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Consider a column with unique end conditions $-$ one end rigidly fixed while the other end is free to rotate, creating a complex mechanical interaction that challenges traditional loadbearing calculations $($see Figure $1\mathrm{.}1).$

The mathematical relationship governing such a column's behaviour is elegantly captured through a transcendental equation that reveals the critical points where structural integrity might be compromised. These equations are not simple linear relationships but involve trigonometric and hyperbolic functions that capture the nuanced deformation characteristics of engineering materials. For the case in Figure $1\mathrm{.}1,$ this equation is given by:

$tanx-x=0$

The bulking load $(P_c)$ is given as: $(E\frac{\text{I}}{L^2})x^2,$ where $L$ is the length of the column, $E$ is Young's modulus of elasticity and I is the moment of inertia of the cross$-$section.

$1\mathrm{.}1$ Without the use of Excel or any computerised tool, estimate the least load $($it should not be zero$),$ using Newton's method, that the column can carry without buckling. Your starting point as $x_o=4\mathrm{.}3.$

Note: The specific values of the parameters are not given, therefore, you must express your answer in terms of these parameters.

$1\mathrm{.}2$ Now consider this column to be subjected to a time$-$varying axial load $P(t),$ the lateral deflection $y(x,t)$ of the column can be described by the following second$-$order differential equation:

$El(d^2\frac{y}{d}{x}^2)+P(t)y=0$

where: $y$ is the lateral displacement, $x$ is the position along the column length, t is time and the applied load $P(t)=P_0(1-e^{-t})$

Given the following initial conditions:

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Fixed at base: $y(0,t)=0$

Free end: $y^{\text{'}}(L,t)=0$

Initial deformation: $y(x,0)=0\mathrm{.}01L**sin(\frac{x}{2L})$

Initially at rest: $d\frac{y}{d}t(x,0)=0$

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