Is My Calculation for Standard Error of a Stress Calculation Correct?

[charlesltl]

Hello Everyone,

I am trying to compute the standard error of a stress related calculation.

Let's consider the following:

A ± ΔA
B ± ΔB

where A and B are the mean values while ΔA and ΔB are the respective standard errors.

The common combination of errors formulas are as follows:

y = A + B then, Δy = √(ΔA2+ΔB2)

y = k. A then, Δy = k.ΔA

Now my equation is,

y = k2 . [ A + k1 . ( A + B + C ) ]

May I know if my following working is correct?

( A + B + C )
= √(ΔA2+ΔB2 + ΔC2)

k1 . ( A + B + C )
= k1 . √(ΔA2+ΔB2 + ΔC2)

A + k1 . ( A + B + C )
= √[ΔA2 + (k1 . √(ΔA2+ΔB2 + ΔC2))2]
= √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

k2 . [ A + k1 . ( A + B + C ) ]
= k2 . √[ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]I am confused because I was suggested that it should be,
k2 . [ A + k1 . ( A + B + C ) ]
= k2 . √ [k1. ΔA2 + k12 . (ΔA2+ΔB2 + ΔC2)]

Your input in highly appreciated.

Charles

 

[ehild]

The function f=y = k₂ . [ A + k₁ . ( A + B + C ) ] is the same as
y = k₂ [ A(k₁+1) + k₁ B + k₁C ) ]. You can take y as linear combination of A, B, C with the constants p and q (p=k₂(1+k₁), q=k₁k₂):

y= pA+qB+qC

ehild

 

[charlesltl]

I am sorry but I noticed that there are some mistakes in my first post. The subscripts and superscripts are not clearly shown. So the correct equations are as follows:

The common combination of errors formulas are as follows:

y = A + B then, Δy = √(ΔA²+ΔB²)

y = k. A then, Δy = k.ΔA

Now my equation is,

y = k₂ . [ A + k₁ . ( A + B + C ) ]

May I know if my following working is correct?

( A + B + C )
= √(ΔA²+ΔB² + ΔC²)

k₁ . ( A + B + C )
= k₁ . √(ΔA²+ΔB² + ΔC²)

A + k₁ . ( A + B + C )
= √[ΔA² + (k₁ . √(ΔA²+ΔB² + ΔC²))²]
= √[ΔA² + k₁² . (ΔA²+ΔB² + ΔC²)]

k₂ . [ A + k₁ . ( A + B + C ) ]
= k₂ . √[ΔA² + k₁² . (ΔA²+ΔB² + ΔC²)]


I am confused because I was suggested that it should be,
k₂ . [ A + k₁ . ( A + B + C ) ]
= k₂ . √ [k₁. ΔA² + k₁² . (ΔA²+ΔB² + ΔC²)]

------------------------------------------------------------------------------------
Thanks ehild,
I tried you suggestion and ended up as follows:
Taking,
p=k₂(1+k₁),
q=k₁k₂

pA+qB+qC
= √[p²ΔA²+q²ΔB² + q²ΔC²]
= √[(k₂(1+k₁))²ΔA²+(k₁k₂)²ΔB² + (k₁k₂)²ΔC²]
= k₂√[(1+k₁)²ΔA²+(k₁)²ΔB² + (k₁)²ΔC²]
= k₂√[(1+2k₁+k₁²)ΔA²+(k₁)²ΔB² + (k₁)²ΔC²]
= k₂√[ΔA² + 2k₁ΔA² +k₁²(ΔA²+ΔB² + ΔC²)]

It ends up to be different than the earlier solutions that I found and was suggested. Please could you (or anyone) enlighten me.

Thank you in advance.

 

[ehild]

As far as I know, that is the correct expression of the error of your function.

If you have a function f(x,y,z) of variables x, y, z and you know the mean values and standard deviations X±Δx, Y±Δy, Z±Δz, than the error of the function is $$Δf=\sqrt{(\frac{\partial f}{\partial x}\Delta x)^2+(\frac{\partial f}{\partial y}\Delta y)^2+(\frac{\partial f}{\partial z}\Delta z)^2}$$

ehild

 

[haruspex]

= k₂√[ΔA² + 2k₁ΔA² +k₁²(ΔA²+ΔB² + ΔC²)]

I agree with ehild. That is the correct answer.

 

[charlesltl]

Thank you for your help ehild and haruspex.