Calculating the Stress on a Cantilever beam

[sbaseball]

Homework Statement

I have a piece of metal with one end fastened to a table by a clamp and the other end off the table. I only know the length of the beam and the strain. I am trying to figure out the stress in order to figure out youngs module so the metal can be identified. How do I do this?

Homework Equations

I have tried:
stress = force/area

The Attempt at a Solution

By using the above equation and then calculating youngs module I get something that is not nearly close to the characteristics of a metal.

 

[Spinnor]

See the second example,

http://www.codecogs.com/reference/engineering/materials/beams/cantilever_beams.php

 

[sbaseball]

What does the Z term in the equation represent?

l = the length of the cantilever (m)
x = the length to the bend (m)
W = load (N)

 

[Spinnor]

From near the top of page 6 of,

http://www.freestudy.co.uk/statics/beams/beam tut1.pdf

it looks like Z = I/y where y is half the beam height.

 

[sbaseball]

Thank You for all your help. The links have been very helpful. This is all the work that I have done and I still seem to be getting the wrong answer

The beam is .2 meters long and .025m wide with a height of .003m

Stress = W/Z ( L – X)
Z = I/y
I = (B*D^3)
B = .2
D = .003

I = (.2*.003^3)/12
I = 4.5e-10

Z = I/y
y = .5 (height)
Z = 4.5e-10/.0015
Z = 3e-7

Stress = (49.05 N/3e-7) (.2 -.08)
Stress = 19620000 Pa

E = Stress/Strain
E for particular aluminum = 73.1 GPa
Strain = 2675e-6
Stress = E*Strain = 195323200

Stress measured = 19600000 Pa
Stress expected = 195323200 Pa

As you can see I am not getting the expected value of the stress. Do you see anything that I am doing wrong? Thank you

 

[Spinnor]

I am guessing you are off by a factor of ten %^) . Will look for one.

 

[Spinnor]

From page 6 of,

http://www.freestudy.co.uk/statics/beams/beam tut1.pdf

I = BD^3/12


Not quite a factor of 10 %^(

 

[sbaseball]

Yes you are right i forgot to type that into the equation but in my calculation I did divide by the 12 so the results should be right and yes i am off by a factor of 10 which leads me to believe it is not in the equations but maybe the use of a wrong number

 

[nvn]

sbaseball: (1) In post 5, why did you say the beam width is 25 mm, but then you used a beam width of 200 mm in your calculations? What is the actual width of your beam? Correct this mistake, and try again. Show your work, so we can check your math.

(2) What is the exact thickness (height) of your beam? If your thickness value is even slightly inaccurate, then your answer will be inaccurate.

(3) Where did your strain (eps = 2675e-6) come from? How did you obtain that value? If eps (epsilon) is inaccurate, then your answer will be inaccurate.

(4) By the way, numbers less than 1 must always have a zero before the decimal point. E.g., 0.025 m, not .025 m.

(5) Always leave a space between a numeric value and its following unit symbol. E.g., 0.025 m, not 0.025m.​

 

[sbaseball]

I guess then I am confused as to what number to use where. I looked back through all the links in post 2 and 4 and I thought

B = length = 0.2 m
D = height = 0.003 m
y = half the height = 0.0015 m
l = length = 0.2 m
x = length to bend = 0.08 m

This is what I thought the links that Spinnor gave to me meant. Also the value of strain was measured by placing 5 kg weight at the end of the cantilever beam.

 

[nvn]

sbaseball: Parameter B (usually called lowercase b) is cross-sectional width, not beam length. What is the actual width of your beam? Correct this mistake, and try again.

Also, see item 2 in post 9.

 

[sbaseball]

The beam is 0.2 meters long and 0.0235m wide with a height of 0.0015m

Stress = W/Z ( L – X)
Z = I/y
I = (B*D^3)/12
B = 0.0235
D = 0.0015

I = (.0235*.0015^3)/12
I = 6.609e-12

Z = I/y
y = .5 (height)
Z = 6.609e-12/.00075
Z = 8.81e-9

Stress = (49.05 N/3e-7) (.23 -.08)
Stress = 8.35e8 Pa

E = Stress/Strain
E for particular aluminum = 73.1 GPa
Strain = 2675e-6
Stress = E*Strain = 195323200

Stress measured = 8.35e8 Pa
Stress expected = 195323200 Pa

I now have the right order of magnitude but as you can see the value is still off by a fair amount. I have doubled checked all the measurements and adjusted them accordingly
the height of the beam is 0.0015 m. Could this error be due to some sort of experimental error or is there still a chance that I am doing something wrong.

 

[nvn]

sbaseball: It seems you did not double-check, nor even check, your numbers. Why do you have contradictory values for beam length listed in post 12? What is the correct beam length? And why do you keep changing your values? Please proofread what you type.

Also, why did your beam height change from 3 mm to 1.5 mm? That does not make sense. Did you really measure it? Also, why are your height values exactly 3.00 mm, or exactly 1.50 mm, if you really measured your beam accurately? Are you using a micrometer? See item 2 in post 9. Also, please note the following.

(1) Always leave a space between a numeric value and its following unit symbol. E.g., 0.0235 m, not 0.0235m. See the international standard for writing units (ISO 31-0).

(2) Numbers less than 1 must always have a zero before the decimal point. E.g., 0.5, not .5. See the above links.​

Also, why do you think everything needs to be converted to meters here? Instead, use convenient units (N, mm, MPa).

 

[sbaseball]

Yes I am using a micrometer and I remeasured everything like you suggested and found that some of my values were off as you thought they could be. The beam is 0.23 m long I merely forgot to change the number. I used the correct value in the calculations. I changed what values I was using becasue I was using the beam length where I should have been using the width. Also I convert everything into meters becasue that is the easiest way for me to calculate everything and think about things. Is converting it to standard meters wrong? The thickness of the beam is 0.0015 m.

 

[nvn]

sbaseball: Your "stress measured" in post 12 is not even close to the expected stress. Therefore, it appears you measured something wrong. I currently suspect your thickness is grossly wrong. Are you sure you are reading your micrometer correctly? If you were, why would the beam be exactly 1.500 mm thick? And before, it was exactly 3.000 mm thick? There is nothing wrong with your calculations. And your discrepancy is too large to be experimental error. Therefore, it currently appears one or more of your measurements is wrong.

 

[sbaseball]

I have remeasured everything and I believe my error was in the length to the bend. The value I was using was much to larege and by then adjusting that and making my dimensions more precise I got a much more accecptalbe answer