Calculations
File: cantilbeam.xls
The formula for the elastic deflection of an end-loaded, circular cantilevered rod is:
where: y = max deflection
64 FL3
y=
F = applied force to the end of the rod Note the use of Insert/Object/Equation
3 E pd 4
L = length of the rod
E = Youngs modulus
d = diameter of rod
Given the following expected input mean values: F = 15 (N)
L = 0.25 (m)
Note the extensive use of text boxes E = 2.04E+11 (Pa)
d = 6.00E-03 (m)
Results in the following calculated mean deflection: y= 6.02E-03 (m) = 64*F*L^3/(3*E*PI()*d^4)
The smallest division of the available measuring instrument for each of the above measured variables are
small_F = 0.5 (N) small_E = 1.00E+10 (Pa)
small_L = 0.01 (m) small_d = 1.00E-04 (m)
We can estimate the standard errors of the individual variables:
Sf = 0.144 (N) =small_F/SQRT(12)
SL = 0.003 (m) =small_L/SQRT(12)
Se = 2.89E+09 (Pa) =small_E/SQRT(12)
Sd = 2.89E-05 (m) =small_d/SQRT(12)
and then find the standard error of the deflection (Sya) analytically as follows:
dya/dF = 4.0E-04 (m/N ) = 64*L^3/(3*E*PI()*d^4)
dya/dL = 0.07 = 64*F*L^2/(E*PI()*d^4)
dya/dE = -2.95E-14 (m/N ) = -64*F*L^3/(3*E^2*PI()*d^4)
dya/dd = -4.01 = -256*F*L^3/(3*E*PI()*d^5)
Sya = 2.6E-04 (m ) = SQRT((dya_dF*Sf)^2+(dya_dL*Sl)^2+(dya_dE*Se)^2+(dya_dd*Sd)^2)
An approximation for standard error of the deflection (Syn) is determined numerically as follows:
dF = 0.144 (N)
dL = 0.003 (m) Note these delta amounts are the same as the standard errors as
dE = 2.89E+09 (Pa) they are based on the individual measuring device's accuracy Note the use of normal Excel formulas in the first two calcs
dd = 2.89E-05 (m) and Visual Basic functions in the last two.(see Module1)
dyn_F = 5.79E-05 (m ) = (64*(F+dF)*L^3)/(3*E*PI()*d^4)-((64*F*L^3)/(3*E*PI()*d^4))
dyn_L = 2.11E-04 (m) = (64*F*(L+dL)^3)/(3*E*PI()*d^4)-((64*F*L^3)/(3*E*PI()*d^4))
dyn_E = #NAME? (m ) = y(F, L, E+dE, d ) - y( F, L, E, d )
dyn_d = #NAME? (m) = y(F, L, E, d+dd ) - y( F, L, E, d )
Syn = #NAME? (m) = SQRT((dyn_F)^2+(dyn_L)^2+(dyn_E)^2+(dyn_d)^2)
Which is 13% below the analytical value. What caused this large a difference between the analytical and the numerical approximation for the estimated standard
error?
Note that the estimated standard error for y is of the same order of magnatued as y. Therefore, a better experimental plan should be initiated. The above results
indicate that the standard error is least sensitive to the estimated errors for F and L and most sensitive to the estimated errors in d and E. Hence, improving the
accuracy of the d and E measurements should increase the calculated accuracy in y. What measuring accuracy is required for d and E to reduce the estimated
standard error to one tenth the value of y?