# Principle Stresses in Cantalever beams

• KJohnston
In summary: The maximum and minimum normal stresses on the circle correspond to the principle stresses.In summary, the group is discussing a problem with cantilever beams and comparing theoretical values to simulated values in Ansys. They have successfully calculated the deflection using an equation and bending stress using a spreadsheet, but are unsure how to calculate the principle stresses. They discuss the possibility of disregarding sigma y and using Mohr's circle to find the principle stresses.
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

KJohnston said:
Hi all,

I have a problem with cantalever beams, the beam is 1m long by 0.1m by 0.1m.

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

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 a lot of time on this problem and have of yet failed to find a solution.

Thanks
KJ

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.

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

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?

Last edited by a moderator:
Its Solid finite elements, thanks and in Dundee University scotland UK its spelt principal stress. thanks

KJ

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.

Last edited:
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

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.

## 1. What are principle stresses in cantilever beams?

Principle stresses in cantilever beams refer to the maximum and minimum stresses that occur at a specific point on the beam, perpendicular to the cross-section. These stresses are caused by external loads acting on the beam and can greatly affect its structural integrity.

## 2. How are principle stresses calculated in cantilever beams?

The calculation of principle stresses in cantilever beams involves using the bending moment and the second moment of area of the beam's cross-section. These values are then used in the flexure formula to determine the maximum and minimum principle stresses at a given point on the beam.

## 3. What factors affect the principle stresses in cantilever beams?

The principle stresses in cantilever beams are affected by several factors such as the magnitude and direction of external loads, the shape and size of the beam's cross-section, and the material properties of the beam itself. The location of the load on the beam can also have a significant impact on the principle stresses.

## 4. How do principle stresses impact the design of cantilever beams?

The principle stresses play a crucial role in the design of cantilever beams as they determine the maximum load that the beam can withstand without failure. Engineers must carefully consider the principle stresses when designing cantilever beams to ensure that they are strong enough to support the intended loads.

## 5. Are there any limitations to the calculation of principle stresses in cantilever beams?

Yes, there are limitations to the calculation of principle stresses in cantilever beams. The calculations assume that the beam is made of a homogeneous, isotropic material and that the load is applied gradually. In reality, beams may be made of composite materials and may experience sudden or dynamic loads, which can affect the accuracy of the calculated principle stresses.

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