UNIVERSITY OF MAURITIUSUNIVERSITY OF MAURITIUSUNIVERSITY OF MAURITIUSUNIVERSITY OF MAURITIUS

FACULTY OF ENGINEERINGFACULTY OF ENGINEERINGFACULTY OF ENGINEERINGFACULTY OF ENGINEERING

FIRST SEMESTER EXAMINATIONS

NOVEMNOVEMNOVEMNOVEMBBBBER/ER/ER/ER/DECEMBER DECEMBER DECEMBER DECEMBER 2010201020102010

PROGRAMME BEng (Hons) Civil Engineering – Part-Time (Year 3)

MODULE NAME Numerical Methods

DATE Thursday

2 December 2010

MODULE CODE CIVE 2210

TIME 13.30 – 15.30 Hrs DURATION 2 hours

NO. OF

QUESTIONS SET

3 NO. OF QUESTIONS TO

BE ATTEMPTED

3

INSTRUCTIONS TO CANDIDATES

There are 3 Questions in this Paper. All Questions are compulsory.

All Questions carry equal marks. Runge Kutta 4th Order Algorithm is attached.

Page 1 of 2

NUMERICAL METHODS – CIVE 2210

SECTION A

Question 1 (a) Given dy/dx = x2+y2, y(0) = 1; find values of y at x=0.1 and x=0.2 to 3 decimal

places using Euler Modified Method. [10 marks] (b) The deflection of the mast of a sail boat subjected to wind force obeys the

following equation;

d2y/dx2 = F(L – x)2 / 2EI where;

F= wind force E= Modulus of Elasticity of the mast L= mast length I= moment of inertia.

Calculate the deflection at the top of the mast for a mast length of 60 m, when

F= 60kN/m, E=1.25 × 108 kN/m2 and I= 0.05 m4 given that when x=0, y=0 and dy/dx =0. Use Runge Kutta method of the fourth order. Give your answer correct to 3 decimal places. The R.K 4th order algorithm is given at annexure 1. [15 marks]

Question 2

The data collected for the flow of water over a triangular notch is tabulated below;

H/(m) 1.2 1.4 1.6 1.8 2.0 2.4

Q/(m3/s) 4.2 6.1 8.5 11.5 14.9 23.5

Where Q = flow over notch H= height of water over notch. The relation between Q and H is Q= CHn. You are required to find C and n by (a) curve fitting [12 marks] (b) the method of group averages. [8 marks] Comment on each of the above methods. [5 marks]

Page 2 of 2

NUMERICAL METHODS – CIVE 2210

Question 3 (a) Find the positive root of x3-x-1 to four decimal places by the bisection method.

(Give your answer at the fourth iteration). [10 marks] (b) You are required to design a spherical tank (as shown in figure 1) to hold water

for a small village in a developing country. The volume of liquid it can hold can be computed as

V= π h2 (3R-h) / 3

Where V= volume of water (m3) H=depth of water in tank (m) R= radius of tank. (m)

If R= 3m, to what depth must the tank be filled so that it holds 30 m3 of water. Use three iterations of the Newton-Raphson method to determine your answer to 3 decimal places. [15 marks]

END OF QUESTION PAPER

/ph

tank

support h

Figure 1

Page 3 of 2

NUMERICAL METHODS – CIVE 2210

ANNEXURE 1- RUNGE KUTTA 4th

ORDER ALGORITHM

K1 = h f1 (x0, y0, z0)

K2= h f1 (x0 + h/2, y0 + K1 / 2, z0 + L1 / 2)

K3 = h f1 (x0 + h/2, y0 + K2/ 2, z0 + L2 / 2)

K4 = h f1 (x0 + h, y0 + K3, z0 + L3)

L1 = h f2 (x0, y0, z0)

L2 = h f2 (x0 + h/2, y0 + K1 / 2, z0 + L1 / 2)

L3 = h f2 (x0 + h/2, y0 + K2/ 2, z0 + L2 / 2)

∆y = 1/6 (K1 + 2 K2 + 2 K3 + K4)