CES 512 / Assignment 1 2015

CES512-Assignment 1-MAM-270815

CES 512 - Computational Analysis Methods for Engineers Assignment 1: This assignment consist a civil engineering problems that need to be solved by using knowledge of roots of equation, Numerical solution of simultaneous equations and Optimization.

i. usebracketing and open methods to solve civil engineering problemsmanually and using software.

Assignment outcome: At the end of this assignment, student should be able to:

ii. use methods learn in Numerical solution of simultaneous equations to solve civil engineering problems manually and using software.

iii. Use optimization to solve the constrained problem.

Tasks: Question 1

Figure Q1(a)

Figure Q1(b)

CES 512 / Assignment 1 2015

CES512-Assignment 1-MAM-270815

Figure 1(a) shows the overhanging beam loaded by uniform distribution load, W and Point load, P. It is supported by pinned and roller at A and B respectively. Figure 1(b) represents the bending moment diagram of the beam which developed based on the load cases in Table Q1.

Table Q1

a) By considering distance, x from node A, derive the equation of reaction (VA& VB) and moment, MAB

(2 marks)

in term of W, P, and xfor beam span AB.(CO1-PO2, C5)

b) Based on derived equation in (a), derive the function, f(x)=0 to correlate the load cases in Table Q1 for the intercept point of bending moment as depicted in Figure Q1(b). From the derived function f(x), is it possible to determine the exact solution for the function? (CO1-PO2, C5)

(3 marks)

c) By using the function, f(x) established in (b), Use the bisection method to determine for the position, x at where the bending moment in both cases is intercept. Carry out the iteration up to approximate percent relative error 𝜀𝜀𝑎𝑎 is less than 𝜀𝜀𝑠𝑠= 0.1%. Calculate the true percentage error, 𝜀𝜀𝑡𝑡as well. Tabulate your answer. [Hint: the initial guesses of lower (xl) and upper (xu

(5 marks)

) of x need to be determined based on rough estimation in bending moment diagram as plotted in Fig 1(b).](CO1-PO2, C4)

d) For the same condition as in (c), apply the Newton-Raphson method to solve the problem as mentioned in (c). Initial guesses, xo

(3 marks)

must be assumed based on rough estimation the plotted graph in bending moment diagram as plotted in Fig 1(b).]Tabulate your answer.(CO1-PO2, C4)

e) Compare your answers obtained in (c) and (d) by using the built-in-function fzero and writing programming (script M-file) in MAT-LAB.(CO2-PO4, C2)

(2 marks) f) Comment and discuss for the following point: (CO2-PO4, C2)

i. What is your opinion to improve the accuracy of the final answer and shorten the

iteration by using both methods in (c) & (d) towards the exact solution in (b). ii. What is the importance of graphical method in finding solution using methods in (c)

and (d) (5 marks)

Load Case W(kN/m) P (kN) 1 1 1 2 2.5 4

CES 512 / Assignment 1 2015

CES512-Assignment 1-MAM-270815

Question 2

Figure Q2 Member 2 of the truss shown in Figure Q2 is subjected to an increase in temperature of 83.3oC. This temperature consequent the axial displacements in entire members in the truss and the system of linear equations to determine the elongation of the members are given by:

0.135 D1 + 0.035 D2 = - 0.000689325

0.035 D1 + 0.135 D2 - 0.1 D4 = - 0.000689325 0.135 D3 – 0.035 D4 + 0.035 D5 = 0 -0.1 D2 – 0.035 D3 + 0.135 D4 – 0.035 D5 = 0 0.035 D3 – 0.035 D4 + 0.135 D5

a) Manually solve for the member displacement for the truss using Gauss Elimination (Consider up to 5 decimal places) (CO1:PO2) – C3 : 10m

= 0

(10 marks)

b) By using the built in function in the MAT-LAB, compare your results with the manualas

calculated in (a). (CO2:PO4) – C2 : 10m (5 marks)

CES 512 / Assignment 1 2015

CES512-Assignment 1-MAM-270815

Question 3

Figure Q3 Figure Q3 shows the simply supported beam imposed by point load, P and UDL, w. Thecorresponding total displacement at the mid-span of beam can be determined by following equation:

𝛿𝛿𝑡𝑡 = 𝛿𝛿𝑤𝑤 + 𝛿𝛿𝑝𝑝

𝛿𝛿𝑡𝑡 = 5𝑤𝑤𝐿𝐿4

384𝐸𝐸𝐸𝐸 + 𝑃𝑃𝐿𝐿3

48𝐸𝐸𝐸𝐸 Where δw= displacement due to UDL,

δp= displacement due to point load If the total beams displacement, δt is constrained byconditions as shown in Table Q3:

Table Q3

Constrained condition Capacity

Point Load, P (kN) 40

Uniform distribution load UDL, w (kNm) 10

Shear force at support, V (kN) 30

Bending moment at mid-span, M (kNm) 60

Determine the value of point load, P and UDL, w at maximum deflection, δt

a) Formulate the linear programming problem.(CO1:PO2) – C5 : 5m

of the beam to satisfy the condition in Table Q3 by:

E, I

w kN/m

L/2 L/2

P kN

CES 512 / Assignment 1 2015

CES512-Assignment 1-MAM-270815

b) Solve the problem byusing Graphical method.(CO1:PO2) – C3 : 6m c) Solve the problem with an excel solver. (CO1:PO2) – C3 : 4m d) Discuss on the overall finding from the result analysis obtained.(CO2:PO4) – C2 : 5m e) Determine the percentage contribution of the δw and δptowards the total displacement of

the beam,δt

f) Compare the total displacement, δand conclude your finding. (CO2:PO4) – C6 : 5m

t obtained in the part (b & c) with the allowable displacement of the beam,δa=3 x 10-3 m. If the displacement, δt

(given L= 5m, E= 200GPa,andI = 1.07 x 10

is exceeded the allowable displacement, what is your suggestion to satisfy the requirement without violent the loading P and w.(CO2:PO4) - C5 : 5m

-3 m3)