Derivation of Formula for Sum of Years Digit Method (SYD)

The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:
 

Depreciation Charge:

$d_m = (FC - SV) \dfrac{n - m + 1}{SYD}$

 

Total depreciation at any time m

$D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}$

 

Where:
FC = first cost
SV = salvage value
n = economic life (in years)
m = any time before n (in years)
SYD = sum of years digit = 1 + 2 + ... + n = n(1 + n)/2
 

Separation of Variables | Equations of Order One

Given the differential equation
 

$M(x, y)\,dx + N(x, y)\,dy = 0 \,\, \to \,\,$ Equation (1)

 
where $\,M\,$ and $\,N\,$ may be functions of both $\,x\,$ and $\,y\,$. If the above equation can be transformed into the form
 

$f(x)\,dx + f(y)\,dy = 0\,\, \to \,\,$ Equation (2)

 
where $\,f(x)\,$ is a function of $\,x\,$ alone and $\,f(y)\,$ is a function of $\,y\,$ alone, equation (1) is called variables separable.
 

Solution to Problem 532 | Economic Sections

Problem 532
A beam simply supported at the ends of a 25-ft span carries a uniformly distributed load of 1000 lb/ft over its entire length. Select the lightest S section that can be used if the allowable stress is 20 ksi. What is the actual maximum stress in the beam selected?

Solution to Problem 531 | Economic Sections

Problem 531
A 15-ft beam simply supported at the ends carries a concentrated load of 9000 lb at midspan. Select the lightest S section that can be employed using an allowable stress of 18 ksi. What is the actual maximum stress in the beam selected?

Solution to Problem 529 | Economic Sections

Problem 529
A 10-m beam simply supported at the ends carries a uniformly distributed load of 16 kN/m over its entire length. What is the lightest W shape beam that will not exceed a flexural stress of 120 MPa? What is the actual maximum stress in the beam selected?