed at the end of the beam deflecting it down. My lecturer simmulated the problem in Ansys and we all got to do it with him using more dense nodes and solutions to find a more refined answer of the deflection in y and principle stresses in x and y direction.

The results are clear from the program but we have been asked to calculate the theoretical values and compare. Deflection is okay i derived the equation

v = 1/EI(5000z^2-1666.67z^3)

this has given me an answer which is quite similar to the simulated. My question is how do i calculate the principle stresses. I have calculated the Bending stress using

Bending stress = My/ I

but i fear this is not what i need,because my values are very different. how do i relate bending stress to sigma (X) and sigma (y) or calculate them from the force and dimensions given?

Thanks
KJ

pongo38

Do you mean 0.1 x 0.1 solid square section, circular section, or hollow section?

KJohnston

Sorry to clarify it is a solid square section and since then i have discovered that sigma (X) is indeed calculated using

Sigma (x)= MC/I

Using this result in a excell spreadsheet gives a 1% difference from the actual stress and theoretical stress given by ansys using a 4 by 40 node anaylsis which is what is expected, but i am unsure how to calculate sigma (y), i would really appreciate any suggestions as i have spent alot of time on this problem and have of yet failed to find a solution.

Thanks
KJ

Studiot

I am puzzled by your question 'principle stresses in a cantilever'

Stresses in a cantilever, or any structural member, vary from point to point.

But which point?

Please note the spelling of cantilever.

KJohnston

yes excuse the spelling mistake, i am looking for the maximum stress in y direction at any point in the beam,i believe the maximum bending stress (sigma x)occurs on the top surface of the beam(positive) at the restraint, this is also where ansys says it occurs, it also says this is where sigma y is maximum? but i am unsure how to calculate it

here is what i have done so far, i hope i made it easily understood

Cantalever Beam

Young Modulus (E) (Pa)= 2.00E+11
2nd moment of Area (I)(m^4)= 8.33E-06
I = bd^3/12 (b = 0.1) (d = 0.1)
Force (F)(N)= 1.00E+04
Distance from NA (Y)(M)= 5.00E-02
Deflection (v)(m) V=1/EI(5000z^2-1666.67z^3) Bending Moment (M) (Nm) M= 10000z - 10000
Stress in X (σ)(Pa) σ = (M*Y)/I
Stress in Y = ????????????????

(z) (v) (M) (σ)x

0 0 -1.00E+04 -6.00E+07
0.05 7.37795E-06 -9.50E+03 -5.70E+07
0.1 2.90116E-05 -9.00E+03 -5.40E+07
0.15 6.41507E-05 -8.50E+03 -5.10E+07
0.2 0.000112045 -8.00E+03 -4.80E+07
0.25 0.000171944 -7.50E+03 -4.50E+07
0.3 0.000243097 -7.00E+03 -4.20E+07
0.35 0.000324755 -6.50E+03 -3.90E+07
0.4 0.000416166 -6.00E+03 -3.60E+07
0.45 0.000516581 -5.50E+03 -3.30E+07
0.5 0.00062525 -5.00E+03 -3.00E+07
0.55 0.000741421 -4.50E+03 -2.70E+07
0.6 0.000864345 -4.00E+03 -2.40E+07
0.65 0.000993272 -3.50E+03 -2.10E+07
0.7 0.00112745 -3.00E+03 -1.80E+07
0.75 0.001266131 -2.50E+03 -1.50E+07
0.8 0.001408562 -2.00E+03 -1.20E+07
0.85 0.001553995 -1.50E+03 -9.00E+06
0.9 0.001701679 -1.00E+03 -6.00E+06
0.95 0.001850864 -5.00E+02 -3.00E+06
1 0.002000798 0.00E+00 0


these are my theoretical, can you help calculate sigma y??

sorry the table columns are a bit messed up, it looked real nice in the editing window lol

KJ

nvn

KJohnston: Please check your spelling of principal. Basically, for a cantilever, sigma_1 = sigma_x (if Mx = 0 N*mm), and sigma_y = 0 MPa, if you are modeling your cantilever with beam finite elements. Are you using beam, shell, or solid finite elements?

KJohnston

Its Solid finite elements, thanks and in Dundee University scotland UK its spelt principal stress. thanks

KJ

nvn

KJohnston: sigma_y for a beam is zero except near boundary conditions (BCs). Your solid finite element model might show a nonzero, localized sigma_y stress in the vicinity of the cantilever support, which rapidly dissipates as you move away from the support, per St. Venant's principle. You might also see some localized, nonzero sigma_y underneath the applied load. The sigma_y stress could be difficult to calculate. It depends on bearing area of the BCs, how you model the beam, etc.

KJohnston

That is exactly what Ansys results show, i think sigma y could be disregarded from my answer as it does not contribute to deflection, thanks for the help.

Ps it would have been nice to work it out,just out of curiosity

KJ

TheJCBand

Principle stresses can be found easily using Mohr's circle. You can read about it in a mechanics of materials textbook. Basically, Mohr's circle is a plot of all the shear and normal stresses in all planes of an element, and the angle on the circle is related to the angle of the plane.