In the CBR method of pavement design, the mixed commercial vehicles with different axle loads are to be converted in terms of the cumulative number of standard axle load Ns, using the equation (where A is number of vehicles/day for completed construction for number of lanes, r is annual growth rate of commercial vehicles, n is design life of pavement and F is vehicle damage factor.)
- ANs = 365A[(1 + r)ⁿ − 1] / r × FCorrect
- BNs = 365A[(1 + r)ⁿ + 1] / r × F
- CNs = 365A[(1 − r)ⁿ − 1] / r × F
- DNs = 365A[(1 − r)ⁿ + 1] / r × F
Solution
Concept
Cumulative standard axles
Formula
Ns = 365 A [(1 + r)ⁿ − 1] / r × F
Step-by-step solution
1. The sum of a geometric series of yearly traffic growing at rate r over n years is 365A[(1 + r)ⁿ − 1]/r. 2. Multiplying by the vehicle damage factor F converts commercial vehicles to standard axles.
Explanation
Traffic grows each year by r, so the total over the design life is the sum of the geometric series.
Why the other options are wrong
The plus sign in the bracket and the (1 − r) forms do not give the sum of a growing series.
Shortcut
Ns = 365A[(1+r)ⁿ − 1]/r × F.
Common mistake
Using (1 − r).
Correct answer: A) N_s = 365A[(1 + r)ⁿ − 1] / r × F