A simply supported beam is subjected to a non-uniform load whose intensity increases linearly along its length. Let (x) be the distance along the beam axis. The beam has a length (L) and is supported at (x = 0) and (x = L). The deflection that the beam undergoes in the vertical direction is a function of the position (x). The point (x) at which the deflection is maximum is a solution of the equation:
-(w_max / (360 * E * I * L)) * (15x^4 - 30L^2x^2) = (w_max / (360 * E * I * L)) * 7L^4
where:
- (x) is the distance along the beam axis (in centimeters [cm]);
- (L) is the length of the beam (in meters [m]);
- (w_max) is the intensity of the load in the vertical direction per unit length (in Newton per centimeter [N/cm]);
- (E) is the Young's modulus (in Newton per square centimeter [N/cm²]);
- (I) is the moment of inertia of the cross-sectional area of the beam (in centimeter to the fourth power [cm⁴]).
Consider a beam with parameters (L = 450) cm, (w_max) = 1.75 \times 10^3) N/cm, (E = 50 \times 10^6) N/cm², and (I = 30 \times 10^3) cm⁴. Calculate the point (x) where the deflection is maximum by developing the following steps:
- Find the function (H(x)) for the calculation of (x).
- Write a code that plots (H(x)) and use it to obtain an initial approximation (x_0) for (x).
- In the same code, implement the Newton's method to find an approximation for (x). Use (x_0) as the initial estimate and (|H(x)| < 10^{-9}) as the stopping criterion.