Background

Recall $($or re$-$watch$)$ the derivation of the differential equation for deflection. During that derivation a simplification was made to produce the interim differential

equation $\frac{d^{2}y}{d{x}^2}=\frac{-M}{EI},$ which in turn led us to the differential equation for "small deflections of a beam within its elastic limit$".$ We will call the approximation th

allowed the simplification the "small deflections condition".

An approximation being accurate to $n$ decimal places means that the approximation is within $+-5*{10}^{-(n+1)}$ of the true value.

Question

Suppose we have a uniform beam that is $2\mathrm{.}57$ metres long, has a flexural rigidity of $31080N{m}^2,$ and is supported by simple supports at both ends.

What is the maximum number of decimal places of accuracy of the small deflection condition for this beam if it is under a load of

$f(x)=8\mathrm{.}58(x+0\mathrm{.}35)(3\mathrm{.}4083-x)\frac{N}{m}?$

Answer

The small deflection condition is accurate to

decimal places, and no more.