Question

How to calculate uncertainty for deflection and stress? using the results from the table. v=-P(L^2/ 2EI) , PLymax/ I. I is 6.8x10^-11m^4.

Mass (kg)	Deflection (mm)	Stress (MPa)
0.000	-0.020	0.030
0.068	-0.270	3.418
0.118	-0.460	5.822
0.167	-0.640	8.252
0.217	-0.830	10.636
0.267	-1.010	12.998
0.316	-1.200	15.297
0.366	-1.370	17.705
0.415	-1.570	20.156
0.465	-1.750	22.601
0.515	-1.930	24.887

Answer 1

Standard Deviation (S.D.) is the best measure for Uncertainty. Standard Deviation needs to be calculated separately for deflection and stress using the values given in the table.

Formulae required for manual S.D. calculations are as follows:

$S.D.=\sqrt{\left \{ \frac{\Sigma d^{2}}{n} -\left ( \frac{\Sigma d}{n} \right )^{2}\right \}}$

where,

n- total number of values

c = individual values of stress (or deflection) in this case

Sum of all values of x ΣΧ -mean-

$d=x-\bar{x}$

Σ01-sum of all values of d

d- sum of all values ofd

Manual Calculations:

S.D. OF DEFELECTION

From Table 1,

$n=10$

zー一1.0045454545

$\Sigma d=0$

12 3.8724727273

S.D. in deflection using formula = 0.6222919514 mm

Table 1. Uncertainty in Deflection

Deflection, x	d=x-mean	d2
-0.02	0.984545455	0.969329752
-0.27	0.734545455	0.539557025
-0.46	0.544545455	0.296529752
-0.64	0.364545455	0.132893388
-0.83	0.174545455	0.030466116
-1.01	-0.005454545	0.000029752
-1.2	-0.195454545	0.038202479
-1.37	-0.365454545	0.133557025
-1.57	-0.565454545	0.319738843
-1.75	-0.745454545	0.555702479
-1.93	-0.925454545	0.856466116

S.D. OF STRESS

From Table 2,

$n=10$

$\bar{x}=12.8910909091$

$\Sigma d=0$

$\Sigma d^{2}=651.6623789091$

S.D. in stress using formula = 8.0725608013 MPa

Table 2. Uncertainty in Stress

Stress, x	d=x-mean	d2
0.03	-12.8610909091	165.4076593719
3.418	-9.4730909091	89.7394513719
5.822	-7.0690909091	49.972046281
8.252	-4.6390909091	21.5211644628
10.636	-2.2550909091	5.0854350083
12.998	0.1069090909	0.0114295537
15.297	2.4059090909	5.7883985537
17.705	4.8139090909	23.1737207355
20.156	7.2649090909	52.7789040992
22.601	9.7099090909	94.2823345537
24.887	11.9959090909	143.9018349174

Alternatively:

Spreadsheet Software like Microsoft Excel have built-in function for Standard Deviation. Table 3 gives Standard Deviations for deflection and stress using such a software.

Uncertainty in deflection = 0.6222919514 mm

Uncertainty in stress = 8.0725608013 MPa

Table 3. Uncertainty using spreadsheet software

Mass (kg)	Deflection (mm)	Stress (MPa)
0	-0.02	0.03
0.068	-0.27	3.418
0.118	-0.46	5.822
0.167	-0.64	8.252
0.217	-0.83	10.636
0.267	-1.01	12.998
0.316	-1.2	15.297
0.366	-1.37	17.705
0.415	-1.57	20.156
0.465	-1.75	22.601
0.515	-1.93	24.887
S.D.	0.6222919514	8.0725608013