LO3 Approximate solutions of contextualised examples with graphical and numerical
methods
Assignment Brief and Guidance
Task 1
The engineering department has developed the following equation for the bending moment of a
beam and you have asked to investigate its behaviour
M ( x) = x3 − 3x2 − 4
The Beam is 4m long and the design team suspect the is problem if the bending moment is zero
in the range between 3-4m and you have been asked to
a) Plot the bending moment at 0.5m interval for the range 0 ≤ x ≤ 4m 0 ≤ x ≤ 4 and
determine if the bending moment is zero in range 3m ≤ x ≤ 4m
b) Use the graph to estimate where the bending moment is zero
c) Use the bisection method to numerically estimate the exact location where the bending
moment is zero
d) Newton-Raphson method to obtain the required location
e) Compare the results of the above method to determine which gives a best solution
Task 2
The following offsets are taken from a chain line to an irregular boundary towards right side
of the chain line.
chainage 0 25 50 75 100 125 150
Offset ‘m’ 3.6 5.0 6.5 5.5 7.3 6.0 4.0
Common distance d =25m
You have been asked to estimate the area using the following methods and compare and
comment on their difference and accuracy.
a) Trapezium Rule
b) Simpson’s Rule
Also, critique the numerical estimation methods including Newton-Raphson, bisection method,
Trapezium and Simpson’s Rule with the same application and comments on the accuracy of
them, i.e., which of these methods is more accurate in your perspective providing exhaustive
rationale.
Task 3
The equation governing a body travelling in a water channel is given by the following equation
dv = 1 − v 2
dt
Plot the velocity time graph for the object and determine the final velocity and the time taken to reach this velocity
