Find the exact value for the deflection of the beam at x 9

Question

Underpinning Science and Mathematics

Apply mathematical methods, tools and notations pro?ciently in the analysis and solution of engineering problems Engineering Analysis

Synthesize appropriate mathematical principles for the analysis of key engineering processes

Develop and use computer-based toolkits in the design and analysis of engineering problems.

Apply quantitative methods and computer software relevant to civil and structural engineering, in order to solve engineering problems.

TASK 1: FINITE DIFFERENCE METHOD

The deflection y in a simply supported beam with a uniform load w and a tensile axial load T (see figure 1 below) is given by the following second order differential equation.

d²y/dx² - Ty/EI = wx(L - x)/2EI

where:

x, is the location across the beam (m)

w, is the uniform loading intensity (N/m)

L, is the length of the beam (m)

E, is the Young's modulus of elasticity of the beam (N/m²)

I, is the second moment of area (m4)

T, is the tension applied (N)

Figure 1

Given T = 55000N, w =1100 N/ m , L =15m, E = 2800000 N/m², and I = 50m⁴,

a) Apply the finite difference method to find the deflection of the beam at x= 9m

b) Find the exact value for the deflection of the beam at x = 9 m, and compare it with your finding from the finite difference approximation.

TASK 2: NUMERICAL SOLUTION OF ODES

For the differential equation x²dv/dx + 2xy = y³ with boundary condition y(I) = I develop an

Excel spreadsheet that uses:

i. Euler's method
ii. Euler-Cauchy Method
iii. Runge-Kutta (4th order)

to approximate values of y, when x = 2.5 and x = 3.1

Also, find the particular solution of the equation and the exact values of y for the values of x stated above. Discuss your findings by comparing the results from the numerical methods with those from the exact solution.

TASK 3: VBA PROGRAMMING

Your task is to develop three VBA modules that implement the Euler's, Euler-Cauchy and Runge-Kutta (4th order) numerical methods respectively. Your code should be able to apply the aforementioned methods for any differential equation.

In order to test the functionality of the VBA code you may use the differential equation given in Task 2 of the assignment.